RECENT DEVELOPMENTS IN DYNAMICAL SYSTEMS: THREE PERSPECTIVES
F. BALIBREA Universidad de Murcia, Dpto. de Matem´aticas, 30100-Murcia, Spain E-mail: [email protected] T. CARABALLO Universidad de Sevilla, Dpto. Ecuaciones Diferenciales y An´alisis Num´erico, Apdo. de Correos 1160, 41080-Sevilla (Spain) E-mail: [email protected] P.E. KLOEDEN Institut f¨ur Mathematik, Goethe-Universit¨at, D-60054 Frankfurt am Main, Germany E-mail: [email protected] J. VALERO Univ. Miguel Hern´andez de Elche, Centro de Investigaci´on Operativa, Avda. Universidad s/n, 03202 Elche (Alicante), Spain E-mail: [email protected]
Received September 15, 2009; Revised October 30, 2009 Dedicated to the memory of Valery S. Melnik
The aim of this paper is to give an account of some problems considered in past years in the setting of Dynamical Systems, some new research directions and also state some open problems
Keywords: Topological entropy, Li-Yorke chaos, Lyapunov exponent, continua, non- autonomous systems, difference equations, ordinary and partial differential equations, set-valued dynamical system, global attractor, non-autonomous and random pullback attractors. Mathematics Subject Classifications (2000): 35B40, 35B41, 35K40, 35K55, 37B25, 37B30, 37B40, 37B45, 37B55, 37E05, 37E15, 39A20 58C06, 60H15.
1. Introduction nected with celestial mechanics. In particular, the theory of discrete dynamical systems, which mainly uses iteration theory, is one of the most relevant The theory of dynamical systems has a long prehis- topics in the subject. tory, but essentially started in its present form with the work of Poincar´eand Birkhoff on problems con- The purpose of this tutorial is to give a par-
1 2 Balibrea, Caraballo, Kloeden & Valero tial account of the progress obtained in discrete and centers and depth of centers [Ye, 1993, Kato, 1995, continuous systems is recent years, to present some Kato, 1998, Efremova & Makhrova, 2003] on open and new problems. It consists of four sections. trees, graphs and dendrites, and on the In Sections 2 and 3 we consider some results in the structure of ω-limit sets [Kocan et al., 2010, topological dynamics in non-autonomous discrete Balibrea & Garc´ıa Guirao, 2005], in particular, systems and in Section 4 we give a rather complete ω-limit sets on hereditarily locally connected review of multi-valued dynamical systems arising continua [Spitalsk´y, 2008], etc. from models involving partial differential equations. Within dynamical systems theory one of the Finally, in Section 5 we provide an overview of many most interesting topics is that of minimal systems, problems in non-autonomous and random dynami- i.e., systems that do not contain non-trivial sub- cal systems, in particular on new concepts of non- systems. A system (X,f) is minimal if there is no autonomous and random attractors. proper subset Y X which is non-empty, closed ⊆ and f-invariant, i.e. satisfies f(Y ) Y . It is im- ⊆ 2. Autonomous discrete dynamical systems mediate that (X,f) is minimal if and only if the forward orbit of all points in X are dense in X. We The development of the theory of topological dy- will say also that in this case f is also minimal. namics began in the earlier part of the last century. Here we will concentrate on the progress on mini- It focused in particular on problems related to au- mal systems in the case that the phase space X is tonomous discrete dynamical systems given by the one-dimensional. pair (X,f), where X is a topological space and f a continuous map of X into itself. The crucial prob- The topological characterization of minimal lem was the study of properties of all orbits of all sets of one-dimensional X has been carried out for points in the space state X. For x X, the orbit intervals, circles, trees, finite graphs and dendrites. n ∈ n of x by f is the sequence (f (x))n∞=0, where f = In the interval case, these are finite and Cantor n 1 0 f(f − ) for all n 1 and f = idX (identity on X). sets (see [Block & Coppel, 1992]), while for the cir- ≥ In most cases X is a compact metric space and, in cle case the circle itself can also be minimal. These particular, extensive results were obtained when X results can be generalized to graphs, where minimal = I = [a, b] because many phenomena from social, sets are characterized as finite sets, Cantor sets and natural and economical sciences can be formulated also unions of finitely many pairwise disjoint circles as systems evolving with time in a discrete way in [Balibrea et al., 2003, Mai, 2005]. such spaces. Moreover, when X = Rd, or a subset thereof, we generally speak of problems on differ- A dendrite is defined as a locally con- ence equations. nected continuum which contains no simple closed Although the number of new results has been curve. In this case, besides partial results in impressive, we will include here some of them from [Balibrea et al., 2003], a complete characterization one of the subjects more active in the field in last has been given recently in [Balibrea et al., 2009] as years, dynamical systems on continua. a consequence of a more general result based on the new notion of almost totally disconnected spaces. A space X is almost totally disconnected if the set of 2.1. Autonomous dynamical systems on its degenerate components is dense in X. The main continua results says that an almost totally disconnected One line of research that has been very active space admits a minimal map if and only if it is ei- in recent years is that of dynamical systems on ther a finite set or has no isolated point. By a brain continua, i.e., where X is a continuum (a compact we mean a cantoroid whose degenerate components and connected topological space) and f C(X, X). are dendrites and form a null family (for any ε> 0, ∈ For definitions and detailed account of results see only a finite number of its members have diameters [Nadler, 1995]. Problems such as periodic struc- greater than ε). A cantoroid is a compact metric ture using methods from combinatorial dynamics and almost totally disconnected space without iso- have been studied on circles, trees and finite lated points. With these ingredients we can now graphs (see [Alsed`a et al., 2000]), as well as on state the characterization in [Balibrea et al., 2009]. Three perspectives 3
Theorem 2.1. Let D be a dendrite and let M be crete system, which they denoted by h(f1, ), using ∞ a subset of D. Then M is a minimal set for some the technique of open covers of X as in the orig- dynamical system (D,f) if and only if M is either inal paper [Adler et al., 1965] on autonomous sys- a finite set or a brain. tems. In the case that X is metric or metrizable, they also used separated and spanning sets as in In addition, the following characterization was [Bowen, 1970]. It is easy to see that such defini- given in [Balibrea et al., 2009] for general almost tions give similar results when the system is in fact totally disconnected spaces. autonomous. What is really interesting, is that it is now possible to define the entropy h(f1, ,Y ), when ∞ Theorem 2.2. An almost totally disconnected Y is a subset of X which is not necessarily compact compact metric space admits a minimal dynamical or invariant under f(f1, ). Such an extension was ∞ system if and only if it is either a finite set or a necessary to deal with other problems considered in cantoroid. [Kolyada & Snoha, 1996]. An interesting and surprising consequence of Question 2.3. Is it possible to give a topological the paper [Kolyada & Snoha, 1996] was a proof of characterization of minimal sets in other families the commutativity of the entropy autonomous dy- of one-dimensional continua like arc-like, tree-like namical systems, i.e., the entropy of the composi- or circle-like ? (For definitions see [Nadler, 1995]). tion of two continuous self-maps on a compact space does not depend on the order in which they are 3. Non-autonomous discrete systems taken, i.e., h(f g) = h(g f). The first time that ◦ ◦ the commutativity of the entropy was mentioned We now consider the situation in which the map seems to have been in [Dinaburg, 1970]. describing the evolution of the dynamics is itself al- Inspired by [Kolyada & Snoha, 1996], the com- lowed to change with time. This admits the follow- mutativity or non-commutativity of other types ing formulation. Given a compact topological space of entropy such sequence entropy were proved in X and a sequence of continuous self-maps (f ) n n∞=1 [Balibrea et al., 1999a, Balibrea et al., 1999b]). = f1, from X into itself, the pair (X,f1, ) will be ∞ ∞ called a non-autonomous discrete system where the It is well known (see [Misiurewicz,1989]) that orbit of a point x X is described by the sequence for interval maps, the entropy is positive if and ∈ only if one of its iterates has a structure called x, f1(x), f2(f1(x)),...,fn(fn 1)...(f2(f1(x)))...) horseshoe. An interval map f has a horseshoe if − there are two disjoint intervals J and K such that We will use the notation f(J) f(K) J K. It is difficult to extend this n ∩ ⊃ ∪ f1 = fn fn 1 ... f2 f1 definition to non-autonomous interval systems be- ◦ − ◦ ◦ ◦ cause it is associated to a unique map and f1, is ∞ which was introduced in [Kolyada & Snoha, 1996]. a sequence. If such notion were possible while re- When all of the maps are the same, i.e., fn = f taining properties similar to those of horseshoes for for all n N, then we have a autonomous discrete ∈ autonomous dynamical interval systems, then one dynamical system or, simply, a discrete dynamical can ask: system, which we considered above. There is not a large literature on truly non- Question 3.1. If a non-autonomous interval sys- autonomous discrete systems and, as consequence, tem f1, has positive entropy, does it possess a there are a few results. In the following subsec- ∞ structure of horseshoe type? tions we will survey some interesting developments. Other results can be found in the expository article [Kloeden, 2000]. Another way to obtain non-autonomous dy- namical systems is to allow not only maps to change, but also spaces. This leads to the following 3.1. Topological entropy generalization: a non-autonomous dynamical sys- The authors of [Kolyada & Snoha, 1996] defined tem is the pair (X,f) given by X = (Xn)n∞=1 and the topological entropy of a non-autonomous dis- f = (fn)n∞=1), where each Xn is a compact metric 4 Balibrea, Caraballo, Kloeden & Valero space and fn : Xn Xn for n N. Such a no- Parrondo, Harmer & Abbott, 2000]. A partial → +1 ∈ tion is used for example in [Kolyada et al., 1999] to analysis of when the paradox is impossible is given construct a class of smooth triangular maps on the in [C´anovas, 2010]. It would be interesting to unit square of type 2∞ having positive topological consider examples in two dimensional settings and entropy and thus extending a previous result from analyse whether the paradox is possible or not. [Balibrea et al., 1995]. Another example is to con- Motivated by Sharkovskii’s results on the co- sider skew-product maps which were introduced in existence of periodic orbits of certain periods [Bowen, 1970] and are often called triangular maps ([Sharkovskii, 1994, Sharkovskii, 1964]) and forc- when defined on [0, 1]2, i.e. with ing relationships, we wonder if the same type of such relationships can be obtained in the F (x,y) = (f(x), g(x,y)) = (f(x), gx(y)), non-autonomous case. There are several papers on this for interval maps ([Alsharawi et al., 2006, where f C(I,I) and gx C(I,I) are continuous. ∈ ∈ Alves, 2009, C´anovas & Linero, 2006]). It is stated Dynamical systems generated by continuous maps in [Alves, 2009] that if the condition on hereditarily locally connected continua such as dendrites can also be interpreted as a sequence of Card x I,fi(x)= fj(x) < , i = j (mod p) spaces and maps between them and hence as as gen- { ∈ } ∞ ∀ 6 eralized non-autonomous systems. The sequence of holds, then if the system has a periodic orbit of lcm(r,p) spaces can be, for example, trees approaching den- period r such that p is odd and larger than 1, drites (see [Nadler, 1995]). Results on entropy can then the system has periodic orbits of all periods be also obtained in such cases. forced by the Sharkovskii ordering. When p = 1 we The above generalization was used also to ob- recover the Sharkovskii’s original result. To clarify tain a formula for entropy in terms of the num- the situation, the sets ber of pieces of monotonicity of the functions f n 1 + + Aq = n Z : lcm(n,p)= pq for q Z when all the Xn are compact real intervals and ∈ ∈ maps are piecewise monotone. This is a general- were introduced in [Alsharawi et al. , 2006]. If de- ization of the well-known Misiurewicz-Szlenk for- ≺ notes Sharkovskii’s ordering, then we have mula for a continuous piecewise interval map (see [Misiurewicz & Slenk,1980]). + Theorem 3.2. If Al P = for some l Z , ∩ 6 ∅ ∈ then Aq P = for all l q. 3.2. Periodic non-autonomous systems ∩ 6 ∅ ≺ It is necessary distinguish the periods of the p- We will restrict now our attention to the case Xn = I or R for each n N since a lot of results have periodic equation in Aq. To this end we introduce ∈ been obtained in this setting. Let us suppose that Q = n P : p . n . there exists a positive integer p (prime period) such { ∈ } + that fn+p = fn for each n 0. Then the system Note, by the definition of Aq, we have Aq pZ = ≥ + ∩ pq for every q Z . As a consequence Aq P = xn+1 = fn(xn) { } ∈ ∩ pq whenever Aq P = and Aq Q = . Finally, { } ∩ 6 ∅ ∩ ∅ if l Z+, then is called a p-periodic non-autonomous system. ∈ In the setting of economical theory, the two + Sl = pq Z : q l, or Aq Q = periodic case has an interpretation in terms of { ∈ ≺ ∩ 6 ∅ Parrondo’s paradox [C´anovas et al., 2006]. This With such notation we reformulate the former the- paradox says that two losing games can result, orem. under random or periodic alternation of their + dynamics, in a winning game. That is, it can Theorem 3.3. If Al P = for some l Z , ∩+ 6 ∅ ∈ happen that combination of losing + losing then S Sl = (Q pZ ) Sl can result winning. For more information on \ ∪ \ this paradox see [Harmer & Abbott, 1999a, Reasons for interest of knowing the ele- Harmer & Abbott, 1999b, ments of Q are given in [Alves, 2009] and the Three perspectives 5 case of two periodic equations is studied in We can extend the notion of Lyapunov expo- [C´anovas & Linero, 2006], where it is shown that nent for an autonomous dynamical system (X,f) if such an equation has an odd period larger than to a non-autonomous systems (X,f1, ) by the for- ∞ 1, then mula + P = Q 2Z . 1 ∪ λ(x) = lim log (fn ...f2 f1)′(x) n Such results underline the role played by the set Q. →∞ n | ◦ ◦ | n 1 1 − ′ Question 3.4. Is it possible to obtain a general = lim log f (x(j)) (2) n n | j | formula for all periods of a p-periodic difference →∞ j=1 equation similar to that above? X As a consequence we can give a notion of chaotic 3.3. Lyapunov exponents in periodic non- behavior for systems. We will say that a non- autonomous systems autonomous discrete system has chaotic behavior if there is a Lebesgue measurable set of points L in Let us start with an example coming from us- the space state X with positive Lyapunov exponent ing the Poincar´emap to understand the behavior (in the extended sense). In turn this is interpreted of orbits in non-linear and non-autonomous differ- in the application considered in the above paper ential equations of second order (see for example as the existence of homoclinic chaos, i.e., the exis- [Chac´on, 2001]), namely tence in the phase space of the non-linear second order differential equation of a homoclinic orbit in xn+1 = [α + ε(bn + βcn)]xn = anxn (1) the neighborhood of a separatrix. The system is non-chaotic if there is no such set L. with α > 1, 0 <β < 1, bn = √2sin n, cn = √ 2[2K(m)(n + Θ)/π; m]. That is, for the sake Exercise 3.5. Propose new examples and applica- of clarity and possibility of comparing results we tions of the extension of Lyapunov exponents and choose the main resonance regime Tsin = Tsn = 2π, develop a complete theory in the setting of (X,f1, ) ∞ where sn denotes the Jacobian elliptic function of with X = I or X = R. parameter m, while K(m) is the complete elliptic integral of the first kin and Θ is the initial phase, 3.4. Li-Yorke chaos in non-autonomous i.e. with 0 Θ 2π. ≤ ≤ systems The model is a particular case of general non- autonomous systems xn+1 = fn(xn), where f1, f2, Since the notion of chaos in the sense of Li and . . . are continuous maps of X into itself, and X = Yorke is given in terms of the behavior of orbits of R or X = I = [0, 1]. The form of the perturba- points in X, we can extend it to the setting of non- tion is designed to capture only the effect of weak autonomous systems given by sequences of maps non-autonomous excitations on the stability of a f1, ∞ generic unstable limit cycle. The elliptic function sn is chosen to introduce in the control excitation in Definition 3.6. The system (X,f1, ) is chaotic in ∞ a simple way the effect of the excitation waveform the sense of Li and Yorke (resp., δ-chaotic) if there on the control scenario. is an uncountable set S X (called a scrambled ⊂ The behavior of the system is investigated by set) such that for all pairs x, y X with x = y ∈ 6 exploiting the connection between the sensitivity n n (i) limsup d(f1 (x),f1 (y)) > 0 (> δ, resp.) to initial conditions and Lyapunov exponents in n autonomous systems: when a point x X has →∞ ∈ (ii) lim inf d(f n(x),f n(y)) = 0. a positive Lyapunov exponent, then its orbit and n 1 1 that of a point nearby diverge at a positive ex- →∞ ponential rate. This connection was explored in As a consequence, we can state similar prob- [Abraham et al., 2004], while the connection be- lems in this new setting similar to what has been tween Lyapunov exponents and positive metric en- proved in the case of autonomous discrete dynami- tropy was clarified in [Barrio, 2007]. cal systems. 6 Balibrea, Caraballo, Kloeden & Valero
Exercise 3.7. Prove or disprove that, when X = given by the sequences (xn)n∞=0 where x0 is the ini- n I, if there is a Li-Yorke pair, then there is an un- tial point and xn = f (x0) with the same meaning countable number of pairs. that in previous paragraphs. In many recent applications we have to consider For topological entropy we have xn+1 = fn(xn), where (fn)n∞=0 is a sequence of maps of X into itself, which need not be continuous. The Question 3.8. If h(f1, ) > 0, is it true that f1, traditional problems to be solved are: boundedness, ∞ ∞ is Li-Yorke chaotic? periodicity, convergence, local and global stability of solutions. Equations with delays which are also There are many problem for autonomous sys- important in applications are formulated by tems with zero topological entropy which can also be considered in the setting of non-autonomous sys- xn+1 = fn(xn,xn 1, ..., xn k), tems. − − k+1 where fn : R R is a continuous map for each → 3.5. Li-Yorke chaos in Rn with n> 1 n = 0, 1,. . ., k 1 and x k, x k+1, . . ., x0 ≥ − − ∈ R1. These can be reformulated as first order (i.e. The notion of snap-back repeller for maps f without delay) vector valued difference equations in 1 n n ∈ k+1 C (R , R ) was introduced in [Marotto, 1978] as terms of Xn := (xn k,xn k+1,...,xn) R . follows. Assume that x⋆ is an expanding fixed point − − ∈ ⋆ ⋆ ⋆ of f in the ball Br(x ), i.e. where f(x ) = x , and 3.6(A). Examples of non-autonomous rational all eigenvalues of Df(x) are greater than one in ⋆ ⋆ equations norm for every x Br(x )). Then x is a snap- ∈ ⋆ back repeller for f if there is a point x Br(x ) Non-autonomous rational equations are non- 0 ∈ with x = x⋆ such that f p(x ) = x⋆ for some pos- autonomous differential equations involv- 0 6 0 itive integer p and the determinant det Df p(x = ing rational functions. They are common 0 6 0. in applications in population dynamics (see In the same paper, Marotto proved that the [Kulenovic & Ladas, 2002]). existence of a snap-back repeller is a sufficient condition to have Li-Yorke chaos (in a general- 1. The non-autonomous delay Pielou logistic ized sense of the usual notion). Marotto’s the- equation: orem was improved after by a theorem of Shi anxn and Chen in [Shi & Chen, 2004]. Other exten- xn+1 = , 1+ xn k sions to the Banach space setting can be found in − [Kloeden & Li, 2006] . where (an)n∞=0 is a positive periodic sequence with period p, i.e. with Exercise 3.9. It would be interesting to know if such theory could be extended to non-autonomous a = a , n = 0, 1, .... systems. That would require also extending the no- n+p n tion of snap-back repeller to another notion avail- When a = α the equation is autonomous able in the new setting. n and the asymptotic behavior and oscillation of its positive solutions have been studied in 3.6. Non-autonomous difference equations [Kulenovic & Ladas, 2002].
