PHYSICA Time-Reversal Symmetry in Dynamical Systems

PHYSICA ELSEVIER Physica D 112 (1998) 1-39

Time-reversal symmetry in dynamical systems: A survey

Jeroen S.W. Lamb a,., John A.G. Roberts b a Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK b School of Mathematics, La Trobe University, Bundoora, Vic. 3083, Australia

Abstract In this paper we survey the topic of time-reversal symmetry in dynamical systems. We begin with a brief discussion of the position of time-reversal symmetry in physics. After defining time-reversal symmetry as it applies to dynamical systems, we then introduce a major theme of our survey, namely the relation of time-reversible dynamical sytems to equivariant and Hamiltonian dynamical systems. We follow with a survey of the state of the art on the theory of reversible dynamical systems, including results on symmetric periodic orbits, local bifurcation theory, homoclinic orbits, and renormalization and scaling. Some areas of physics and mathematics in which reversible dynamical systems arise are discussed. In an appendix, we provide an extensive bibliography on the topic of time-reversal symmetry in dynamical systems.

1991 MSC: 58Fxx Keywords: Dynamical systems; Time-reversal symmetry; Reversibility

1. Introduction scope of our survey. Our survey also does not include a discussion on reversible cellular automata. For fur- Time-reversal symmetry is one of the fundamen- ther reading in these areas we recommend the books tal symmetries discussed in natural science. Con- by Brush [6], Sachs [22] and Hawking [14], and the sequently, it arises in many physically motivated survey paper by Toffoli and Margolus [23]. dynamical systems, in particular in classical and Our survey is largely self-contained and accompa- quantum mechanics. nied by an extensive bibliography in Appendix A. The aim of this paper is to give a brief and com- However, in areas where other good recent surveys pact survey of the state of the art with regards to time- are available (most of them in this special volume: reversal symmetry in dynamical systems theory. That [Sevryuk, 1998; Champneys, 1998; Hoover, 1998]), is, we consider ordinary differential equations and our discussion will be brief and will refer to those pa- diffeomorphisms possessing (what we call) reversing pers for more details. We will focus here on survey- symmetries. ing areas of research that are complimentary to those We are aware that the interest in time-reversal sym- reviewed elsewhere. metry goes beyond this confined setting. For instance, Needless to say, we aim to give a balanced ac- there is extensive work on time-reversal symmetry in count of the work and interests in the field of time- statistical and quantum mechanics that falls outside the reversal symmetry in dynamical systems. However, we realize very well that our survey is subjective and * Corresponding author. we would like to apologize to those colleagues who