The field of difference equations has developed When (an)n∞=0 is bounded and persistent (be- quickly over the last thirty years. In many cases, low bounded), [Kocic & Ladas, 1993] gives problems can be formulated in terms of autonomous sufficient conditions for boundedness and the systems given by the pair (X,f) where X is either global attractivity of solutions. In the peri- (0, ) or R and f : X X is a non necessarily odic case, sufficient conditions in the coeffi- ∞ → continuous map. The main problem is the study cients which ensure that there are periodic of properties of solutions of the equation which are solutions are also given. Three perspectives 7
2. The previous model is an extension of the pe- with positive initial values x k, x k+1, . . ., − − riodically forced Beverton-Holt model x0, that the solutions are bounded, persist rK x and have certain periodicity properties under x = n n , n+1 some conditions on the sequences (pn)n∞ and Kn + (r 1)xn =1 − (qn)n∞=1, which represents a population with inherent growth rate r > 1 and a carrying capacity The following open problem was stated in represented by the periodic positive sequence [Papaschinopoulos & Schinas, 2008]. (Kn)n∞=1 which is assumed to have a prime period p 2. Question 3.10. Consider the difference equation ≥ (3) with k = 3,4, . . . where (pn)∞ and (qn)∞ 3. Non-autonomous Lyness equation: Let n=1 n=1 are positive sequences such that (pn)n∞=1 is bounded (bn)n∞=1 be a sequence of positive numbers. and either The non-autonomous Lyness equation
xn + bn pn pn k 0 and qn = qn k, n = k, k + 1,... xn+1 = , x0 > 0, x1 > 0, − − ≥ − xn 1 − or is a non-autonomous version of the Lyness
equation given by pn pn k 0 and qn = qn k, n = k, k + 1,... − − ≤ − xn + b xn+1 = , x0 > 0, x1 > 0, Is every solution of (3) bounded and persistent? xn 1 − which is well-known in the literature. See for example [Camouzis & Ladas, 2007], where a 4. Multi-valued dynamical systems complete report of such equation is given and 4.1. Equations without uniqueness stated some interesting open problems. In the non-autonomous case, it has been conjec- The theory of semigroups of operators has been a tured that solutions are bounded and persist powerful tool for studying the properties (and, in (i.e. are bounded away form zero and infin- particular, the asymptotic behavior) of solutions ity) if (bn)n∞=1 has the same properties. One of autonomous differential equations in partial can ask which other more general sequences derivatives with uniqueness of the Cauchy problem. (bn)n∞=1 ensure that the solution (xn)n∞=0 is In this situation the existence and properties of bounded and persistent. An interesting ex- global attractors have been established for a ample in [Angelis, 2004] shows that (xn)n∞=0 wide type of equations such as reaction-diffusion can even be non-bounded when (bn)n∞=1 at- systems, the two-dimensional Navier-Stokes tains only two values and proves that is suffi- equations, wave equations and many others cient for (bn)n∞=1 to be a monotone sequence (see, e.g., [Babin & Vishik, 1992, Hale, 1988, to prove that the conjecture is true. Ladyzhenskaya, 1991, Sell & You, 1995, Temam, 1988]). 4. It is well-known [Camouzis & Ladas, 2007] However, in many situations concerning sys- that the solutions of the difference rational tems of physical relevance either uniqueness fails equation with two delays or it is not known to hold. In such cases we cannot xn + xn 2 define a classical semigroup of operators, so that xn+1 = − , (3) xn 1 another theory involving multi-valued maps is nec- − tends to a period-four solution. It was essary, namely, the theory of multi-valued dynami- shown for extension of the model in cal systems. Often a system is really multi-valued [Papaschinopoulos & Schinas, 2008], (example of this cases will be given) but, in other cases, we are simple not able to prove the unique- pn + qnxn + qn xn+1 = , ness of solutions. Hence, the theory of multi-valued xn k − dynamical systems allows us to continue when the n = 0, 1,..., k = 1, 2,..., proof of uniqueness fails due to technical problems, 8 Balibrea, Caraballo, Kloeden & Valero and it is still possible to study the asymptotic be- 2. For every u H there exists at least one 0 ∈ havior of solutions no matter we have uniqueness weak solution of (4) which exists in the whole or not. half-line [0, + ). ∞ There exists now in the literature a great num- ber of equations of physical interest for which Therefore, strong solutions are unique, but they uniqueness of solutions fails. Among these we can cannot be defined globally in time. On the other cite, for example, the three-dimensional Navier- hand, weak solutions exist globally in time, but it Stokes system, the Ginzburg-Landau equation, the is not known whether uniqueness holds or not. Lotka-Volterra system with diffusion, the wave Thus, if we intend to study the asymptotic be- equation or differential inclusions (including some havior of solutions as time goes to infinity, then we models from climatology). This is, of course, one need to work with weak solutions and we have a important reason justifying the interest of multi- problem for which uniqueness fails. In the two-dimensional case, i.e., Ω R2, valued dynamical systems. Another (not less im- ⊂ portant) reason is the fact that there is usually a uniqueness of weak solutions is true (see, e.g., gap between the conditions that we need to im- [Temam, 1979]), so a semigroup of operators can pose to obtain existence of solutions and the con- be defined in the phase space H, and the ex- ditions necessary to prove uniqueness. Hence, we istence of a finite-dimensional global attractor is can weaken the conditions imposed on a differen- a well known result (see [Ladyzhenskaya, 1982, tial equations and then consider more general situ- Temam, 1988, Babin & Vishik, 1992]). ations. We shall give now in more detail some examples 4.1(B). Reaction-diffusion systems of equations in which either uniqueness is not true Let d > 0 and N 1 be integers and let Ω RN ≥ ⊂ or is not known to hold. be a bounded open subset with smooth boundary. We denote by the norm in the space Rd (or R), |·| 4.1(A). Three-dimensional Navier-Stokes equa- and by ( , ) the scalar product in Rd. Consider the · · tions problem 3 Let Ω R be a bounded open subset with smooth ∂u ⊂ a∆u + f(u)= h(x), (t,x) (0, T ) Ω, boundary. For given ν > 0 we consider the Navier- ∂t − ∈ × Stokes system ∂u u x ∂Ω = 0 or x ∂Ω = 0 , | ∈ ∂ν | ∈ ∂u ν∆u + (u )u = p + f, in (0, T ) Ω, ∂t − ·∇ −∇ × u (0,x)= u0(x), (6) div u = 0, where T > 0, x Ω, ν is the unit outward nor- 1 ∈ d u ∂Ω = 0, u (0,x)= u0 (x) , mal, u = (u (t,x),. . ., u (t,x)), a is a real d d | 1 t× (4) matrix with a positive symmetric part (a + a ) 2 ≥ 2 d where u (t,x) = (u1 (t,x) , u2 (t,x) , u3 (t,x)) is the βI with β > 0 and h L (Ω) . Moreover, f = ∈ velocity of an incompressible fluid, p is the pressure (f 1,. . .,f d) is a continuous function satisfying the and f is an external force. following conditions: If we consider the usual function spaces d d pi i p −1 i pi = u (C (Ω))3 : div u = 0 , f (u) i C1(1 + u ), (7) 0∞ | | ≤ | | V { ∈ } i=1 i=1 H = cl(L2(Ω))3 , V = cl(H1(Ω))3 , (5) X X V V d and assume that f H, then the following results (f(u), u) α ui pi C , (8) ∈ ≥ | | − 2 are well known (see e.g. [Lions, 1969, Temam, 1979, i=1 Temam, 1988]): X where pi 2, α, C1,C2 > 0. ≥ 2 d 1 d 1. For every u0 V there exists a unique strong Let H = L (Ω) , V = H0 (Ω) for Dirichlet ∈ 1 d solution of problem (4) which exists in some boundary conditions and V = H (Ω) for Neu- interval [0, T ( u )). mann boundary conditions. Also, for p = (p , ..., pd) k 0kV 1 Three perspectives 9 define the spaces where t = r + 2 and v (t,x) is a solution on r √K r [t , T ] with v (t ,x) = ψ1(x) . It is easy to see that p def p p r r r K L (Ω) = L 1 (Ω) L d (Ω) , 1 ×···× the function uτ (t,x) is a solution of (6) until 4 (t def 2 − Lp (0, T ; Lp (Ω)) = Lp1 (0, T ; Lp1 (Ω)) r) ψ1(x) 1, and it is clear that ur(t,x) 1 for × ≤ | | ≤ p p all t [r, tτ ]. Hence, ur (t,x) is another solution of L d (0, T ; L d (Ω)) . ∈ ×···× problem (6). This example shows that conditions By a globally defined weak solution of (6) (7)-(8) are not enough to give uniqueness. we mean a function u ( ) which belongs to · Let us consider some models for which condi- L (0, T ; H) L2 (0, T ; V ) Lp (0, T ; Lp (Ω)) for all ∞ ∩ ∩ tion (9) fails. T > 0 and satisfies the equation in the sense of distributions. As the regularity of u implies that u C ([0, ),H), the initial condition makes sense ∈ ∞ [Chepyzhov & Vishik, 1996]. Under these condi- tions there exists at least one globally defined weak The complex Ginzburg-Landau equation d solution for every initial condition u L2 (Ω) The complex-valued Ginzburg-Landau equation is 0 ∈ [Chepyzhov & Vishik, 2002b; p.283]. the following: Uniqueness is true, for example, if the following conditions holds (see e.g. ∂u [Chepyzhov & Vishik, 2002b; p.283]): the function = (1+ iη)∆u + Ru (1+ iβ) u 2 u + g (x) , f is continuously differentiable and ∂t − | | u = 0, u (x, 0) = u (x) , 2 d ∂Ω 0 (fu(u)w, w) C w , u, w R , (9) | ≥− 3 | | ∀ ∈ (10) where u = u (t,x) = u1 (t,x)+ iu2 (t,x) for (x,t) where C 0 and fu denotes the Jacobian matrix ∈ 3 ≥ Ω [0, T ] and g (x)= g1 (x)+ig2 (x) L2 (Ω, C) for of u f (u) . × ∈ i 7→ η,β R and R> 0. We assume that g L2 (Ω). If we consider a bit more general situation ∈ ∈ where f = f (x, u) is a Carath´eodory function, then For v = u1, u2 and u = u1 + ıu2, equation (10) can be written as the real-valued system we can give an example for which at least two weak solutions corresponding to a given initial condition exist. Let λ1 > 0 be the first eigenvalue of ∆ in 1 − 1 H0 (Ω), and ψ1 be the corresponding eigenfunction. 1 η g (x) ∂v − Without loss of generality we can assume that ψ1(x) = ∆v + ∂t η 1 g2 (x) > 0 for any x Ω. It is known that ψ1 C(Ω), so ∈ ∈ 2 2 that max ψ1(x) K. We put Ru1 u1 + u2 u1 βu2 x Ω | | ≤ − − ∈ + 2 2 Ru2 u1 + u2 βu1 + u2 − f(x, u)= λ1u ψ1(x)√u, u [0, 1], − − ∈ and conditions (7)-(8) hold with p = (4, 4). p 2 λ1u ψ1(x)u + u (u 1), u [0, 1], Also, condition (9) holds if β √3 − − − 6∈ | | ≤ [Chepyzhov & Vishik, 2002b; p.42]. and h 0. It isp easy to check that f(x, u) satisfies ≡ conditions (7)-(8) with p = 4. Suppose that a = 1, Hence, if β √3, there exists a unique so- | | ≤ 2 2 u0 = 0. Then u(t,x) 0 is a trivial solution of the lution for every initial data in L (Ω) . We note ≡ Cauchy problem (6). For fixed r 0 we define that, if N = 1, 2, uniqueness is proved for every ≥ β [Temam, 1988; p.224]. However, if N 3 and ≥ 0, 0 t r, we do not assume the condition β √3, then it ≤ ≤ | | ≤ 1 2 2 is not known whether this equation possesses the ur(t,x) := 4 (t r) ψ1(x), r t r + √ , − ≤ ≤ K property of uniqueness of the Cauchy problem or vr (t,x) tr t T, not. ≤ ≤ 10 Balibrea, Caraballo, Kloeden & Valero
The Lotka-Volterra system with diffusion [Otani, 1977], and then we can guarantee the global We consider the system existence of strong solutions for every u L2 (Ω) 0 ∈ (see also [Rossi et al., 2008], where the case of more 1 ∂u 1 1 1 2 3 regular initial data is considered). However, we can- = D1∆u + u a1 u a12u a13u , ∂t − − − not expect to have uniqueness when the function 2 ∂u 2 2 2 1 3 f2 is not continuous. Indeed, consider the following = D2∆u + u a2 u a21u a23u , ∂t − − − equation 3 ∂u 3 3 3 1 2 = D3∆u + u a3 u a31u a32u , ∂u ∂2u ∂t − − − H (u) , on (0, T ) (0, 1) , 2 0 (11) ∂t − ∂x ∈ × (15) with Neumann boundary conditions u ∂Ω= 0, u t=0= u0, | | 1 2 3 ∂u ∂u ∂u where ∂ = ∂ = ∂ = 0, ∂ν | Ω ∂ν | Ω ∂ν | Ω 1 if u< 0, i i where u = u (x,t) 0 and ai > 0. Also, the Di − ≥ H0(u)= [ 1, 1] if u = 0, are positive constants and Ω R3. − ⊂ 1 if u> 0, Conditions (7)-(8) hold for u R3 with p = ∈ + (3, 3, 3). is the Heaviside function. If we consider the initial Uniqueness of the Cauchy problem for this sys- condition u (x) = 0, then obviously u(t,x) 0, for 0 ≡ tem has been proved only if we consider solutions all t 0, is a solution, but it is not the only one. ≥ confined in an invariant region (for example, in a In fact, it was shown in [Arrieta et al., 2006] that parallelepiped = u1, u2, u3 : 0 ui ki ) D { ≤ ≤ } problem (15) possesses an infinite (but countable) (see [Marion, 1987] and [Smoller, 1983]). However, number of stationary points, and for each of these 3 in the general case for initial data just in L2 (Ω) points there exists at least one solution with initial it is still an open problem. data u (x) = 0 converging to it as t + . Hence, 0 → ∞ there exists in fact an infinite number of solutions 4.1(C). Parabolic equations with a discontinu- corresponding to the initial data u0(x) = 0. ous nonlinearity We now consider some models of physical in- Let Ω RN be a bounded open set with smooth terest. ⊂ boundary ∂Ω and consider the differential inclusion A model of combustion in porous media ∂u ∆u + f (u) f (u) h, in (0, ) Ω, Consider the equation ∂t − 1 − 2 ∋ ∞ × 2 u ∂Ω= 0, u t=0= u0, ∂u ∂ u | | f (u) λH (u 1) , in (0, T ) (0,π) , (12) ∂t − ∂x2 − ∈ − × 2 R where h L (Ω) and fi : R 2 for i = 1 and 2 ∈ → u (0) = u (π) = 0, u t=0= u0, are maximal monotone maps with domain D (fi)= | R and satisfying where f : R R is continuous and non-decreasing, → λ> 0 and where sup y K1 + K2 s , (13) y f2(s) | |≤ | | ∈ 0, if z < 0, 2 H(z)= [0, 1] if z = 0, (y1 y2) s ( λ1 + ε) s M, yi fi (s) (14) − ≥ − − ∀ ∈ 1, if z > 0. for i = 1 and 2, where λ1 is the first eigenvalue of ∆ in H1 (Ω), for some K , K , M 0 and ε> 0. Suppose also that there exist K 0, 0 K < − 0 1 2 ≥ 1 ≥ ≤ 2 As shown in [Valero, 2001] this equation is a 1 such that f (s) K + K s . This equation | | ≤ 1 2 | | particular case of an abstract differential inclusion models a process of combustion in porous media generated by a difference of sub-differential maps (see [Feireisl & Norbury, 1991]) and it is easy to see of proper convex lower semicontinuous functionals that conditions (13)-(14) hold [Valero, 2001]. Three perspectives 11
A model of conduction of electrical impulses and the incident solar energy at the point x on the in nerve axons Consider the equation Earth surface. Obviously, β(u(x,t)) depends on the nature of the Earth surface. For instance, it is well ∂u ∂2u + u H (u a) , in (0, T ) (0,π) , known that on ice sheets β(u(x,t)) is much smaller ∂t − ∂x2 ∈ − × than on the ocean surface because the white color u (0) = u (π) = 0, u t=0= u0, | of the ice sheets reflects a large portion of the in- cident solar energy, whereas the ocean, due to its where a 0, 1 . In this case f (s) = s and f (s) ∈ 2 1 2 dark color and high heat capacity, is able to absorb = H (s a). It is clear that (13)-(14) are satisfied − a larger amount of the incident solar energy. [Valero, 2001]. This inclusion is used as a model We point out that this model is very close to of conduction of electrical impulses in nerve axons (12). In fact, if we consider that the function f can (see [Terman, 1983, Terman, 1985]). 2 depend on x, then it would be a particular case of that equation. Therefore, we cannot expect to have 4.1(D). A model from climatology uniqueness for problem (16) either. We now consider a climate energy balance model proposed in [Budyko, 1969], and which 4.1(E). Wave equation has been studied from the dynamical point of N view in several works (see e.g. [D´ıaz et al., 1997, Let Ω R be a bounded open set with smooth ⊂ D´ıaz & D´ıaz, 2002, D´ıaz et al., 2002]). The prob- boundary ∂Ω and let us consider the nonlinear wave lem is the following: equation
∂u ∂2u ∂2u ∂u + Bu QS(x)β(u)+ h(x), 2 2 ∆u + β + f (u) = 0, ∂t − ∂x ∈ ∂t − ∂t (t,x) R+ ( 1, 1), (t,x) (0, T ) Ω, ∈ × − (16) ∈ × + ux( 1,t)= ux(1,t) = 0, t R , u ∂Ω= 0, − ∈ | u(x, 0) = u0(x), x ( 1, 1), ∂u ∈ − u (x, 0) = u0 (x) , (x, 0) = u1 (x) , x Ω, ∂t ∈ where B, Q and ε are positive constants, S, h (19) 2 ∈ L∞ ( 1, 1), u0 L ( 1, 1) and β is a maximal where β > 0 and f : R R is continuous and − ∈ 2 − → monotone graph in R , which is bounded, i.e., there satisfies the sign condition exist m, M R such that ∈ f (u) m z M, for all z β(s), s R. (17) lim inf > λ1, u u − ≤ ≤ ∈ ∈ | |→∞ We also assume that with λ > 0 the first eigenvalue of ∆ in H1 (Ω). 1 − 0 0
G (t, u )= u (t) : u ( ) (u ) . (26) 1. attracts any B (X), i.e., 0 { · ∈ D 0 } A ∈ B It is proved in [Kapustyan & Valero, 2006; Lemma dist (G (t,B) , ) 0 as t + . (28) A → → ∞ 9] that the map G is a strict multi-valued semi- flow. The inclusion G (t + t ,x) G (t , G (t ,x)) 2. is negatively semi-invariant. 1 2 ⊂ 1 2 A is a consequence of the fact that the translation u ( + τ) of any weak solution u ( ) is a weak solution The main property of a global attractor is thus · · for any τ > 0. Also, the converse inclusion follows the attraction property. If we consider for example from the property of concatenation of solutions: if the semi-flow G given by (26), then this property u ( ), v ( ) (u0), then means that all the weak solutions starting at the · · ∈ D bounded set B uniformly converge to the set as A u (t) if 0 t t1, time goes to + . The second property implies for z (t)= ≤ ≤ ∞ system (6) (using the property of concatenation of ( v (t t1) if t t1, − ≥ solutions) that for any initial data inside the global is a new weak solution. We note that the last prop- attractor backward solutions exist. erty can fail sometimes (as we will see later for Usually, the global attractor is assumed to be the example of the three-dimensional Navier-Stokes also compact. However, as this property can fail Three perspectives 15 in some applications, we prefer to give this more assume that G is asymptotically compact and that general definition. We note that if is compact, a bounded absorbing set B exists. Then a global A 0 then it is the minimal attracting set, that is, for any compact minimal attractor exists. Moreover, if A closed set C satisfying the attracting property (28) G is strict, then the global attractor is invariant. we have C and, moreover, is unique. A ⊂ A Let us introduce now some definitions that are Remark 4.8. If we assume that the map G (t, ) · necessary to obtain a global attractor. has closed graph for any t 0 and eliminate ≥ the assumption of upper semi-continuity, then the Definition 4.3. The m-semi-flow G is called above results remains true [Melnik & Valero, 2008; asymptotically compact if any sequence ξn Lemma 3]. ∈ G (tn,B) with tn + is precompact in X. Also, we observe that in fact the conditions → ∞ given in [Melnik & Valero, 1998; Theorem 3 and In [Melnik & Valero, 1998] a slightly weaker Remark 8] are weaker, since, instead of supposing definition was used. Namely, the property of com- the existence of a bounded absorbing set, it is as- pactness of ξn G (tn,B) was assumed to be true sumed only that G is point dissipative, which means ∈ only for those sets B (X) such that γ+ (B) that there exists a bounded set B for which ∈ B T (B) 0 (X) for some T (B) R+, i.e., for eventu- ∈ B ∈ dist (G (t,x) ,B ) 0 as t + , (29) ally bounded sets. We think that Definition 4.3 0 → → ∞ is better, since, in fact, it was supposed later in for all x X. We remark that in this situation if we [Melnik & Valero, 1998] that all the sets are eventu- ∈ change the upper semi-continuity of G (t, ) by the ally bounded, so that the property is satisfied for all · assumption of having closed graph, then we need the bounded sets. Of course, the asymptotic com- to suppose also that the semi-flow G is strict (see pactness as given in Definition 4.3 implies eventu- [Melnik & Valero, 2008]). ally boundedness of every B (X), which allows ∈ B to use less assumptions and make the exposition of The global attractor given in Theorem 4.7 the results easier for the reader. can be characterized as the union of all the ω- The following result is useful and easy to prove: limit sets for bounded sets. Indeed, as shown in [Melnik & Valero, 1998; Theorem 1] the conditions Lemma 4.4. If every B (X) is eventually ∈ B of Theorem 4.7 imply that for any B (X) the bounded and the map G (t, ) is compact for some ∈ B · ω-limit set ω (B) is non-empty, negatively semi- t > 0 (i.e., it maps bounded sets into precompact invariant and the minimal closed set attracting B. ones), then G is asymptotically compact. Hence, as the global attractor attracts every B A (X), we have ω (B) . Also, since ω (B0) Definition 4.5. The set B0 is said to be absorbing ∈ B ⊂ A attracts B , where B is the absorbing set, then for if for any B (X) there exists T (B) such that 0 0 ∈ B τ T ( ) and any ε > 0 we have ≥ A G (t,B) B , t T. ⊂ 0 ∀ ≥ G (t, ) G (t, G (τ, )) A⊂ A ⊂ A Definition 4.6. A multi-valued map F : X → G (t,B0) Nε (ω (B0)) if t T (ε, B0) , P (X) is called upper semicontinuous if for all u ⊂ ⊂ ≥ 0 ∈ X and any neighborhood O (F (u )) there exists δ 0 so that ω (B0). Hence, > 0 such that F (u) O(F (u )), as soon as ρ(u, u ) A⊂ ⊂ 0 0 < δ. = ω (B )= ω (B) . A 0 B (X) We have: ∈B[ One important topological property of the Theorem 4.7. [Melnik & Valero, 1998; Theorem global attractor is its connectivity. For this we need 3 and Remark 8] Let G (t, ) : X P (X) be up- additional assumptions. · → per semicontinuous for any t 0 and have closed The map x( ) : R+ X is said to be a tra- ≥ · → values (i.e. G (t,x) C (X) for all (t,x)). Also, jectory of the m-semi-flow G corresponding to the ∈ 16 Balibrea, Caraballo, Kloeden & Valero initial condition x if x(t + τ) G(t,x(τ)) for all Lemma 4.4 implies that G is asymptotically com- 0 ∈ t, τ R+ and x(0) = x . The m-semi-flow G : pact. Also, it follows that the graph of G is closed ∈ 0 R+ X P (X) is said to be time-continuous if it and that G has compact values. Finally, by a con- × → is the union of continuous trajectories for all x tradiction argument (see the proof of Corollary 7 in 0 ∈ X, i.e., [Kapustyan & Valero, 2006]) it can be shown that G (t, ) is upper semi-continuous. · x(t) : x( ) is a trajectory On the other hand, it is proved in G(t,x0)= · . ( and x( ) C(R+,X) ) [Kapustyan & Valero, 2009a] that the map G · ∈ has connected values, and the other conditions in Theorem 4.9. [Melnik & Valero, 1998; Theorem Theorem 4.9 are obviously satisfied. 5] Let us suppose that the conditions of Theorem Hence, by Theorems 4.7, 4.9 we have: 4.7 are satisfied. Assume also that G is a strict time-continuous m-semi-flow with connected values Theorem 4.12. The m-semi-flow G generated by (i.e., G (t,x) is a connected set for every (t,x)). If (6) possesses a global compact minimal attractor, the space X is connected, then the global attractor which is invariant and connected. is connected. A Let us consider now the method of generalized Remark 4.10. We can avoid the assumptions that semi-flows. G is strict and time continuous, if we assume the One of the difficulties of the method of multi- existence of a bounded connected set in X contain- valued semi-flows is that we have to work with ing the global attractor [Amig´o et al., 2009; The- A multi-valued maps. This is avoided in this other orem 4.5]. We note that in order to prove the con- method. Also, the method of generalized semi-flows nectivity is essential to keep the property of upper has the advantage of working directly with solu- semi-continuity of G (t, ) . · tions. Following [Ball, 1997, Ball, 2000] we have:
Let us apply these results to the m-semi-flow Definition 4.13. A generalized semi-flow S on X (26). is a family of maps ϕ : [0, ) X (called solu- ∞ → First note that the following estimate is true tions) which satisfy: [Kapustyan & Valero, 2000; p.625]: (H1) For each x X there exists at least one ϕ S 2 δt 2 ∈ ∈ u (t) e− u + K, (30) such that ϕ (0) = x; k k ≤ k 0k (H2) (Translation) If ϕ S and τ 0, then ϕτ for any weak solution of (6), where the constants δ ∈ ≥ ∈ S, where ϕ (t) = ϕ (τ + t). and K > 0 are universal. Then a bounded absorb- τ ing set exists (for example, a ball in X of radius (H3) (Concatenation) If ϕ, ψ S with ψ (0) = + ∈ √1+ K centered at 0) and γ (B) (X) for all ϕ (t ) , then φ S, where 0 ∈ B 1 B (X). ∈ ∈ B ϕ (t) , 0 t t1, Also, the following lemma holds: φ (t)= ≤ ≤ ψ (t t1) , t t1. Lemma 4.11. [Kapustyan & Valero, 2006; − ≥ Lemma 2] Let un be an arbitrary sequence (H4) If ϕj S withϕj (0) z, then there exist ϕ { } ∈ → of solutions of (6) with un(0) u0 weakly in H. S with ϕ (0) = z and a subsequence such → ∈ Then for any tn t0, where tn,t0 (0, T ], there that ϕj (t) ϕ (t), for all t 0. → ∈ → ≥ exists a subsequence such that un(tn) u(t ) in → 0 H, where u( ) is a weak solution of (6) and u(0) = We note here that we can define a multi-valued · u . semi-flow G : R+ X P (X) by 0 × → G (t,x)= ϕ (t) . (31) As a consequence of this lemma we have that ϕ S, the map G (t, ) is compact for every t > 0. Hence, [∈ · ϕ(0)=x Three perspectives 17
Then conditions (H2) (H3) imply that G is a strict S with ϕj (0) bounded, any sequence ϕj (tj), where − multi-valued semi-flow and condition (H4) implies tj + , is precompact in X. → ∞ that the map x G (t,x) is upper semi-continuous 7→ and has compact values. Therefore, Definition 4.13 It is clear that S is asymptotically compact if contains stronger conditions than Definition 4.1. and only if the associated multi-valued semi-flow is asymptotically compact. If we consider the reaction-diffusion system (6), then we have seen that the weak solutions satisfy Definition 4.16. The generalized semi-flow S is properties (H1) (H3). Also, by Lemma 4.11 we said to be point dissipative if there exists a bounded − have (H4). Hence, taking S as the union of all weak set B such that for any ϕ S there exists T (ϕ) 0 ∈ solutions we obtain a generalized semi-flow. such that ϕ (t) B for all t T. ∈ 0 ≥ It is obvious that the associated multi-valued semi-flow defined in (31) coincides with the semi- This definition of point dissipativity is weaker flow given in (26). than the one given for multi-valued semi-flows in (29). The difference is that if (29) holds for the as- We introduce in this case the concepts of ω- sociated semi-flow G, this implies that all the solu- limit set and global attractor. For B B (X) we tions starting at ϕ (0) entry uniformly in a bounded ∈ set set B0 after some time, that is, there exists a time ω (B)= T (ϕ (0)) such that for any solution ϕ S we have ∈ ϕ (t) B0 for all t T. z X : there exist ϕj S, with ϕj (0) B, ∈ ≥ ∈ ∈ ∈ The following lemma implies that an asymp- and tj + such that ϕj (tj) z. → ∞ → totically compact and point dissipative generalized It is clear that ω (B) coincides with the omega-limit semi-flow satisfies property (29). set (27) for the associated multi-valued semi-flow given in (31). Lemma 4.17. [Ball, 2000; Lemma 3.5] Let S be an asymptotically compact and point dissipative Definition 4.14. The set is said to be a global generalized semi-flow. Then there exists a bounded A attractor for the generalized semi-flow S if the as- set B1 such that given any compact K X there ⊂ sociated multi-valued semi-flow G satisfies the at- exist ε = ε (K) > 0, t1 = t1 (K) > 0 such that traction property (28) and G (t, ), that is, it G (t, Nε (K)) B1 for all t t1. A ⊂ A ⊂ ≥ is negatively semi-invariant. It is important to remark that property (H4) In other words, is a global attractor for S is crucial in the proof of this lemma. A if it is a global attractor for the associated multi- We now state the theorem concerning the exis- valued semi-flow G. In [Ball, 1997, Ball, 2000] it tence of a global attractor. is assumed also that the global attractor has to be compact and invariant, but, as we have commented Theorem 4.18. [Ball, 2000; Theorem 3.3] Let S before, we prefer to give a more general definition be an asymptotically compact and point dissipative and add these properties in the statement of the generalized semi-flow. Then there exists a compact theorem. The reason is that, although usually in global attractor given by applications these additional properties hold (as is = ω (B) , the case of the reaction-diffusion system (6)), some- A B (x) times they can fail. When the global attractor is ∈B[ compact, one can check that it is also minimal and which, moreover, is invariant. unique. As before in order to obtain the existence of a With respect to the connectivity of the attrac- global attractor we need some previous definitions. tor we have:
Definition 4.15. The generalized semi-flow S is Theorem 4.19. [Ball, 2000; Corollary 4.3] As- asymptotically compact if for any sequence ϕj sume the conditions of Theorem 4.18. Let X be ∈ 18 Balibrea, Caraballo, Kloeden & Valero a connected space. Assume that the operator G has are added when they become necessary. On the connected values and that every ϕ S belongs to other hand, the method of generalized semi-flows ∈ C ((0, ) ; X). Then the global attractor is con- allows to use a weaker definition of point dissipa- ∞ A nected. tivity. These approaches have been very useful We note that the definition of the generalized and productive and have allowed results about semi-group S implies that the associated semi-flow existence and properties of attractors to be G is strict and that the map G (t, ) is upper semi- obtained for a wide class of dissipative sys- · continuous. Also, it follows from the fact that every tems without uniqueness. For example, these ϕ S belongs to C ((0, ) ; X) that G can be repre- results have been applied fruitfully to reaction- ∈ ∞ sented as the union of trajectories which are contin- diffusion equations [Iovane & Kapustyan, 2006, uous on t (0, ). Thus the conditions imposed Kapustyan, 2002, Kapustyan & Shkundin, 2003, ∈ ∞ in this theorem are almost the same as in Theo- Kapustyan & Valero, 2006, rem 4.9 (in fact, the only difference is that there it Morillas & Valero, 2005], the three-dimensional is assumed that the trajectories are continuous on Navier-Stokes equation [Ball, 1997, Ball, 2000, t [0, ). Cheskidov, 2006, Cheskidov & Foias, 2006, ∈ ∞ Kapustyan et al., 2007, Kapustyan & Valero, 2007, Let us apply these results to equation (6). As Kloeden & Valero, 2007, Rosa, 2006], we have seen before it follows from (30) and Lemma the wave equation [Ball, 2004, 4.11 that S is an asymptotically compact and point Horban & Stanzhyts’kyi, 2008, dissipative generalized semi-flow. Also, as the weak Iovane & Kapustyan, 2005, Wang & Zhou, 2007], solutions of (6) are continuous, it is clear that every differential inclusions [Carvalho & Gentile, 2003, ϕ S belongs to C ((0, ) ; X). The space X = Cheban & Mammama, 2006, ∈ d ∞ L2 (Ω) is connected and we have seen that G Kapustyan & Valero, 2000, has connected values. Kenmochi & Yamazaki, 2001, Kloeden & Li ,2005, Hence, by Theorems 4.18, 4.19 we have: Li et al., 2008, Melnik & Valero, 1998, Segatti, 2006, Valero, 2000, Valero, 2001, Theorem 4.20. The generalized semi-flow S gen- Yamazaki, 2004], delay ordinary differ- erated by (6) possesses a global compact minimal ential equations [Caraballo et al., 2005, attractor, which is invariant and connected. Caraballo et al., 2007], degenerate parabolic equations [Elmounir & Simonolar, 2000], lattice Remark 4.21. Since the associated multi-valued systems [Morillas & Valero, 2009], the phase-field semi-flow G coincides with the semi-flow defined in equation [Kapustyan, 1999, Kapustyan et al., 2003, (26) and the global attractor is unique, we see that Morillas & Valero, 2008, Rossi et al., 2008, the global attractors given in Theorems 4.12, 4.20 Valero, 2005] or the Cahn-Hilliard equa- are the same. tion [Segatti, 2007, Schimperna, 2007]. See also the monographs [Kapustyan et al., 2008, Remark 4.22. Under some conditions the global at- Cheban, 2004]. tractors given in Theorems 4.7, 4.18 are described Of course, in the case of uniqueness of solutions as the union of all bounded complete trajectories all results in the two methods coincide with the of the semi-flow [Ball, 2000, Kapustyan et al., 2008, classical ones. Simsen & Gentile, 2008]. 4.2(B). The method of trajectory attractors In conclusion, we can say that these two meth- In order to avoid the problem of non- ods are very close and allow rather similar results uniqueness of the Cauchy problem in to be obtained. The main difference is that in the the papers [Chepyzhov & Vishik, 1996, method of generalized semi-flows several extra con- Chepyzhov & Vishik, 1997, ditions are included in the definition, whereas in the Chepyzhov & Vishik, 2002a, Malek & Necas, 1996, method of multi-valued semi-flows such conditions Sell, 1996] (see also [Chepyzhov & Vishik, 2002b, Three perspectives 19