might find their work under- or misrepresented. Also, 2.1. Time-reversal symmetry in classical mechanics rather than trying to discuss results in this field in detail, we have aimed at giving the reader a taste of the The conventional notion of time-reversal symmetry state of the art. Our bibliography in Appendix A is relates to observations of physical phenomena. inevitably incomplete and the process of compiling To fix the discussion, consider the example of the the bibliography is without end. 1 However, we hope dynamics of a classical ideal pendulum that experi- that our bibliography will provide an encouragement ences no energy loss due to friction. and opportunity for the reader to explore further from We now propose the following experiment: we let there. the pendulum swing, film it, and watch it using a pro- The paper is organized as follows. In Section 2 we jector that plays the film backward (in the reverse di- briefly discuss the position of time-reversal symmetry rection). So we see the pendulum moving backward in physics. In Section 3 we introduce the setting of our in time. If we are not familiar with the original film, survey, defining time-reversal symmetry in dynami- then as a viewer it would be impossible to tell that the cal systems and sketching its relation to equivariant film was actually played in reverse. This is because and Hamiltonian dynamical systems. In Section 4 we the motion on the reverse film also corresponds to a survey the state of the art on the theory of reversible possible motion of the same pendulum. Namely, the dynamical systems, including results on symmetric pe- reverse motion satisfies the same laws of motion as riodic orbits, local bifurcation theory, homoclinic or- the forward motion. The only difference between the bits, and renormalization and scaling. In Section 5 we motion depicted on the forward and reverse versions briefly discuss some areas of applications in physics of the film is the initial position and speed of the pen- and mathematics that have stimulated the research into dulum at the point where we start showing the movie. reversible dynamical systems. Our concluding section If for a motion picture of a mechanical system one is devoted to an outlook. Appendix A contains an ex- cannot decide whether it is shown in the forward or re- tended bibliography on time-reversal symmetry in dy- verse direction, the system is said to have time-reversal namical systems. References to this bibliography are symmetry. separated by style ([author, year]) from other refer- When we consider the more realistic physical situ- ences ([number]). ation of a swinging pendulum in the presence of fric- As a guide to the reader, we note that Sections 1-3 tion, we can tell the difference between a forward and provide a nontechnical introduction, aimed at a non- a reverse film of this pendulum. Namely, the original specialized audience. In contrast, Sections 4 and 5 (forward) film will show the swinging pendulum los- contain more details and references. ing amplitude until it comes to a standstill. However, the film in reverse direction shows a swinging pendulum whose amplitude is increasing in time. The latter film is clearly unphysical as it does not satisfy the 2. Time-reversal symmetry in physics natural laws of motion anymore (assuming there is no hidden source of energy feeding the pendulum). The Before addressing the topic of time-reversal symme- presence of friction breaks the time-reversal symmetry try in dynamical systems in Section 3, in this section of the ideal pendulum. we will briefly discuss the position of time-reversal The time-reversal symmetry described in this exam- symmetry in physics, i.e. in classical mechanics, ther- ple arises very frequently in classical mechanics. Al- modynamics and quantum mechanics. though in nature we hardly ever encounter mechanical systems with perfect time-reversal symmetry, in the- try a truly isolated pendulum has time-reversal sym- 1 Regular updates of this bibliography will be made available metry. The friction and energy transfer are merely due at ht tp : //www. maths, warwick, ac. uk/~lamb. to the coupling of the pendulum with its environment. J.S.W. Lamb, J.A.G. Roberts/Physica D 112 (1998) 1-39