Malek & Prazak, 2002]) a new method for studying ΠM fn( ) ΠM f( ) in C([0, M]; E ), M > 0, · → · 0 ∀ the asymptotic behavior of solutions was proposed, where Π is the operator of restriction on [0, M]. which is now known as the method of trajectory M We observe that the translation semi- attractors. group is continuous in the space C(R+; E ) The main idea of this method is the follow- 0 [Chepyzhov & Vishik, 2002a; Proposition 1.1]. ing. Let us consider some collection of solutions + of an equation, denoted by = ϕ (s) ,s 0 . Definition 4.23. The set U + is called a tra- K { ≥ } ⊂ K These solutions belong to some topological space jectory attractor (with respect to the space of tra- W , which usually is a space with some local con- + + + jectories in the topology C(R ; E0), if: vergence topology on any interval [t1,t2] R . If K + ⊂ + we assume that for any ϕ we have that ϕτ ( ) 1. U is compact in C(R ; E0) and bounded in + ∈ K · + := ϕ( + τ) K for all τ 0, then the translation L∞(R ; E); · ∈ ≥ semi-group S (t, ) : W W given by · → 2. U is invariant, that is, T (t)U = U for all t ≥ S (t) ϕ ( )= ϕt( )= ϕ ( + t) 0; · · · is well defined. 3. U is an attracting set, that is, for every set B This method has the advantage that a +, bounded in L (R+; E), we have that ⊂ K ∞ global attractor for such problems as the three- for any M > 0, dimensional Navier-Stokes equations can be con- structed, whereas this question remains still open dist (ΠM T (t)B, ΠM U) 0, (32) C([0,M];E0) → using the theory of multi-valued semi-flows or gen- as t . eralized semi-flows. However, as pointed out in → ∞ [Ball, 2000], the disadvantage is that the direct con- The function ϕ( ) L (R; E) C(R; E ) is nection with the evolution of the system in the · ∈ ∞ ∩ 0 called a complete trajectory for +, if physical phase space is lost. K In the cited papers different approaches are + Π+ϕh( ) , h R, used in order to construct a trajectory attractor, · ∈ K ∀ ∈ i.e., a global attractor for the translation semi- where ϕh(s) = ϕ(s + h) and Π+ is the operator group. We shall describe briefly the methods of restriction on [0, ). Let K be the union of all ∞ given in [Chepyzhov & Vishik, 2002a, Sell, 1996] complete trajectories for +. and their application to the three-dimensional K Navier-Stokes system (4). Theorem 4.24. [Chepyzhov & Vishik, 2002a] If In [Sell, 1996] it is considered that W is a com- there exists an attracting set P +, which is + ⊂ K + plete metric space. Then the standard theory of compact in C(R ; E0) and bounded in L∞(R ; E), attractors for classical semi-groups is used. then there exists a trajectory attractor U P and ⊆ On the other hand, in [Chepyzhov & Vishik, 2002a] a bit different U = Π+K. (33) approach is used. Let us take some Banach spaces In the case where uniqueness of the Cauchy E and E0 such that E E0 with continuous ⊆ problem holds (and then a semi-group S can be embedding (it is possible to have E = E0), and let given), in [Chepyzhov & Vishik, 2002a; Corollary + + W = L∞(R ; E) C(R ; E0). 2.1] it is shown a relation between the trajectory ∩ + attractor U and the global attractor of the semi- We note that if ϕ( ) W , then ϕ( ) Cw(R ; E) · ∈ · ∈ group S. Namely, assume that for any z E there (i.e., it is continuous with respect to the weak topol- ∈ exists a unique trajectory ϕ ( ) + such that ϕ (0) ogy of E) and · ∈ K = z. Then the semi-group given by S (t, z) = ϕ (t) ∞ ϕ(t) E ϕ L (R+;E), t 0. for all t 0 is well defined. Then if we assume k k ≤ k k ∀ ≥ ≥ also that the set t 0S (t,B) is bounded in E for The convergence in C(R+; E0) is given by ∪ ≥ any set B, which is bounded in E, then under the + fn( ) f( ) in C(R ; E ) assumptions of Theorem 4.24 the set = U (0) is · → · 0 ⇐⇒ A 20 Balibrea, Caraballo, Kloeden & Valero a global (E, E0)-attractor for S. This means that diffusion systems [Chepyzhov & Vishik, 1996, the set is bounded in E, compact in E , it is in- Efendiev & Zelik, 2001], differential inclusions A 0 variant (i.e., S (t, )= for all t 0) and attracts [Hetzer, 2001, Hetzer & Tello, 2002], the wave A A ≥ every bounded set of E in the metric of E0. equations [Chepyzhov & Vishik, 1997] or equations This result can be generalized to the case of with delay [Chepyzhov et al., 2006]. non-uniqueness, that is, when we assume that for any z E there exists at least one trajectory 4.3. Technical difficulties and open prob- ∈ ϕ ( ) + such that ϕ (0) = z. In such a lems in multi-valued dynamical sys- · ∈ K case we define a multi-valued semi-flow G given by tems G (t, z) = ϕ (t) : ϕ ( ) + such that ϕ (0) = z { · ∈ K } In the previous subsection we reviewed the main and assuming that t 0G (t,B) is bounded in E for ∪ ≥ any set B, bounded in E, we can obtain the same points of two different theories for multi-valued dy- result [Kapustyan & Valero, 2009a]. namical systems. It is natural to ask about addi- tional technical problems which arise if we compare Let us consider system (4). In the paper them with the classical case for which uniqueness [Chepyzhov & Vishik, 2002a] the set + is the holds. K union of all weak solutions of (4) satisfying a suit- able energy inequality. We take E = H and 4.3(A). Asymptotic compactness and the 3D δ σ Navier-Stokes system E0 = H− , where H is given in (5) and H stands for the scale of Hilbert spaces corresponding to the One of the difficulties which appears as a conse- Hilbert space H. Then it is proved that an attract- quence of the lack of uniqueness is that, unlike the + δ ing set U, which is compact in C(R ; H− ) and case with uniqueness, there are strong restrictions + bounded in L∞(R ; H), exists. Therefore, Theo- in the kind of estimates that we can obtain. This oc- rem 4.24 implies the existence of a trajectory at- curs because, usually, we do not have a regular ap- tractor. proximation for each solution of the equation and, Now let us consider Sell’s approach [Sell, 1996]. as a result, we cannot obtain formally the necessary There the set + is the union of all weak solutions estimates and justify them through suitable approx- K of (4) satisfying some regular assumptions and suit- imations. We are only able to obtain the estimates able dissipation and energy inequalities, as well. that the regularity properties of the solutions allow. The space W is defined as the set + endowed In particular, this problem arises when we need K with the topology of L2 (0, ; H), which is a com- to prove the property of asymptotic compactness loc ∞ plete metric space. Then, using the classical theory of a multi-valued semi-flow or a generalized semi- of attractors for semi-groups [Ladyzhenskaya, 1991, group which, as we have seen, it is crucial in order Sell & You, 1995], it is proved that the translation to prove the existence of a global attractor. semi-group has a compact global attractor U in For example, let us consider the reaction- W , that is, U is compact in W , it attracts every diffusion system (6). In this case the asymptotic bounded set of W and is invariant (S (t) U = U for compactness of the multi-valued semi-flow (or the all t 0). single-valued semi-group if we have uniqueness) is ≥ proved by checking that the map G (t, ) is compact · As the two previous methods, the method for any t > 0. If the monotonicity assumption (9) of trajectory attractors has been fruitfully ap- holds (so that we have uniqueness), then for any plied to different partial differential equations. weak solution we can obtain (formally) an estimate For example we can cite applications to the of the type three-dimensional Navier-Stokes equation and related systems [Chepyzhov & Vishik, 1997, u (t) C (t, u0 ) , t> 0, (34) k kV ≤ k kH ∀ Chepyzhov & Vishik, 2002a, Cutland, 2005, 1 d 1 d Feireisl, 2000, Flandoli & Schmalfuß, 1999, where V = H (Ω) or H0 (Ω) and H = d Malek & Necas, 1996, Malek & Prazak, 2002, L2 (Ω) , and C (t, u ) as t 0+ k 0kH → ∞ → Norman, 1999, Sell, 1996, [Chepyzhov & Vishik, 1996; p.67]. Then using the Vorotnikov & Zvyagin, 2007], reaction- compact embedding V H we obtain that the ⊂ Three perspectives 21 semi-group S (t) is compact for any t > 0. Since and in order to obtain yn y (so yn y k k → k k → the solution is unique, we can justify this result strongly) it is enough to show that as follows: we take a suitable regular approxima- tion sequence for which the estimate is correct (for lim sup yn y . n k k ≤ k k example, a Galerkin approximative sequence) and →∞ then passing to the limit we obtain the estimate for Hence, estimates in more regular spaces are the weak solutions of the equation. avoided. The two methods diverge in the way of When condition (9) is not satisfied, this ar- proving the last inequality, as one can check in the gument fails, as we cannot state that every weak cited papers. solution can be approximated by a Galerkin se- We observe that these methods fail in the case quence: we just can say that at least one weak of the three-dimensional Navier-Stokes system (4), solution can be approximated. Hence, we cannot so that the problem of obtaining the property of obtain (34) for all weak solutions. The same prob- asymptotic compactness for this equation remains lem appear in other equations as the phase-field sys- open so far. For example, the method of mono- tem (21) or the wave equation (19), although it is tonicity does no work because the weak solutions worth to point out that in some multi-valued sys- are not known to be continuous (just weakly contin- tems we have enough regularity in order to obtain uous) and this property is necessary in this method. estimates of the kind (34), as for example equation Nowadays, if we consider an arbitrary external force f H, then the question about the asymp- (12) [Valero, 2001]. ∈ To avoid this problem two appropriate tools totic compactness for the solutions of (4) is open. have been used in several papers: the method In the papers [Ball, 2000, Cheskidov & Foias, 2006, of the energy equation and the monotonicity Kapustyan & Valero, 2007, Rosa, 2006] some con- method. For example, the first one has been ap- ditional results have been obtained in this di- plied in [Ball, 2004] to the wave equation (19) rection (that is, under assumptions that are not and in [Morillas & Valero, 2005] to the reaction- known to hold). On the other hand, in the papers diffusion system (6) in unbounded domains, and [Birnir Svanstedt, 2004, Bondarevski, 1997] under the second one in [Kapustyan et al., 2003] to the some quite restrictive conditions on the external phase-field system (21) and to the reaction- force f, it is shown that for every bounded set B diffusion system (6) in [Morillas & Valero, 2005, all the weak solutions starting at B become regular Kapustyan & Valero, 2006]. We note that the after a time T (B), and in this way it is proved the method of the energy equation has been also suc- existence of a global attractor. cessfully used in many papers for equations with On the other hand, there can be also difficulties uniqueness of solutions (see e.g. [Ghidaglia, 1994, in obtaining an absorbing set. Although this prob- Lu & Wang, 2001, Moise et al., 1998, Rosa, 1998, lem does not appear in reaction-diffusion systems or Wang, 1999]). in the phase-field system (as we have enough regu- larity to obtain similar estimates as in the case of The key idea in these methods is the follow- uniqueness), again some extra effort is needed for ing. Let the phase space X be a Hilbert space (of the three-dimensional Navier-Stokes system. If we course, more general situations are possible). In multiply equation (4) by u and make some standard such spaces if xn x weakly and xn x , → k k → k k operations, one can obtain (formally) an estimate then xn x strongly. If we consider the property of → in the norm of the space H of the kind: asymptotic compactness, then we take a sequence yn G (tn,xn) with xn (X), where G is the 1 ∈ ∈ B 2 νλ1t 2 2 multi-valued semi-flow generated by the equation, u (t) e− u (0) 2 2 f k k ≤ k k − ν λ1 k k and we need to prove that yn is precompact. As- { } suming that the a priori estimates of the equation 1 2 + 2 2 f , t 0. (35) give us that this sequence is bounded, we obtain ν λ1 k k ∀ ≥ passing to a subsequence that yn y weakly in X. → However, we cannot state that this holds for every Then weak solution, and just for the one that was ob- y lim inf yn n tained via the Galerkin approximations. Neverthe- k k≤ →∞ k k 22 Balibrea, Caraballo, Kloeden & Valero less, estimate (35) can be proved if we take a suit- Kapustyan et al., 2008] with solutions satisfy- able subset of the set of weak solutions. Namely, ing (37) and having a bounded absorbing set by in [Ball, 2000] it is taken only the subset of weak restricting the phase space to a ball of radius R 2 1 2 solutions satisfying the following energy inequality: R0, where R0 = 2 2 f . Namely, put BR = ≥ ν λ1 k k u H : u R and for any R R0 define the V (u (t)) V (u (s)) , t s, a.a. s> 0 and s = 0, { ∈ k k≤ } R + ≥ ≤ ∀ ≥ multi-valued map G : R BR P (BR) as (36) × → where R G (t, u0) := 1 V (u (t)) := u (t) 2 2 k kH u(t) : u( ) is a globally defined weak solution · t t with u (0) = u0 such that (37) holds and 2 + ν u (r) dr (f (r) , u (r)) dr. u(r) R, for all r 0. k kV − k k≤ ≥ Z0 Z0 This map is well defined since at least one For such solutions (35) holds [Ball, 2000] and then such solution exists [Kapustyan & Valero, 2007, a bounded absorbing set exists. Kapustyan et al., 2008] and, as the phase space is However, such a choice of solutions has an- bounded, GR has a bounded absorbing set. Then other problem. If we define the translation u (t)= τ the existence of a global attractor can be proved u (t + τ), then property (36) fails for u ( ) at s = 0. · in the space BR endowed with the weak topology Thus, property (H2) of the definition of general- of H. We note that in [Kapustyan & Valero, 2007] ized semi-flow is not satisifed. Also, for the asso- inequality (38) is used also in the definition of GR, ciated multi-valued map G (t, ) the key property · but this is not necessary as (37) implies (38). G (t + s,x) G(t, G (s,x)) fails. ⊂ We could try to modify inequality (36) in the 4.3(B). The structure and fractal dimension following way: of attractors of multi-valued dynamical V (u (t)) V (u (s)) (37) systems ≤ We have seen so far that a lot of results have been holds for all t s and a.a. s > 0. Then the map ≥ proved about the existence of global attractors for u (t) also satisfies this property and a multi-valued τ different multi-valued dynamical systems. More- semi-flow can be correctly defined. However, in over, some topological properties such as its con- such case we cannot prove (35). One can check that nectedness has been studied. However, not much is the following holds [Kapustyan et al., 2008; Lemma known about the structure of these attractors and 5.29]: also about their fractal dimension. Comparing with
2 νλ1(t s) 2 1 2 the classical single-valued semi-groups these prob- u (t) e− − u (s) 2 2 f k k ≤ k k − ν λ1 k k lems are much more difficult to solve in the multi- 1 2 valued case. + ν2λ2 f , t s and a.a. s> 0, (38) 1 k k ∀ ≥ Let us consider the simplest structure of a but this is not enough to obtain an absorbing set. global attractor. If we consider a continuous classi- Also, we note that in this case the concatenation cal semi-group having a global compact connected of two solutions do not have to satisfy necessarily attractor and a Lyapunov (also called energy) func- (37) and then we cannot state that the semi-flow is tion, then, when the set of stationary points is strict. Moreover, property (H3) fails and a gener- finite, it is well known that the attractor con- alized semi-flow cannot be defined in this case. sists of the set of stationary points and all the We observe that, without defining a multival- bounded complete trajectories (heteroclinic connec- ued semiflow, the uniform convergence of the weak tions) joining them [Ladyzhenskaya, 1990]. In par- solutions starting in a bounded set to the universal ticular applications we need to determine which attractor (which consists of points in bounded com- connections exist, as this will give us a full descrip- plete trajectories) is proved in [Foias et. al, 2001]. tion of the attractor. The Chafee-Infante equation Finally, a multi-valued semi-flow is is a very well known model in which this problem defined in [Kapustyan & Valero, 2007, has been solved. In this equation the number of Three perspectives 23 stationary points is finite (this number depends on Then, it is shown that v1± are asymptotically sta- a parameter) and a global compact connected at- ble fixed points, and all the other ones are unsta- tractor, together with a Lyapunov function, exist. ble. The fixed point v 0 possesses the following ≡ Moreover, the set of stationary points is ordered remarkable property: for any fixed point vk differ- by the Lyapunov function V . In [Henry, 1985] a ent from 0 there exists a solution, u (t), with ini- complete description of this attractor is given by tial value u (0) = 0, such that u (t) converges to showing that the two stationary points with less vk as t + . We note that the existence of a → ∞ energy are stable, whereas the other ones are un- Lyapunov function implies in this case, as occurs stable, and that the necessary and sufficient condi- for the Chafee-Infante problem, that the global at- tion for the existence of an heteroclinic connection tractor can be described completely by the equilib- from a point v to a point u is that V (v) > V (u), ria and the heteroclinic connections between them. i.e., if the energy of v is greater than the energy The natural question is then to establish which con- of u. More general and complicated cases can be nections actually exist. found, for example, in [Brunovsky & Fiedler, 1989, The global attractor of (39) is approximated Fiedler & Rocha, 1996]. by a sequence of attractors ε corresponding to A n Can we obtain such results in multi-valued sys- an approximating sequence of Chafee-Infante prob- tems? It is possible, of course, but it is much more lems, for which, as we have seen, all existing con- difficult than in the single-valued case. For exam- nections are known. The natural conjecture is that ple, if we consider the property of stability of sta- the connections are the same when we pass to the tionary points, then one can see that the method limit case, that is, that a connection exists from the of linearization is not applicable, as the non-linear fixed point v to the fixed point v∗ if and only if E (v) terms of equation as (6), (12), (19) or (21) are > E (v∗). Of course, since the energy is decreasing not differentiable. Hence, one of the main tools along the trajectories, no connections can exist if for studying stability in non-linear differential equa- E (v) E (v ). It is natural to expect that (39) ≤ ∗ tions cannot be used in multi-valued systems. Nev- is equivalent to a Chafee-Infante problem that has ertheless, some partial results are known so far. undergo all the typical bifurcation cascade of these In [Arrieta et al., 2006] it is studied the follow- type of problems, and thus all connections should ing differential inclusion be present. In [Arrieta et al., 2006] some of these connections are established, but the complete de- ∂u ∂2u 2 H0 (u) 0, on (0, 1) (0, T ) , scription of the attractor remains an open problem. ∂t − ∂x − ∋ × On the other hand, in [D´ıaz et al., 1997, u (0,t)= u (1,t) = 0, D´ıaz et al., 2002] the authors show that equation (16) also has an infinite, but countable, number of u (x, 0) = u0 (x) , (39) equilibria. Probably, the structure of the attractor where of this equation is the same as for (39). For other problems like (6), (19) or (21) nothing 1, if u< 0 is known in this direction. − H (u)= [ 1, 1] , if u = 0, 0 − Finally, let us consider the problem of obtaining 1, if u> 0, estimates of the fractal and Hausdorff dimensions of the global attractor. This question has been stud- is the Heaviside function. This inclusion is a partic- ied fruitfully and widely for nonlinear single-valued ular case of (12). First, it was proved that equation semi-groups and in this way deep results have been (39) has an infinite, but countable, number of equi- proved in a lot of papers. To mention some of libria v0 = 0, v1±, v2±, ..., vk±, ... ,which can be them, for example, estimates of the fractal and ordered using a natural energy (or Lyapunov func- Hausdorff dimensions depending on the physical pa- tion) E (u): rameter of the equation were proved for the two- dimensional Navier-Stokes system, for reaction- E v+ = E v < E v+ = E v2 < ... 1 1− 2 1 diffusion systems and for the wave equation (see + ... < E vk = E vk− < ... < E (v0) . e.g. [Ladyzhenskaya, 1982, Temam, 1988]). 24 Balibrea, Caraballo, Kloeden & Valero
In the multi-valued case almost nothing has Stokes system (4) in the phase space H for been done in this direction. Some abstract the- an arbitrary external force f H. ∈ orems concerning the fractal dimension of multi- 2. Describe the structure of the global attractor valued semi-flows can be found in [Melnik, 1998] in some applications. (see also [Kapustyan et al., 2008]), a result which was applied in [Valero, 2000] to a differential in- 3. Study the stability of stationary points in clusion. However, the conditions imposed in some applications. [Melnik, 1998] are too strong and, in fact, in the application given in [Valero, 2000] the multi-valued 4. Develop a new method for obtaining esti- semi-flow is single-valued when we restrict it to mates of the fractal and Hausdorff dimen- the global attractor. The main difficulty in the sions. multi-valued case is that the technique used in the single-valued one is not suitable due to the fact 5. Non-autonomous and random dynamical that either a Lipschitz property of the semi-group systems [Ladyzhenskaya, 1982, Ladyzhenskaya, 1990] or a strong differentiability property [Temam, 1988] is Undoubtedly, non-autonomous and random sys- required. In particular, such properties do not hold tems are also of great importance and interest as for multi-valued semi-flows. they appear in many applications to natural sci- Nevertheless, from our point of view there is ences. On some occasions, some phenomena are not an objective reason to think that the global at- modeled by nonlinear evolutionary equations which tractors of problems like (6), (19) or (21), for exam- do not take into account all the relevant informa- ple, do not satisfy similar estimates of the fractal tion of the real systems. Instead some neglected dimension as in the case of uniqueness. Thus, we quantities can be modeled as an external force are convinced that we need just a new technique which in general becomes time-dependent (some- in which the semi-flow would not need to be either times periodic, quasi-periodic or almost periodic Lipschitz or differentiable. due to seasonal regimes) or even can contain ran- On the other hand, there are multi-valued prob- dom/stochastic features (also called noise). lems in which we could expect to have an infinite In the finite-dimensional framework (i.e. for dimensional global attractor. An example is given non-autonomous ordinary differential equations in N in [Valero, 2000] for a differential inclusion. Also, R ) the long-time behaviour of non-autonomous we think that, probably, this is true also for the at- dynamical systems has been widely studied by tractor of equation (39). The heuristic reason is the means of the theory of skew-product flows (see the following: we have seen that for any fixed point vk pioneering works [Miller, 1965] and [Sell, 1996]). different from 0 there exists a solution, u (t), with However, most of the progress in the infinite- initial value u (0) = 0, such that u (t) converges to dimensional context, i.e., for non-autonomous vk as t + , and in some sense this means (as partial differential equations and specially for → ∞ the number of fixed points vk is infinite) that the systems appearing in mathematical physics, has unstable manifold of 0 (if it exists, of course) is been done during the last two decades. infinite-dimensional. Also, we note that the global The first attempts to extend the notion of global attractor of (39) is approximated by a sequence of attractor to the non-autonomous case led to the attractors of Chafee-Infante problems with growing concept of the so-called uniform attractor (see dimension. The same could be true for equation [Chepyzhov & Vishik, 1996]). It is remarkable (16). that the conditions ensuring the existence of the uniform attractor parallel those for autonomous In conclusion, we state the following relevant systems. To this end, non-autonomous systems open problems in the theory of multi-valued dy- are lifted in [Vishik, 1992] to autonomous ones by namical systems: expanding the phase space. Then, the existence of uniform attractors relies on some compactness 1. Obtain the existence of a global compact property of the solution operator associated to the attractor for the three-dimensional Navier- system. However, one disadvantage of this uniform Three perspectives 25 attractor is that it needs not be “invariant” unlike autonomous dynamical system. the global attractor for autonomous systems. At the same time, the theory of pullback (or cocycle) 5.2. Processes attractors has been developed for both the non- Consider an initial value problem for a non- autonomous and random dynamical systems (see d [Crauel et al., 1995, Langa & Schmalfuß, 2004, autonomous ordinary differential equation in R , Kloeden & Siegmund, 2005, x˙ = f(t,x) , x(t0)= x0 . (40) Kloeden & Schmalfuß, 1998, Schmalfuß, 2000]), and has shown to be very useful in the under- In contrast to the autonomous case, the solutions standing of the dynamics of non-autonomous may now depend separately on the actual time t dynamical systems. In this case, the concept of and the starting time t0 rather than only on the pullback (or cocycle or non-autonomous) attractor elapsed time t t since starting. For example, the − 0 provides a time-dependent (or random in the scalar initial value problem stochastic case) family of compact sets which x˙ = 2tx, x(t )= x , attracts families of sets in a certain universe − 0 0 (e.g. the bounded sets in the phase space) and has the explicit solution satisfying an invariance property, what seems to 2 2 (t t0) be a natural set of conditions to be satisfied for an x(t)= x(t,t0,x0)= x0 e− − , appropriate extension of the autonomous concept where t2 t2 = (t t )2 + 2(t t )t cannot be of attractor. Moreover, this cocycle formulation − 0 − 0 − 0 0 expressed in terms of t t alone. allows to handle more general time-dependent − 0 terms in the models not only the periodic, quasi Assuming global existence and uniqueness of periodic or almost periodic ones (see, for instance, solutions in forward time, the solutions form a con- tinuous mapping (t,t ,x ) x(t,t ,x ) Rd for [Caraballo & Real, 2004, Caraballo et al., 2004] 0 0 7→ 0 0 ∈ t t with t,t R and x Rd, with the initial for non-autonomous models containing hered- ≥ 0 0 ∈ 0 ∈ itary characteristics). Readers can con- value and evolution properties sult the monographs [Carvalho et al., 2010, d (i) x(t0,t0,x0)= x0 for all t0 R and x0 R , Kloeden & Rasmussen, 2010] for more references, ∈ ∈ examples and information. (ii) x(t2,t0,x0) = x t2,t1,x(t1,t0,x0) for all t t t and x Rd. 0 ≤ 1 ≤ 2 0 ∈ 5.1. Motivation of the process and skew- The evolution property (ii) is a consequence of product flow formalisms the causality principle that the solutions are deter- The formulation of an autonomous dynamical sys- mined uniquely by their initial values (for the given tem as a group or semi–group of mappings depends differential equation). on the fact that such systems depend only on the 5.2(A). Process formulation of non- elapsed time since starting t t0 and not directly − autonomous systems on the current time t or starting time t0 themselves. For a non-autonomous system, both the current Solution mappings of non-autonomous ordinary dif- time t and starting time t0 are important. The ferential equations are one of the main motivations most natural generalization of a semi–group formal- for the process formulation [Dafermos, 1971] (see ism to non-autonomous dynamical systems includes also [Hale, 1988]) of an abstract non-autonomous both t and t instead of only t t . This leads to system on a complete metric state space (X, d) and 0 − 0 the two-parameter semi–group or process formalism time set T, where T = R for a continuous time pro- of a non-autonomous dynamical system. An alter- cess and T = Z for a discrete time process. native method includes an autonomous dynamical system as a driving mechanism which is respon- Definition 5.1. A process is a continuous map- sible for, e.g., the temporal change of the vector ping (t,t ,x ) φ(t,t ,x ) X for t t with 0 0 7→ 0 0 ∈ ≥ 0 field of a non-autonomous dynamical system. This t,t T and x X, with the initial value and 0 ∈ 0 ∈ leads to the skew product flow formalism of a non- evolution properties 26 Balibrea, Caraballo, Kloeden & Valero
d (i) φ(t ,t ,x )= x for all t T and x X, Let Σd denote the subset of R consisting of the 0 0 0 0 0 ∈ 0 ∈ N-dimensional probability row vectors, i.e., p = (ii) φ(t2,t0,x0) = φ t2,t1, φ(t1,t0,x0) for all d (p , ,pd) Σd satisfies pi = 1 with 0 pi 1 · · · ∈ i=1 ≤ ≤ t0 t1 t2 and x0 X. 1 for i = 1, , d. If the states of the Markov chain ≤ ≤ ∈ · · · P at time t satisfy the probability vector p(t ) Σd, 0 0 ∈ A process is often called a 2-parameter semi– then they are distributed according to a probability group on X in contrast with the 1-parameter semi– vector p(t) = p(t0)P (t0,t) at time t t0. group of an autonomous semi–dynamical system ≥ Thus, the mapping φ defined by φ(t,t0,p0) := since it depends on both the initial time t0 and the p0P (t0,t) is a process on the state space Σd, which actual time t rather than just the elapsed time t t0. − is in fact linear in the initial state component p0 and thus continuous in this variable. Continuity in the 5.2(B). Examples of processes time variables is trivial in the discrete time case and
The solution x(t,t0,x0) of the non-autonomous dif- requires the additional assumption of continuity of ferential equation (40) on Rd defines a continuous the transition matrices in both of their variables time process under the assumption of global ex- in the continuous time case. The two-parameter istence and uniqueness of solutions. Indeed, this semi-group property follows from the Chapman- was the motivating example behind the definition Kolmogorov property. of a process. Similarly, a non-autonomous differ- ence equation generates a discrete time process. Finally, all the examples in Section 4 admit non-autonomous variants which generate processes d d Example 5.2. Let fn : R R be continuous map- in the case of uniqueness of solutions. → pings for n Z. Then, the non-autonomous differ- ∈ ence equation 5.3. An interesting property of processes x = f (x ) (41) n+1 n n A process can be reformulated as an autonomous generates a discrete time process φ which is defined semi–dynamical system, which has some interesting d implications. for all x0 R and n,m Z with n m +1 by ∈ ∈ ≥ The extended phase space will be denoted by φ(m,m,x0) := x0 and := T X, and define a mapping π : T+ X × ×X →X φ(n,m,x0) := fn 1 fm(x0) . by − ◦···◦
In particular, note that x0 φ(n,m,x0) is contin- π(t, (t ,x )) := t + t , φ(t + t ,t ,x ) 7→ 0 0 0 0 0 0 uous as the composition of finitely many continuous mappings. for all (t, (t ,x )) T+ . Note that the variable 0 0 ∈ ×X t in π(t, (t0,x0)) is the time which has elapsed since There are processes which do not involve either starting at time t0, while the actual time is t + t0. differential or difference equations. Theorem 5.4. π is an autonomous semi– Example 5.3. Consider a non-homogeneous dynamical system on . Markov chain on a finite state space 1,...,N X { } with d d probability transition matrices × Proof. It is obvious that π is continuous in its vari- ables and satisfies the initial condition P (t0,t)= pi,j(t0,t) i,j=1,...,d for all t ,t T with t t . Such transition matrices π(0, (t0,x0)) = t0, φ(t0,t0,x0) = (t0,x0) . 0 ∈ ≥ 0 satisfy P (t ,t ) = 1, the d d identity matrix, for 0 0 × all t T. They also satisfy the so-called Chapman- It also satisfies the (1-parameter) semi–group prop- 0 ∈ Kolmogorov property erty
P (t ,s)P (s,t)= P (t ,t) for all t s t . π(s + t, (t ,x )) = π s,π(t, (t ,x )) , s,t T+ , 0 0 0 ≤ ≤ 0 0 0 0 ∀ ∈ Three perspectives 27 since, by the evolution property (ii) of the process, Assuming global existence and uniqueness of solutions forwards in time, the system of differen- π(s + t, (t0,x0)) tial equations (42) generates an autonomous semi– n+m = s + t + t0, φ(s + t + t0,t0,x0) dynamical system π on R which we will write in component form as = s + t + t0, φ s + t + t0,t + t0, φ(t + t0,t0,x0) = π s, (t + t0, φ (t + t0,t0,x0) π(t,p0,x0)= p(t,p0),x(t,p0,x0) , = π s,π(t, (t0,x0)) . with initial condition π(0,p0,x0) = (p0,x0). This finishes the proof of this theorem. There are two important points to observe here. Firstly, the p-component of the system is an in- The autonomous semi–dynamical system π on dependent autonomous system in its own right, the extended state space generated by a process i.e., its solution mapping p = p(t,p0) generates X n φ on the state space X has some unusual properties. an autonomous semi–dynamical system on R and In particular, π has no nonempty ω-limit sets and, amongst other properties satisfies the semi–group indeed, no compact subset of is π-invariant. This property X is a direct consequence of the fact that time is a p(s + t,p )= p(s,p(t,p )) for all s,t 0 . (43) component of the state. 0 0 ≥ This has significant implications and means Secondly, the semi–group property for π on Rn+m, that many concepts for autonomous systems need i.e., to be modified appropriately to be of any use in π(s + t,p0,x0)= π(s,π(t,p0,x0)) , the non-autonomous context. For example, note that a π-invariant subset of has the form can be expanded out componentwise as A X = t T(t0, At0 ), where At0 is a nonempty subset A 0 π(s + t,p0,x0)= p(s + t,p0),x(s + t,p0,x0) of X for∈ each t T. Then the invariance property S 0 ∈ π(t, )= for t T+ is equivalent to = p(s,p(t,p0)),x(s + t,p0,x0) , A A ∈ + using (43), and φ t+t ,t , At = At t for all t T and t T . 0 0 0 + 0 ∈ 0 ∈ π(s,π(t,p0)) This will be used later on to motivate the definition of φ-invariant sets for a process φ. = p(s,p(t,p0)),x(s,p(t,p0),x(t,p0,x0)) .