In the Hamiltonian formulation of classical me- tank. When we open up a connection between the two chanics, we describe the system with variables (q, p), compartments, molecules from one compartment will where q is a vector describing the position of the sys- flow to the second compartment and in the end they tem and p a vector describing its momentum. will distribute evenly over the two compartments. If In its simplest form, the Hamiltonian H (q, p) is a one watches a film of this dynamical process in re- function which generates the equations of motion via verse, one observes the gas flowing from an evenly dq OH dp OH distributed state towards a state in which all the gas dt Op dt 0q (2.1) is in one of the two compartments of the tank. De- spite the theoretical possibility of this happening, the The classical notion of time-reversal symmetry as dis- realistic chance of this occurring is extremely small. cussed above is directly related to a symmetry prop- In statistical mechanics, when we want to describe erty of the Hamiltonian the bulk dynamics of many particles (N --+ cxz), there H(q, p) -~ H(q, -p). (2.2) is a true sense of direction of time. The most well- known result in this direction is Boltzmann's second Namely, if the Hamiltonian satisfies (2.2), then the law of thermodynamics, saying that entropy is a mono- equations of motion (2.1) are invariant under the tonically increasing function of time [2]. transformation This result needs careful interpretation, and it is not surprising that this result has led to a lot of confusion. R0 : (q, p, t) ~ (q,-p,-t). (2.3) Loschmidt [ 18] challenged Boltzmann by pointing out In turn, this implies that when (q(t), p(t)) is a tra- that his result "violated" the time-reversal symme- jectory in phase space describing a possible motion try of the (microscopic) equations of motion of the of the system with initial position and momentum particles concerned. In recognition of his critique, the (q0, P0), then so is (q(-t), -p(-t)) with initial con- situation whereby an ensemble of particles with time- clifton (qo,-Po). reversible dynamics displays irreversible behaviour is In configuration (position) space this means that if called Loschmidt's paradox. we have a trajectory q(t), then we also have a trajec- A first solution to the paradox was proposed by tory q(-t). This is precisely what we see when we Gibbs [11], who gave an explanation involving the play a film of a time-reversible system in reverse. course-grained structure of the phase space. However, to the present day, many papers are written that pro- 2.2. Thermodynamics vide an explanation of the paradoxical situation in which a system that has time-reversal symmetry on a Let us now consider a macroscopic number of clas- microscopic scale breaks this symmetry in its collec- sical particles and describe their collective behaviour. tive macroscopic behaviour. A popular resolution of In fact, such a system is in principle described by Loschrnidt's paradox is to argue that despite the re- Hamiltonian equations of motion where q and p de- versibility of the equations of motion, not all solutions scribe the positions and momenta of N particles, where need possess the full time-reversal symmetry. A differ- N is a very large number (e.g. in the order of Avo- ent view is presented by Kumicak and de Hemptinne gadro's number 1024). [17] in this issue. For a historical account, see [6]. Despite the time-reversal symmetry property of the A general discussion of this area is outside the equations of motion, the collective behaviour of a scope of our survey (but we refer the reader to macroscopic number of classical particles displays a the work of Prigogine and the Brussels school on clear direction of time, i.e. if q(t) is a likely trajectory reversibility/irreversibility and arrows of time in of the system then q(-t) is not necessarily! As an ex- physics and chemistry [20,21]). A recent approach ample, consider the motion of a macroscopic number in which irreversible dynamics of nonequilibrium of gas molecules in one of the two compartments of a thermodynamical systems is modelled by reversible J.S.W. Lamb, J.A.G. Roberts/Physica D 112 (1998) 1-39 dynamical systems has attracted a lot of interest, see, are taken to be flows of vector fields. Discrete time e.g. Section 5 and the papers [Dellago and Posch, dynamical systems are taken to be generated by an 1998; Gallavotti, 1998; Hoover, 1998] in this volume. invertible map f. In most applications of interest will be a manifold, e.g. ~2 = ~n. 2.3. Time-reversal symmetry in quantum mechanics In the continuous time context we consider autonomous ordinary differential equations of the form In the 1930s, Wigner [25] successfully introduced a dx -- = F(x) (x ~ ~2), (3.1) quantum mechanical version of the classical conven- dt tional time-reversal operator. With it, he explained the where F : S2 ~+ TI2 is a (smooth, continuous) vec- twofold degeneracy of energy levels that was reported tor field. The dynamics of (3.1) is given by a one- by Kramers [16] in systems with an odd number of parameter family of evolution operators electrons in the absence of a magnetic field. In the presence of a magnetic field the time-reversal symme- ~ot : S'2 ~--> ~2, try is broken and the degeneracy disappears yielding a qgt : x(z) v-+ qgt(x(r)) = x(z + t), (3.2) splitting of energy bands, cf. also [15]. Time-reversal symmetry is also important in quantum field theories such that 2 for elementary particle physics, cf. [22]. qgtl o (Pt2 = ~Otl+t2 for all tl, t2 E N. (3.3) With the growing interest in chaotic dynamics in the 1980s, there was also a growing interest in quan- We now say that an invertible (smooth, continuous) tum mechanical systems whose classical limit displays map R : S2 w-> ~2 is a reversing symmetry of (3.1) chaotic behaviour. In this field (called quantum chaol- when ogy by Berry [1 ]), time-reversal symmetry enters in the dR(x) analysis of level statistics. Random matrix theory pre- -- F(R(x)), (3.4) dt dicts that the rate of energy-level repulsion is classified or equivalently, when by the presence or absence of time-reversal symmetry. These predictions have been discussed by many, dR Ix" F(x) = --F(R (x)), (3.5) and for an introduction, see [13]. We note that, despite the success of group representation theory in quan- where dR Ix denotes the (Frechet) derivative of R in x. In terms of the evolution operator q~t, (3.4) and (3.5) tum mechanics [26], the role of comparable symmetry imply methods in quantumchaology appears to be relatively unexplored. For a recent exception see Cvitanovic and R o ~ot = ¢P-t o R = ~Ot I o R Eckhardt [8], who discuss the use of symmetries (but for all t 6 R. (3.6) not time-reversal symmetries!) in the calculation of zeta-functions. In the context of classical mechanics, where the ordinary differential equations are derived from a Hamiltonian H(q, p), the conventional reversing 3. Time-reversal symmetry in dynamical systems symmetry is given by