These are identical for all s,t 0 and all (p0,x0) ≥ ∈ 5.4. Motivation of skew product flows Rn+m. Equating for the second components gives
To motivate the concept of a skew product flow, x(s + t,p0,x0)= x s,p(t,p0),x(t,p0,x0) , (44) first a triangular system of ordinary differential equations is considered in which the uncoupled for all s,t 0, which is a generalization of the semi– ≥ component can be considered as the driving force group property and known as the cocycle property. in the equation for the other component. Given a solution p = p(t,p0) of the p- component of the triangular system (42), the x- 5.4(A). Triangular autonomous differential component becomes a non-autonomous ordinary m equations differential equation in the x variable on R of the form Consider an autonomous system of ordinary differ- dx ential equations of the form = g(p(t,p ),x) where t 0 and x Rn . dt 0 ≥ ∈ dp dx (45) = f(p) , = g(p,x) , (42) Here p = p(t,p ) can be considered as “driving” the dt dt 0 non-autonomous system here, i.e., as being respon- where p Rn and x Rm, i.e., with a triangular sible for the changes in the vector field with the ∈ ∈ structure. passage of time. 28 Balibrea, Caraballo, Kloeden & Valero
The solution x(t) = x(t,p0,x0) with initial intuitive. In contrast, the skew product flow for- value x(0) = x0 (which also depends on the choice mulation is more abstract, but it contains more in- of p0 as a parameter through the driving solution formation about how the system evolves in time, p(t,p0)) then satisfies the following. especially when the driving system is on a compact space P and in the case of a random perturbations, (i) Initial condition: when P is a probability space. x(0,p0,x0)= x0. 5.5. Skew product flows (ii) Cocycle property: x(s + t,p0,x0)= x s,p(t,p0),x(t,p0,x0) . Let (X, dX ) and (P, dP ) be metric spaces. A non- autonomous dynamical system (θ, ϕ) is defined in (iii) Continuity condition: terms of a cocycle mapping ϕ on a state space X (t,p ,x ) x(t,p ,x ) is continuous. 0 0 7→ 0 0 which is driven by an autonomous dynamical sys- tem θ acting on a base or parameter space P and The mapping x : R+ Rn Rm Rm is called × × → the time set T = R or Z. a cocycle mapping. It describes the evolution of the Specifically, the dynamical system θ on P is a solution of the non-autonomous differential equa- group of homeomorphisms (θt)t T under composi- tion (45) with respect to the driving system. Note tion on P with the properties that∈ that the variable t here is the time since starting at the state x0 with the driving system at state p0. (i) θ (p)= p for all p P , 0 ∈ θs(p) X θs t(p) X p X { }× + (ii) θs+t = θs(θt(p)) for all s, t T, { }× { }× ∈ φ(s,p) φ(t,θs(p)) b (iii) the mapping (t,p) θt(p) is continuous, φ(s + t,p,x) 7→ x b φ(s,p)x b and the cocycle mapping ϕ : T+ P X X = φ(t,θs(p), φ(s,p)x) × × → satisfies
φ(s + t,p) (i) ϕ(0,p,x)= x for all (p,x) P X, b ∈ × b θ (p) b (ii) ϕ(s + t,p,x) = ϕ s,θ (p), ϕ(t,p,x) for all p s P t θ (p) s,t T+, (p,x) P X, s+t ∈ ∈ × = θs(θt(p)) (iii) the mapping (t,p,x) ϕ(t,p,x) is continu- 7→ ous. Fig. 1. The cocycle property The mapping π : T+ P X P X defined × × → × As mentioned above, the product system π on by n m R R is an autonomous semi–dynamical sys- π(t, (p,x)) := θt(p), ϕ(t,p,x) (46) × tem and is known as a skew product flow due to forms an autonomous semi–dynamical system on the asymmetrical roles of the two component sys- = P X. tems. This motivates an alternative definition of a X × non-autonomous dynamical system, called the skew Definition 5.5. The autonomous semi–dynamical product flow formalism, where, for various reasons, system π on = P X defined by (46) is called X × the driving system p is usually taken to be a re- the skew product flow associated with the non- versible dynamical system, i.e., forming a group autonomous dynamical system (θ, ϕ). rather than a semi–group. This will happen for example, if the driving differential equation is re- 5.5(A). Examples of skew product flows stricted to a compact invariant subset of Rn. The process formulation of a non-autonomous Non-autonomous difference equations and differen- dynamical system defined by the solution mapping tial equations provide a rich source of examples for of a non-autonomous differential equation is quite skew product flows. Three perspectives 29
Example 5.6. The solution mapping x(t) = known about how the different mappings fn vary x(t,t ,x ) of a non-autonomous differential equa- with n Z, it is often possible to have a compact 0 0 ∈ tion (40) with initial value x(t0,t0,x0)= x0 at time base space. t0 defines a process. Theorem 5.4 shows that such a process can be reformulated as a skew product Example 5.8. Suppose that the mappings fn in the flow with the cocycle mapping ϕ on the state space non-autonomous difference equation (41) are chosen d X = R defined by ϕ(t,t0,x0) := x(t,t0,x0) and in some way from a finite number of continuous d d the driving system θ on the base space P = R de- mappings Ri : R R for i 1,...,r . Then → ∈ { } fined by the shift operators θt(t ) := t t . The the difference equation has the form 0 − 0 disadvantages of this representation were discussed x = R (x ) , (47) above. n+1 in n where the in 1,...,r for all n N. It gen- ∈ { } ∈ The advantages of the skew product flow for- erates a discrete time skew product flow over the mulation reveals itself, for instance, when the gen- parameter set P = 1,...,r Z of bi-infinite se- { } erating non-autonomous differential equation is pe- quences p = (in)n Z in 1,...,r with respect to ∈ { } riodic or almost periodic in time, because the base the group of left shift operators (θm)m N, where ∈ space is then compact. θm (in)n Z = (in+m)n Z. The cocycle mapping ∈ ∈ ϕ(n, , ) is defined by Example 5.7. The skew product formulation of a · · ϕ(0,p,x) := x, ϕ(n,p,x) := Ri − Ri (x) non-autonomous differential equation (40) is based n 1 ◦···◦ 0 d on the fact that whenever x(t) is a solution of the for all n N, x R and p = (in)n N P . The ∈ ∈ Z ∈ ∈ differential equation, then the time-shifted solution parameter space P = 1,...,r here is a compact { } xτ (t) := x(τ + t) (for some fixed τ) satisfies the metric space with the metric non-autonomous differential equation ∞ n d(p,p′)= (r + 1)−| | in i′ , x˙ τ (t)= fτ (t,xτ (t)) := f(τ + t,x(τ + t)) . − n n= X−∞ Consider the set of functions fτ ( , ) := f(τ + , ) : · · · · and the mappings p θn(p) and (p,x) ϕ(n,p,x) τ R . Its closure in an appropriate topol- 7→ + 7→ ∈ F are continuous for each n Z . To see this, ogy is called the hull of the vector field in the ∈ note that d(p,p ) δ < 1 requires ij = i for ′ ≤ j′ non-autonomous differential equation (40). See j = 0, 1,..., N(δ). Then take δ small enough ± ± [Sell, 1971] for examples and typical topologies. For corresponding to a given ε> 0 and fixed n. example, is a compact metric space for periodic F or almost periodic differential equations. More generally, the difference equation may in- Introduce a group of shift operators θτ : volve a parameter q Q, which varies from iterate F 7→ Fd by θτ (f) := fτ for each τ R, define = R ∈ ∈ X F × to iterate, by choice or randomly, and write ϕ(t,f,x0) for the solution of (40) with ini- tial value x0 at initial time t0 = 0. Finally, define π : xn+1 = f(xn,qn) . (48) + R by π(t,x ,f) := θt(f), ϕ(t,f,x ) . ×X →X 0 0 Then, π = (θ, ϕ) is a continuous-time skew Example 5.9. Consider the parametrically depen- product flow on the state space . To see this, ob- dent difference equation (48) with the continuous X mapping f : R [ 1, 1] R, given by serve that the second component of the semi–group × − → identity π(t+s,f,x )= π(t,π(s,f,x ) expands out 0 0 f(x,q)= fq(x) := νx + q , as the cocycle property where ν [0, 1) and q [ 1, 1]. Let P = [ 1, 1]Z ∈ ∈ − − ϕ(t + s,f,x0)= ϕ t,θs(f), ϕ(s,f,x0) . be the space of bi-infinite sequences p = (qn)n Z taking values in [ 1, 1], which is a compact metric∈ − Non-autonomous difference equations (41) gen- space with the metric erate discrete time skew product flows, the simplest coming from discrete time processes via Theorem ∞ n d(p,p′)= 2−| | qn q′ , − n 5.4 and have Z as their base space. When more is n= X−∞
30 Balibrea, Caraballo, Kloeden & Valero and let (θn)n Z be the group of the left shift op- Then for any t0 T and a0 At0 , there exists ∈ ∈ ∈ erators on this sequence space (cf. Example 5.8). an entire solution ξ through a0 which is contained Finally, define the mappings ϕ(n, , ) by in , i.e., with ξ(t ) = a and ξ(t) At for all · · A 0 0 ∈ t T. ϕ(0,p,x ) := x, ϕ(n,p,x) := fq − fq (x) ∈ 0 n 1 ◦···◦ 0 When the subsets in a φ-invariant family are for all n N, x R and p = ( qn)n N P . Specifi- ∈ ∈ ∈ ∈ cally, compact, it follows from the continuity of a con- n 1 tinuous time process that the set-valued mapping n − n 1 j ϕ(n,p,x)= ν x + ν q t At is continuous in t R with respect to the − − j 7→ ∈ j=0 Hausdorff metric h, since X for all n N. ∈ h(At, At0 )= h φ(t,t0, At0 ), φ(t0,t0, At0 ) 0 The mappings p θn(p) and (p,x) 7→ 7→ → ϕ(n,p,x) are obviously continuous here for each as t t by the continuity of the processφ in its n N. Thus, (θ, ϕ) is a discrete time skew product → 0 ∈ first variable. flow on R with the compact base space P . Similar definitions hold for positive and nega- tive invariant families of sets. 5.5(B). Entire solutions and invariant sets The definition of an entire solution of a non- Definition 5.13. Let φ be a process on a metric autonomous dynamical system is an obvious gen- space (X, d). A family A = (At)t T of nonempty ∈ eralization of the autonomous case. subsets of X is said to be positive invariant with respect to φ, or φ-positive invariant, if Definition 5.10. An entire solution of a process φ on a metric space (X, d) with time set T is a φ (t,t0, At0 ) At for all t t0 , ⊂ ≥ mapping ξ : T X such that → and negative invariant with respect to φ, or φ- ξ(t)= φ(t,t0,ξ(t0)) for all t,t0 T with t t0 . negative invariant, if this holds with instead of ∈ ≥ ⊃ . The discussion following Theorem 5.4, in which ⊂ a process φ on X was formulated as a skew product Analogous definitions hold for skew product flow on the extended state space R X, suggests flows. × that it is more appropriate to consider the invari- ance of a family of time-dependent subsets rather Definition 5.14. An entire solution of a skew than of a single set. This motivates the following product flow (θ, ϕ) on a metric phase space (X, d) definition. and a base set P with time set T is a mapping ξ : P X such that Definition 5.11. Let φ be a process on a metric → space (X, d). A family = (At)t T of nonempty ξ(θt(p)) = ϕ(t s,θs(p),ξ(θs(p)) A ∈ − subsets of X is said to be invariant with respect to φ, or φ-invariant, if for all p P and s,t T with s t . ∈ ∈ ≤
φ (t,t0, At0 )= At for all t t0 . (49) Definition 5.15. Let (θ, ϕ) be a skew product flow ≥ on a metric phase space (X, d) and a base set P . A A simple example of a φ-invariant family = family A = (Ap)p P of nonempty sets of X is said to A ∈ (At)t T is given by an entire solution of φ, i.e., hav- ∈ be invariant with respect to (θ, ϕ), or ϕ-invariant, ing the singleton subsets At = ξ(t) for each t T. { } ∈ if In fact, φ-invariant families consist of entire solu- tions. ϕ(t,p,Ap)= A for all t 0 and p P . θt(p) ≥ ∈
Lemma 5.12. Let = (At)t T be a nonempty φ- For positive and negative invariant families, replace A ∈ invariant family of subsets of X of a process φ. = here by or , respectively. ⊂ ⊃ Three perspectives 31
The compact set-valued mapping t A 5.6(A). Non-autonomous sets 7→ θt(p) induced by a ϕ-invariant family (Ap)p P of compact ∈ Let φ be a process on a metric space (X, d), and subsets is continuous in t R with respect to the ∈ consider a family ˜ = (Mt)t T of subsets of X. Hausdorff metric for each fixed p P . M ∈ ∈ For a more compact and elegant formulation, such families ˜ will henceforth be viewed equivalently 5.6. Attractors for non-autonomous and M as subsets of the extended phase space T X, and random dynamical systems × the translation is as follows. The family ˜ induces M Simple generalizations of concepts for autonomous a subset T X, defined by M⊂ × dynamical systems to non-autonomous dynamical := (t,x) : x Mt systems are not always adequate or appropriate. It M ∈ was previously seen that, for non-autonomous dy- and, conversely, a subset of the extended phase M namical systems, it is often too restrictive to con- space T X leads to a family ˜ = (Mt)t T of × M ∈ sider the invariance of just a single set and that subsets of X with instead a family of subsets is more appropriate. A Mt := x X : (t,x) for all t T . similar situation also applies to attractors, which ∈ ∈ M ∈ are the most important examples of invariant sets. The advantage of the new formulation is that is M Attractors of autonomous dynamical systems a set, which makes the direct use of set-valued op- are ω-limit sets, which are invariant sets. Since the erations possible, and the notation becomes easier solution of a process ϕ depends both on initial time to read. Such sets are called non-autonomous M t0 and initial value x0, ω-limit set for a process will sets in the following. It will become clear soon that also depend on both of these two parameters, specif- all interesting objects in the non-autonomous con- ically text are non-autonomous sets, i.e., subsets of the extended phase space, whereas the main interest ω(t0,x0)= x X : lim ϕ(tn,t0,x0)= x ∈ n focusses on subsets of the phase space in an au- n →∞ tonomous setting. The precise definition of a non- for some sequence tn . → ∞} autonomous set is: As in the autonomous case, one can show that ω(t0,x0) is a nonempty compact set when, for ex- Definition 5.16. Let φ be a process on a metric ample, the forward trajectory t t0 ϕ(t,t0,x0) is space (X, d). A subset of the extended phase ∪ ≥ { } M precompact. However, unlike its autonomous coun- space T X is called a non-autonomous set and for × terpart, a non-autonomous ω-limit set ω(t0,x0) may each t T the set not be invariant for the process. ∈ Mt := x X : (t,x) As an example, consider the non-autonomous { ∈ ∈ M} scalar differential equation is called the t-fiber of . A non-autonomous set M t is said to be invariant if φ(t,t0, Mt0 ) = Mt for x˙ = x + e− , M − all t t0. which can be solved with the variation of constants ≥ formula to give the explicit solution An analogous notion of a non-autonomous set
(t t0) t is also used for skew product flows. x(t,t ,x )= e− − x + (t t ) e− . 0 0 0 − 0 This implies that Definition 5.17. Let (θ, ϕ) be a skew product flow on a base set P and a metric phase space (X, d). A lim x(t,t0,x0) = 0 for all (t0,x0) R R , t ∈ × subset of the extended phase space P X is →∞ M × called a non-autonomous set and for each p P so the non-autonomous ω-limit set is given by ∈ the set ω(t0,x0)= 0 for all (t0,x0) R R . Mp := x X : (p,x) { } ∈ × { ∈ ∈ M} However, x(t,t , 0) = (t t ) e t = 0 for all t>t , is called p-fiber of . A non-autonomous set is 0 − 0 − 6 0 M M i.e., the ω-limit set here is not invariant in the sense said to be invariant if ϕ(t,p,Mp) = Mθt(p) for all of autonomous systems. t 0 and p P . ≥ ∈ 32 Balibrea, Caraballo, Kloeden & Valero x In general, a non-autonomous set is said to x( ,t ,x ) M · 0 0 have a topological property (such as compactness ϕ¯ or closedness) if each fiber of has this property. M 5.6(B). Attractors of processes t There are basically two ways to define attraction 0 of a compact and invariant non-autonomous set t A for a process φ on a metric space (X, d) with time set T. The first, and perhaps more obvious, cor- responds to the attraction in Lyapunov asymptotic stability, is called forward attraction and involves Fig. 2. Forward attraction a moving target, while the latter, called pullback x0 attraction, involves a fixed target set with progres- x( ,t0,x0) sively earlier starting time. In general, these two · ϕ¯ types of attraction are independent concepts, while for the autonomous case, they are equivalent.
Definition 5.18. Let φ be a process. A nonempty, t compact and invariant non-autonomous set is t t t A 0 0 0 said to be
(i) forward attracting if Fig. 3. Pullback Attraction lim dist φ(t,t0,x0), At = 0 t →∞ for all x0 X and t0 T , compact and that they attract bounded subsets D ∈ ∈ of initial values in X (rather than just individual (ii) and pullback attracting if points), in the sense that
lim dist φ(t,t0,x0), At = 0 t0 dist φ(t,t0, D), At 0, →−∞ → for all x0 X and t T . either as t with t fixed (forward case), or as ∈ ∈ →∞ 0 t with t fixed (pullback case). Moreover, if the forward attraction in (i) is uniform 0 → −∞ with respect to t T, or equivalently, if the pull- 0 ∈ back attraction in (ii) is uniform with respect to Definition 5.19. Let φ be a process. A nonempty t T, then is called uniformly attracting. and invariant non-autonomous set is called ∈ A A (i) a forward attractor if it forward attracts Figures 5.6(B) and 5.6(B) illustrate forward bounded subsets of X, and pullback attraction, respectively, of a non- autonomous set with singleton sets as fibers At = (ii) a pullback attractor if it pullback attracts ρ(t) , i.e., an entire solution of the process. { } bounded subsets of X, and In an autonomous system, the solutions depend only on the elapsed time t t . Moreover, t t (iii) a uniform attractor if it uniformly attracts − 0 − 0 → when either t with t fixed or as t bounded subsets of X. ∞ →∞ 0 0 → −∞ with t fixed, so pullback and forward convergence are equivalent for an autonomous system. Forward and pullback attractors are indepen- Two types of non-autonomous attractors for dent concepts and one can exist without the other. processes are possible, depending which of the above types of attraction is used. It is required Example 5.20. The non-autonomous set T 0 , × { } that the component subsets of such attractors are i.e., the zero solution, is a forward attractor but Three perspectives 33 not a pullback attractor of the system and is independent of x0, i.e.,
x˙ = 2tx (50) lim x(t,t0,x0) ρ(t) = 0 . − t0 − 2 2 →−∞ (t t0) with the general solution x(t,t0,x0) = x0e− − , Hence, the non-autonomous set having the sin- and a pullback attractor but not a forward attractor A gleton fibers At := ρ(t) is pullback attracting for { } of the system the solution process. x˙ = 2tx (51) Moreover, it is easily shown that ρ(t) is a so- t2 t2 with the general solution x(t,t0,x0)= x0e − 0 . lution of the non-autonomous differential equation (52) and since all solutions converge to each other This example demonstrates that forward at- forward in time, the forward convergence traction can be seen as an attraction concept for lim x(t,t0,x0) ρ(t) = 0 the future of the system, since the coefficient 2t t − − →∞ of (50) is negative for t > 0. On the other hand, also holds. pullback attraction means basically attraction for the past of the system, see the negativity of 2t for Remark 5.22. The non-autonomous set in Exam- t < 0 in (51). The concept of a uniform attrac- A ple 5.21 is both a pullback and forward attractor of tor, however, is concerned with attractivity for the the non-autonomous differential equation (52). It entire time. is not difficult to find other forward attractors of (52) which are not pullback attractors. Example 5.21. Consider the non-autonomous This fact demonstrates that forward attractors scalar ordinary differential equation can be nonunique, and this is quite typical for for- x˙ = x +2sin t . (52) ward attractors. For pullback attractors, however, − the following uniqueness result can be proved. If x1(t) and x2(t) are any two solutions, then their difference z(t)= x (t) x (t) satisfies the homoge- 1 − 2 Proposition 5.23. Suppose that a process φ has neous linear differential equation two pullback attractors and ¯ such that both ¯ A A ¯ z˙ = z t 0 At and t 0 At are bounded. Then = . − ≤ ≤ A A (t t0) S S with the explicit solution z(t)= z(t0)e− − , so Proof. The boundedness of t 0 At implies for all t T that ≤ (t t0) x1(t) x2(t) = x1(t0) x2(t0) e− − 0 ∈ S − − → ¯ ¯ dist At, At = lim dist φ(t,t0, At0 ), At t0 as t , from which it follows that all solutions →−∞ → ∞ ¯ converge to each other in time. What do they con- lim dist φ(t,t0, τ 0 Aτ ), At = 0 . ≤ t0 verge to? →−∞ ≤ S The explicit solution of the non-autonomous Analogously, one shows that dist A¯t, At = 0, differential equation (52) with initial value x(t )= which finishes the proof, since both the sets A and 0 t x0 is A¯t are compact. t (t t0) t s x(t,t0,x0)= x0e− − + 2e− e sin s ds Exercise 5.24. A φ-invariant family of sets such t Z 0 as a pullback attractor A = (At)t T consists of en- (t t0) ∈ = x0 (sin t0 cos t0) e− − tire solutions, see Lemma 5.12. Give an example of − − a process φ which has entire solutions that are not + (sin t cos t) , − contained in the pullback attractor. from which it is clear that the forward limit limt x(t,t0,x0) does not exist. On the other →∞ 5.7. Attractors of skew product flows hand, the pullback limit does exist for all t and x0, i.e., Let (θ, ϕ) be a skew product flow on a base space P and a state space X with time set T = R or Z, lim x(t,t0,x0) = sin t cos t =: ρ(t) , t0 where (P, dP ) and (X, dX ) are metric spaces. →−∞ − 34 Balibrea, Caraballo, Kloeden & Valero
Then π = (θ, ϕ) is an autonomous semi– As for processes, the concepts of forward and dynamical system on the extended state space := pullback attractors for skew products are gener- X P X. A global attractor for an autonomous semi– ally independent of each other, and one can exist × dynamical system π is, specifically, a nonempty without the other existing. If the above limit is compact subset of which is π-invariant, i.e., replaced by limt supp P distX ( , ), then the at- A X →∞ ∈ · · which satisfies π(t, ) = for all t T+ is called tractors are called uniform pullback and uniform A A ∈ a global attractor of π if forward attractors, respectively. If either of the lim- its is uniform in this sense, then so is the other and lim dist (π(t, D), ) = 0 the attractor is both a uniform pullback and a uni- t X A →∞ form forward attractor, so will be called simply a for every nonempty bounded subset D of . Sup- uniform attractor. X pose that P is compact and θ-invariant. Then the The relationship between these different global attractor of π has the form kinds of non-autonomous attractors can be seen A in [Cheban et al., 2002] and in the monograph = (p,x) : x Ap , [Kloeden & Rasmussen, 2010]. A ∈ p P [∈ Example 5.26. Reconsider the non-autonomous where Ap is a nonempty compact subset of X for scalar ordinary differential equation (52), now each p P , and the π-invariance property π(t, )= writing p(t), a periodic function with period 2π, ∈ + A for t T is equivalent to the ϕ-invariance prop- instead of sin t, A ∈ + erty ϕ(t,p,Ap)= A for all t T and p P . θt(p) ∈ ∈ A global attractor of the autonomous system π x˙ = x + 2p(t) , is a possible candidate for an attractor for the non- − autonomous dynamical system described by the with the initial condition x(0) = x0. In the spirit of skew product flow. A disadvantage of this defini- skew product flows as introduced in Section 5.4, the tion is that the extended state space includes general solution of this equation depends on both X the base space P as a component, which often does p and x0. The initial value problem has thus the not have the same physical significance as the state explicit solution x(t)= x(t,p,x0) given by space X. t t t s Other types of attractors consisting of a family x(t)= x0e− + 2e− e p(s)ds . of nonempty compact subsets of the state space X Z0 have also been proposed for a skew product (θ, ϕ). We introduce shift operators on the space P = These are analogues of the forward and pullback p(t + ) : 0 t 2π defined by θt(p( )) = p(t + ) · ≤ ≤ · · attractors of a process. and consider the solution corresponding to the driv- ing term θ τ (p( )) = p( τ + ) at time τ, i.e., Definition 5.25. Let (θ, ϕ) be a skew product − · − · τ flow. A nonempty, compact and invariant non- τ τ s x τ,θ τ (p( )),x0 = x0e− + 2e− e θ τ (p(s)) ds autonomous set is called a pullback attractor of − · − A 0 (θ, ϕ) if the pullback convergence Z τ τ τ s = x0e− + 2e− e p(s τ)ds 0 − lim distX (ϕ(t,θ t(p), D), Ap) = 0 τ Z t − τ s τ →∞ = x0e− + 2 e − p(s τ)ds 0 − holds for every nonempty bounded subset D of X Z 0 τ t and p P , and is called a forward attractor if the = x0e− + 2 e p(t)dt , ∈ τ forward convergence Z− where the substitution t := s τ has been used. lim distX ϕ(t,p,D), Aθ (p) = 0 − t t The pullback limit as τ gives →∞ →∞ 0 holds for every nonempty bounded subset D of X t lim x τ,θ τ (p( )),x0 = α(p( )) := 2 e p(t)dt and p P . τ − · · ∈ →∞ Z −∞ Three perspectives 35 for x in an arbitrary bounded subset D. It con- The pullback attractor thus consists of singleton 0 A sists of singleton fibers Ap = α(p( )) , p P , and fibers Ap = α(p) for p = (qn)n N P . Since the · ∈ { } ∈ ∈ corresponds to the entire solution pullback convergence here is in fact uniform in p ∈ P , the pullback attractor is also a uniform forward ρ(t) := sin t cos t − attractor, and hence a uniform attractor. Moreover, is also the global attractor of the autonomous in the process version of this differential equation in A semi–dynamical system π = (θ, ϕ) on the extended Example 5.21, i.e., ρ(t)= α θt(p( )) for all t R. · ∈ state space P R. The pullback attraction in this example is uni- × form, and the pullback attractor is also a forward attractor, and hence a uniform attractor. Moreover, 5.7(A). Existence of pullback attractors the autonomous semi–dynamical system π = (θ, ϕ) In order to prove the existence of the attractor (in on the extended state space P R has a global at- × both the autonomous and non-autonomous cases) tractor given by the simplest, and therefore the strongest, assump- tion is the compactness of the solution operator as- = p( ), α(p( )) . A · · sociated with the system, which is usually available p( ) P [· ∈ for parabolic systems in bounded domain. How- ever, this kind of compactness does not hold in Remark 5.27. The analysis of this system as a skew general for parabolic equations in unbounded do- product flow is somewhat more complicated and mains and hyperbolic equations on either bounded less transparent than its analysis as a process in or unbounded domain. Instead we often have some Example 5.21. This is typical and is why the pro- kind of asymptotic compactness. In this situa- cess formulation will often be used in subsequent tion, there are several approaches to prove the ex- examples (whenever it is possible). istence of the global (or non-autonomous) attrac- tor. Roughly speaking, the first one ensures the Example 5.28. The difference equation in Exam- existence of the global (resp. non-autonomous) at- ple 5.9 generates a discrete time skew product flow tractor whenever a compact attracting set (resp. with cocycle mapping a family of compact attracting sets) exists. The n 1 second method consists in decomposing the solu- n − n 1 j ϕ(n,p,x)= ν x + ν − − qj for all n N , tion operator (resp. the cocycle or two-parameter ∈ j X=0 semi-group) into two parts: a compact part and (53) another one which decays to zero as time goes to on the state space X = R. The base space P = infinity. However, as it is not always easy to find [ 1, 1]Z is the space of bi-infinite sequences p = − this decomposition, one can use a third approach (qn)n N taking values in [ 1, 1] and θ is the left which is based on the use of the energy equations ∈ − shift operator on this sequence space. which are in direct connection with the concept of Replacing p by θ n(p) in (53) implies − asymptotic compactness. This third method has n 1 been used fo example in [Moise et al., 1998] and n − n 1 j [Lukaszewicz & Sadowski, 2004] to extend to the ϕ n,θ n(p),x = ν x + ν − − q n+j n N , − − ∀ ∈ j=0 non-autonomous situation the corresponding one in X the autonomous framework (see [Rosa, 1998]), but which can be re-indexed as related to uniform asymptotic compactness. Our 1 aim here is to consider the case without uniformity n − k 1 ϕ n,θ n(p),x = ν x + ν− − qk, n N . properties and show how the pullback theory works − ∀ ∈ k= n in this situation. In this sense, it is worth men- X− tioning that the compact case has been treated in Taking pullback convergence gives [Flandoli & Schmalfuß, 1996] in the random case,
1 so our results can be considered as natural com- − k 1 plements (as it happens in the deterministic au- lim ϕ n,θ n(p),x = α(p) := ν− − qk . n − →∞ k= tonomous case). X−∞ 36 Balibrea, Caraballo, Kloeden & Valero
There are generalizations of the theorems en- pullback construction used in (54) gives an in- suring existence of an attractor for autonomous sys- variant set and can be regarded as a proper tems to pullback attractors for processes and skew version of a non-autonomous ω-limit set (see product flows. These are also based on the sup- [Crauel & Flandoli, 1994] for details). posed existence of an absorbing set, which is now absorbing in the pullback sense. Example 5.32. Consider a non-autonomous dy- namical system in Rd given by Definition 5.29. Let φ be a process on a metric x˙ = f(t,x) , (55) space (X, d). A nonempty compact subset B of X is called pullback absorbing if for each t T where f is continuously differentiable and satisfies ∈ and every bounded subset D of X, there exists a the uniform dissipative condition T = T (t, D) > 0 such that x,f(t,x) K L x 2, x Rd,t R (56) ≤ − k k ∀ ∈ ∈ φ(t,t , D) B for all t T with t t T . 0 ⊂ 0 ∈ 0 ≤ − with positive constants K and L. These assump- tions ensure that the differential equation (55) gen- Definition 5.30. Let (θ, φ) be a skew product flow erates a process. on a metric space (X, d). A nonempty compact Moreover, any solution x(t) of (55) satisfies subset B of X is called pullback absorbing if for each p P and every bounded subset D of X, d 2 ∈ x(t) = 2 x(t), x˙(t) there exists a T = T (p, D) > 0 such that dtk k = 2 x(t),f(t,x (t)) K L x(t) 2 , ≤ − k k ϕ t,θ t(p), D B for all t T . − ⊂ ≥ from which, on integrating, it follows that The existence theorems will be presented here 2 2 L(t t0) K L(t t0) x(t) x(t0) e− − + 1 e− − . under basic but restricted assumptions, which can k k ≤ k k L − then be relaxed and generalized. Suppose that for a bounded subset D of Rd with D := supd D d > 1, we have x(t0) D, and k k ∈ k k ∈ 5.7(B). Existence of pullback attractors for pro- define 1 cesses T := ln (L D ) . L k k The following theorem is a simple generalization of Then the corresponding one for attractors of autonomous 1 K K + 1 x(t) 2 + = semi–dynamical systems. k k ≤ L L L for x(t0) D and t0 t T . Thus, the closed ball Theorem 5.31. Let φ be a process on a complete ∈ ≤ − d 2 Bd 0, (K + 1)/L := x R : x (K+1)/L metric space X with a compact pullback absorbing ∈ k k ≤ set B such that is pullback p absorbing and positively invariant. From Theorem 5.31, it follows that the process φ(t,t ,B) B for all t t . 0 ⊂ ≥ 0 generated by the differential equation (55) has a pullback attractor in Rd with components subsets Then there exists a pullback attractor with fibers A At B := Bd 0, (K + 1)/L . in B uniquely determined by ⊂ p 5.7(C). Existence of pullback attractors for At = φ(t,t ,B) for all t T . (54) 0 ∈ skew product flows τ 0 t0 τ \≥ [≤− The counterpart of Theorem 5.31 for skew prod- The formula (54) is a kind of a non-autonomous uct flows is the first part of the following theorem. ω-limit set of the set B. As seen in the in- The second part provides some information about troduction of this chapter, a naive definition of a form of forwards convergence of the cocycle map- a non-autonomous ω-limit set leads to a set, ping, which is different from that in the definition which is not positively invariant. However, the of a forward attractor. Three perspectives 37
Theorem 5.33. Let (θ, ϕ) be skew product flow on for all p P and x ,x Rd with some constant ∈ 1 2 ∈ a complete metric space X with a compact pullback L > 0. Then f satisfies the uniform dissipative absorbing set B such that condition (61) with constants
ϕ(t,p,B) B for all t 0 and p P . (57) 2 L ⊂ ≥ ∈ K′ = sup f(0) and L′ = , L p P k k 2 Then there exists a unique pullback attractor with ∈ A fibers in B uniquely determined by and the closed ball B′ := Bd 0, (K′ + 1)/L′ is pullback absorbing and positively invariant. Thus, Ap = ϕ(t,θ t(p),B) for all p P . (58) p − ∈ τ 0 t τ the skew product flow has a pullback attractor with \≥ [≥ component subsets Ap in this ball. If, in addition, (P, dP ) is a compact metric space, In fact, the fibers of the pullback attractor are then singleton sets. The proof uses the fact that due to the uniform one-sided dissipative Lipschitz condi- lim sup dist ϕ(t,p,D), A(P ) = 0 (59) t →∞ p P tion (62), the system satisfies ∈ d Lt for any bounded subset D of R , where A(P ) := x (t) x (t) e− x , x , (63) k 1 − 2 k≤ k 0 1 − 0 2k Ap B. p P ⊂ ∈ for any pair of solutions with the same initial value S 5.7(D). A continuous time example p P of the driving system. This follows from ∈ Consider a non-autonomous dynamical system in d 2 d x (t) x (t) R given by dt 1 − 2 x˙ = f(p,x) (60) d = x1(t) x2 (t),x1(t) x2(t) with the driving system θ on a compact metric dt − − = 2 x (t) x (t), x˙ (t) x˙ (t) space (P, dP ). Suppose that f is regular enough to 1 − 2 1 − 2 ensure that the differential equation (60) generates = 2 x (t) x (t),f θtp,x (t) f θtp,x (t) 1 − 2 1 − 2 a skew product flow. 2 2L x1(t) x2(t ) , In addition, suppose that f satisfies the uni- ≤− − form dissipative condition which is integrated to give 2 d x,f(p,x) K L x , p P,x R (61) 2 2Lt 2 ≤ − k k ∀ ∈ ∈ x (t) x (t) e− x , x , . k 1 − 2 k ≤ k 0 1 − 0 2k with positive constants K and L. Then, similarly to Example 5.32, a solution x(t) satisfies the differ- Taking square roots yields the desired result. ential inequality Theorem 5.34. The pullback attractor of the d A x(t) 2 K L x(t) 2 , skew product flow (θ, ϕ) generated by the differential dtk k ≤ − k k equation (60) consists of singleton fibers Ap = ap { } from which it follows that the closed ball for each p P when the vector field f satisfies ∈ the uniform one-sided dissipative Lipschitz condi- B := Bd 0, (K + 1)/L tion (62). Moreover, t a , t R, is an entire 7→ θt(p) ∈ p solution of (60) for each p P . is pullback absorbing and positively invariant. ∈ From Theorem 5.33, it follows that the skew prod- d Proof. Since Ap B for all p P , it follows that uct flow has a pullback attractor in R with com- ⊂ ′ ∈ Ap := maxa Ap a R := (K′ + 1)/L′ for each ponents subsets Ap B, p P . k k ∈ k k ≤ ⊂ ∈ p P . Now consider a fixed p P , and suppose Suppose instead that the vector field f satisfies ∈ ∈ that there exists an ε > 0 and points a , a Ap the uniform one-sided dissipative Lipschitz condi- 0 1 2 ∈ such that a1 a2 = ε0. Moreover, choose T > 0 tions k −LT k such that 2Re− = ε0. The ϕ-invariance of the 2 x1 x2,f(p,x1) f(p,x2) L x1 x2 (62) pullback attractor gives ϕ T,θ T (p), Aθ− (p) = − − ≤− k − k − T 38 Balibrea, Caraballo, Kloeden & Valero
Ap, which means that there exist a , a Aθ− p The definitions of pullback convergence and 1′ 2′ ∈ T such that pullback attractor need to be extended accordingly.
ϕ T,θ T (p), a1′ = a1 and ϕ T,θ T (p), a2′ = a2 . − − Definition 5.37. Let (θ, ϕ) be a skew product flow Then, from the inequality in (63), it follows that on P X. A nonempty, compact and invariant non- × autonomous set is called pullback attractor with 0 < ε0 = a1 a2 A k − k respect to an attraction universe D if the pullback = ϕ T,θ T (p), a1′ ϕ T,θ T (p), a2′ − − − convergence 1 LT LT e− a1′ a2′ Re− = ε0 , ≤ k − k≤ 2 lim dist ϕ t,θ t(p), Dθ−t(p) , Ap = 0 t − →∞ which is not possible. Hence, a1 = a2. holds for all p P and D. Finally, from the ϕ-invariance of the pullback ∈ D∈ attractor, ϕ(t,p,ap)= a for all t R and p P , θt(p) ∈ ∈ so the singleton sets forming the pullback attractor Exercise 5.38. Show that a pullback attractor is define an entire solution of the system. It follows unique within a given attraction universe D. from inequality (63) that this entire solution also forward attracts all other solutions with the same The pullback absorbing property now depends on the attraction universe under consideration. initial value of the driving system, so the pullback D attractor is also a forward attractor. (There will Definition 5.39. Let D be an attraction universe be more than one such entire solution when P is of a skew product flow (θ, ϕ) on P X. A nonempty not a minimal subset for the autonomous dynamical × and compact non-autonomous set D is called system θ). B ∈ pullback absorbing with respect to D if for each D∈ The above theorem generalizes the following D and p P , there exists a T = T (p, D) > 0 such ∈ autonomous result in [Stuart & Humphries, 1996]. that
ϕ t,θ t(p), Dθ− (p) Bp for all t T . Corollary 5.35. An autonomous differential − t ⊂ ≥ equation with a vector field f which satisfies a one-sided dissipative Lipschitz condition such as Theorem 5.33 on the existence of a pullback (62) (i.e., without the p variable) has a unique attractor assuming that of a pullback absorbing set globally asymptotically stable equilibrium point. generalizes to attraction universes and pullback ab- sorbing families. 5.7(E). Pullback attractors for absorbing fami- lies and attraction universes Theorem 5.40. Let (θ, ϕ) be a skew product flow on P X, and suppose that the compact non- × To take into account non-uniformities that are autonomous set is pullback absorbing with respect B ubiquitous in non-autonomous dynamical systems, to an attraction universe D. Then (θ, ϕ) has a pull- greater generality can be attained in the defini- back attractor with respect to D, where the fibers tion of a pullback attractor by considering arbitrary A Ap are defined for each p P by non-autonomous sets and instead just a sin- ∈ B D gle compact absorbing set B and single attracted Ap = φ(t,θ t(p),Bθ− (p)). (64) − t bounded set D. This allows local as well as global s>0 t>s attraction to be handled at the same time. The \ [ skew product flows (θ, ϕ) in this subsection are on a The proof is a direct modification of that of metric state space (X, dX ) with a metric base space Theorem 5.33. Similar theorems can be found (P, dP ) and a time set T. in [Schmalfuß, 1992, Flandoli & Schmalfuß, 1996, Kloeden & Schmalfuß, 1996, Kloeden, 1997], see Definition 5.36. An attraction universe D of a also [Crauel et al., 1995, Cheban & Fakhih, 1994]. skew product flow (θ, ϕ) is a collection of bounded non-autonomous sets , which is closed in the sense Remark 5.41. The assumption that the absorbing D that if ( for some D, then D. sets in Theorems 5.33 and 5.40 are compact is no ∅ D′ ⊆ D D∈ D′ ∈ Three perspectives 39 restriction in a state space such as Rd, which is is a flow on Ω, i.e., with θ : R Ω Ω satisfying × → locally compact and thus closed and bounded sub- sets are equivalently compact. This is not true for θ0 = idΩ, θt+τ = θt θτ =: θtθτ for t, τ R, ◦ ∈ a general state space. In particular, for infinite di- mensional spaces, compact subsets are “thin” and is called a non-autonomous perturbation. As an ex- it is much easier to determine an absorbing prop- ample which describes typical non-autonomous per- erty for a closed and bounded subset, such as a unit turbations we consider Ω = R and θtτ = t + τ for τ = ω Ω, t R. ball, rather than a compact subset. Counterparts ∈ ∈ of Theorems 5.33 and 5.40 then hold, if the cocycle Let := (Ω, , P) be a probability space. On P F mapping is assumed to be compact, i.e., the map- this probability space we consider a measurable ping ϕ(t,p, ) : X X maps bounded subsets into non-autonomous flow θ : · → precompact subsets for all t> 0 and p P , or more ∈ generally, asymptotically compact. These general- θ : (R Ω, (R) ) (Ω, ). izations will be considered in the next section on in- × B ⊗ F → F finite dimensional dynamical systems, i.e., with an infinite dimensional state space X, and dealing with In addition, P is supposed to be ergodic with possibly multi-valued non-autonomous and random respect to θ, which means that every θt-invariant set has measure zero or one, t R. Hence P is invariant dynamical systems. It is also remarkable that, when ∈ with respect to θt. The quadruple (Ω, , P,θ) which the problem contains some randomness or stochas- F tic features, the parameter space possesses a mea- is the model for a noise is called a metric dynamical surable structure (i.e., is a probability space), and, system. consequently, the measurability must appear in the If we replace in the definition of a metric dy- namical system the probability space by its com- definitions. This will be highlighted in the next P pletion c := (Ω, ¯, P¯) the above measurability subsection. In addition, it is also worth mentioning P F that not all stochastic differential equation generate property is not true in general, see Appendix A [Arnold, 1998; Appendix A]. But for fixed t R a random dynamical system (see [Arnold, 1998] for ∈ a more detailed exposition on this point). Specif- we have that the mapping ically, in the case of stochastic partial differential θt : (Ω, ¯) (Ω, ¯) equations, it is only known when the noise possesses F → F a very particular form (e.g. either additive or linear is measurable. multiplicative). From now on, let X = (X, dX ) be a Polish space. Let D : ω D(ω) 2X be a multi-valued 5.8. Existence of pullback and random → ∈ mapping. The set of multi-functions D : ω attractors for multi-valued non- → D(ω) 2X with closed and non-empty images is autonomous and random dynamical ∈ systems denoted by C(X). Let also denote by Pf (X) the set of all non-empty closed subsets of the space X. Although we could have developed our theory for Thus, it is equivalent to write that D is in C(X), the existence of pullback attractors for processes or or D :Ω Pf (X) . → cocycles directly in the multi-valued case, we have Let D : ω D(ω) be a multi-valued mapping → preferred to first introduce and establish the theory in X over . Such a mapping is called a random P in the single-valued framework because, in this way, set if it may be clearer for a general audience. Now, we will establish the theory in the more general set-up ω inf dX (x,y) → y D(ω) of multi-valued dynamical systems generalizing the ∈ theory in Section 4 for non-autonomous/random is a random variable for every x X. It is well ∈ dynamical systems (see [Caraballo et al., 2008] for known that a mapping is a random set if and only more details). if for every open set O in X the inverse image A pair (Ω,θ), where Ω is a set and θ = (θt)t R ω : D(ω) O = is measurable, i.e., it belongs ∈ { ∩ 6 ∅} 40 Balibrea, Caraballo, Kloeden & Valero to (see [Hu & Papageorgiou, 1997; Proposition the set U(t,ω,x ) there exists δ > 0 such that if F 0 2.1.4]). dX (x0,y) < δ then Clearly, all this is also valid if we replace P by c and by . If we do not specify which U(t,ω,y) . P F F ∈U probability space we are using ( or c), it will P P U(t,ω, ) is called upper semi-continuous if it is up- mean that the result is valid for both cases. · per semi-continuous at every x in X. It is also evident that if D is a random set with 0 respect to , then it is also random with respect to P Remark 5.44. (i) Note that if a mapping U (t,ω, ) c. · P is upper semi-continuous at x0, then for all ε > 0 We now introduce multi-valued non- there exists δ (ε) > 0 such that autonomous and random dynamical systems. distX (U (t,ω,y) , U (t,ω,x0)) ε, Definition 5.42. A multi-valued map U : R+ ≤ × Ω X Pf (X) is called a multi-valued non- for any y satisfying dX (y,x0) δ (ε). × → ≤ autonomous dynamical system (MNDS) if (ii) The converse is true when U (t,ω,x0) is compact, see [Aubin & Cellina, 1984]. i) U(0, ω, )=idX , · It is not difficult to extend Definition 5.43 if we ii) U(t + τ,ω,x) U(t,θτ ω, U(τ,ω,x)) for all ⊂ consider the upper-semi-continuity with respect to t,τ R+,x X,ω Ω. ∈ ∈ ∈ all variables assuming that Ω is a Polish space. It is called a strict MNDS if We now formulate a general condition ensuring
iii) U(t + τ,ω,x) = U(t,θτ ω, U(τ,ω,x)) for all that an MNDS defines an MRDS. We need the par- t,τ R+,x X,ω Ω. ticular assumption that Ω is a Polish space and F ∈ ∈ ∈ the associated Borel-σ-algebra. An MNDS is called a multi-valued random dy- namical system (MRDS) if the multi-valued map- Lemma 5.45. Let Ω be a Polish space and let F ping be the Borel-σ-algebra. Suppose that (t,ω,x) 7→ (t,ω,x) U(t,ω,x) U (t,ω,x) is upper semi-continuous. Then this → is (R+) (X) measurable, i.e. (t,ω,x) : mapping is measurable in the sense of Definition B ⊗F⊗B { U(t,ω,x) O = (R+) (X) for every 5.42. ∩ 6 ∅}∈B ⊗F⊗B open set O of the topological space X. Proof. Thanks to Proposition 1.2.5 in For the above composition of multi-valued [Hu & Papageorgiou, 1997], we have that for mappings we use that for any non-empty set V each closed subset C X, the set ⊂ ⊂ X, U(t,ω,V ) is defined by 1 U − (C)= (t,ω,x) : U (t,ω,x) C = { ∩ 6 ∅} U(t,ω,V )= U(t,ω,x0). is closed, and thus is a Borel set in (R+) x0 V + B ⊗F⊗ [∈ (X) = (R Ω X). This implies that the B B × × We also note that the above measurability hypothe- inverse of a closed set is measurable and then, by sis is not standard at least for single-valued random Theorem 2.2.4 in [Hu & Papageorgiou, 1997], the dynamical system. However, for MRDS it is more map is measurable. difficult to derive measurability than for single val- ued systems. 5.8(A). Non-autonomous and random attrac- tors for MNDS We now introduce some topological properties of the MNDS U. In this section we generalize the concept of pullback and random attractors to the case of an MNDS and Definition 5.43. U(t,ω, ) is called upper semi- prove a general result for the existence and unique- · continuous at x if for every neighborhood of ness of attractors. 0 U Three perspectives 41
As a preparation we need the following defini- Remark 5.48. In contrast to the theory of random tions. attractors for single valued random dynamical sys- A multi-valued mapping D is said to be nega- tems we have weaker assumptions on the measura- tively, strictly, or positively invariant for the MNDS bility of A. Of course, it is desirable to obtain that A is a random set with respect to , but usually U if P we need stronger assumptions in the applications to obtain this property. ⊂ + D(θtω) = U(t,ω,D(ω)) for ω Ω, t R . ∈ ∈ ⊃ A consequence of the pullback convergence and invariance of P is that it reflexes the forward con- Let be the family of multi-valued mappings D vergence to the attractor with values in C(X). We say that a family K ∈ D is pullback -attracting if for every D and all P lim distX (U(t,ω,D(ω)), A(θtω)) = 0 D ∈ D − t + ω Ω → ∞ ∈ for all sets D such that ω U(t,ω,D(ω)) is mea- → lim distX (U(t,θ tω, D (θ tω)), K(ω)) = 0. surable for t 0. Indeed, we have to replace in t + − − ≥ → ∞ the formula for the pullback convergence ω by θtω. B is said to be pullback -absorbing if for every However, this is only true in the weaker conver- ∈ D D D there exists T = T (ω, D) > 0 such that gence in probability. There exist counterexamples ∈ D which show that in general the forward convergence U(t,θ tω, D (θ tω)) B(ω), for all t T. (65) does not hold almost surely (see [Arnold, 1998; page − − ⊂ ≥ 488]). The following definition provides the main ob- The main tool to prove the existence of an jective of this article. We have to introduce a par- attractor is the pullback-omega-limit set for the ticular set system (see [Schmalfuß, 2000]): let be D MNDS U. For some multi-valued mappings D a set of multi-valued mappings in C(X) satisfying we define a pullback-omega-limit set as an ω- the inclusion closed property: suppose that D ∈ D dependent set Λ(D,ω) given by and let D′ be a multi-valued mapping in C(X) such that D′(ω) D(ω) for ω Ω, then D′ . Λ (D,ω)= U (t,θ tω, D (θ tω)). ⊂ ∈ ∈ D − − s 0 t s \≥ [≥ Definition 5.46. A family A is said to be a ∈ D This set is obviously closed, but in general it global pullback -attractor for the MNDS U if it D can be empty. It is not difficult to prove that satisfies: y Λ (D,ω) if and only if there exist tn ∈ → + and yn U (tn,θ tn ω, D (θ tn ω)) such that 1. A (ω) is compact for any ω Ω; ∞ ∈ − − ∈ limn + yn = y. → ∞ 2. A is pullback -attracting; We then have the following lemma, which is D a generalization of Theorem 6 and Lemma 8 in 3. A is negatively invariant. [Caraballo et al., 2003b] to the case in which we consider the family instead of the bounded sets A is said to be a strict global pullback - D D of X. attractor if the invariance property in the third item is strict. Lemma 5.49. Suppose that the MNDS U(t,ω, ) is · upper semi-continuous for t 0 and ω Ω. Let B A natural modification of this definition for ≥ ∈ be a multi-valued mapping such that the MNDS is MRDS is asymptotically compact with respect to B, i.e., for every sequence tn + , ω Ω every sequence Definition 5.47. Suppose U is an MRDS and sup- → ∞ ∈ yn U(tn,θ tn ω,B(θ tn ω)) is pre-compact. pose that the properties of Definition 5.46 are sat- ∈ − − Then for ω Ω the pullback-omega-limit set isfied. In addition, we suppose that A is a random ∈ Λ (B,ω) is non-empty, compact, and set, with respect to c. Then A is called a random P global pullback -attractor. lim distX (U(t,θ tω,B (θ tω)), Λ (B,ω)) = 0, t + − − D → ∞ 42 Balibrea, Caraballo, Kloeden & Valero
Λ (B,θtω) U (t,ω, Λ (B,ω)) , for all t 0. Projects MTM2005-03860, MTM2005-06098-C02- ⊂ ≥ 01, MTM2008-00088 and MTM2009-11820, by Fun- The main theorem of this section is daci´on S´eneca (Comunidad Aut´oma de la regi´ode Murcia) Project 08667/PI/08, by JCCM Projects Theorem 5.50. Assume the hypotheses in Lemma PAI06-0114 and PBC05-011-3 and the Consejer´ıa 5.49. In addition, suppose that B is pullback de Innovaci´on, Ciencia y Empresa (Junta de An- ∈ D -absorbing. Then, the set A given by daluc´ıa), Proyecto de Excelencia P07-FQM-02468. D A (ω) := Λ (B,ω) References is a pullback -attractor. Furthermore, A is the D unique element from with these properties. Abraham, C., Biau, G. & Cadre,B., [2004] “On Lya- D In addition, if U is a strict MNDS then A is punov exponents and sensitivity,” J. Math. Anal. strictly invariant. Appl. 290, 395-404.
Adler, R.L., Konheim, A.G. & McAn- With respect to the applicability of the mea- drew, M.H. [1965] “Topological entropy,” surability of Definition 5.47 in the applications, we Trans.Amer.Math.Soc., 114, 309-319. suppose for the next lemma a complete probability space c. However, the result is also valid for the Alsed`a, L., Llibre, J. & Misiurewicz, M. [2000] P space . Combinatorial dynamics and entropy in dimen- P sion one, second edition, (World Scientific, Sin- Lemma 5.51. Assume that U is a MRDS. Un- gapore). der the assumptions in Theorem 5.50, let ω → U(t,ω,B(ω)) be a random set for t 0. Assume AlSharawi, Z., Angelos, J., Elaydi, S. & Rakesh, L. ≥ also that U(t,ω,B(ω)) is closed for all t 0 and [2006], “An extension of Sharkovsky’s theorem ≥ ω Ω. Then the set A introduced in Theorem 5.50 to periodic difference equations,” J.Math. Anal. ∈ is measurable, and therefore, it is a random global Appl., 316, 128-141. pullback -attractor. D Amig´o, J.M., Catto, I, Gim´enez, A. & Valero, 5.8(B). Applications and examples J. [2009] “Attractors for a non-linear parabolic equation modelling suspension flows,” Discrete The autonomous examples considered in Sec- Contin. Dyn. Syst. Series B 11, 205-231. tion 4 can all be modified to give examples of non-autonomous multi-valued systems. Some addi- Angelis, V. de. [2004],“Notes on the non- tional specific references are [Caraballo et al., 2006, autonomous Lyness equation,” preprint (2004) Kloeden & Mar´ın-Rubio, 2003, Kloeden & Schmalfuß, 1998, Arnold, L. [1998] Random Dynamical Systems Kloeden & Valero, 2005], which also consider (Springer-Verlag, Berlin). weak attractors, i.e., for which the invariance and attraction properties hold for at least one (rather Arrieta, J., Carvalho, A.N. & Hale, J.K. [1992] “A than all) trajectories emanating from each point. damped hyperbolic equation with critical expo- nent,” Comm. Partial Differential Equations 17, 841-866. Acknowledgements Arrieta, J.M., Rodr´ıguez-Bernal, A. & Valero, J. We are very grateful to our late Professor Valery S. [2006] “Dynamics of a reaction-diffusion equa- Melnik for his encouragement, his constant support tion with a discontinuous nonlinearity,” Inter- and his friendship. We will keep him forever in our nat. J. Bifur. Chaos 16, 2695-2984. hearts. This paper has been partially supported Aubin, J.P. & Celina, A. [1984] Differential Inclu- by Ministerio de Ciencia e Innovaci´on (Spain) sions (Springer-Verlag, Berlin). Three perspectives 43
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