We will now give a more precise mathematical de- R(q, p) = (q, -p). (3.7) scription of time-reversal symmetry in the setting of Note that in this particular case R is an involution (i.e. dynamical systems, as considered in this survey. We R 2 = id), and R is anti-symplectic. also give a historical account of its origin. By analogy to definition (3.6) in the case of flows, We consider two types of dynamical systems, with we call an invertible map R : Y2 ~ 12 a reversing continuous time (t 6 N) and discrete time (t 6 77) on some phase space ~2. Continuous time systems 2 o denotes composition. J.S.W. Lamb, J.A.G. Roberts/Physica D 112 (1998) 1-39

symmetry of an invertible map f : £2 w-~ S2, when- t = a/2 has reversing symmetry R. A similar result ever 3 also applies to local return maps for R-symmetric periodic orbits of autonomous flows with reversing sym- R o f ----- f-1 o R. (3.8) metry R. The notion of reversing symmetries for autonomous After the work of Birkhoff, time-reversal symme- flows extends in a natural way to nonantonomous try did not receive much attention until the 1960s flows, [DeVogelaere, 1958; Heinbockel and Struble, 1965; Moser, 1967; Bibikov and Pliss, 1967; Hale, 1969]. In dx -- = F(x, t). (3.9) particular, Hale described the property of time-reversal dt symmetry as property E. 4 Namely, we call Ra : (x, t) e--> (R(x), -t + a) a re- Devaney [Devaney, 1976] noted that many conse- versing symmetry of (3.9) whenever (3.9) is invariant quences of conventional time-reversal symmetry are under the transformation R a (for some a ~ R), i.e. shared by dynamical systems which have a different dR(x) type of involutory reversing symmetry than the con- -- F(R(x), -t + a). (3.10) dt ventional anti-symplectic one (3.7). This led him to a definition of reversible systems in which the involu- Note that by introducing a new variable v = t - tory nature of the time-reversal operator R was central, a/2, the extended differential equation d(x, v)/dt = together with the fact that R should fix a subspace half (F'(x, v), 1) (with F'(x, r) := F(x, r + a/2)) is au- the dimension of the phase space. Later, Arnol'd and tonomous and has reversing symmetry R0 : (x, v) e--> (R(x), -r). Sevryuk [Arnol'd, 1984; Arnol'd and Sevryuk, 1986] relaxed this further to allow for any involutory re- The presence and importance of time-reversal sym- versing symmetry. The latter definition of reversibility metry was recognized in the early days of dynamical was adopted by many, and hence for a map was taken systems by Birkhoff. He utilized it in his study of the to be synonymous with the decomposition property restricted three-body problem in classical mechanics (3.11). [Birkhoff, 1915]. In particular, he noted that a map f Arnol'd and Sevryuk [Arnol'd, 1984; Arnol'd and with an involutory reversing symmetry R can always Sevryuk, 1986] remarked that systems with reversing be written as the composition of two involutions symmetries need not have an involutory reversing f=RoT, whereR 2=T 2=id. (3.11) symmetry (for some examples, see [Arnol'd and Sevryuk, 1986; Lamb, 1996a; Baake and Roberts, In this context, note that when R is not an involution 1997]). In quantum mechanics, the importance of one readily verifies that the decomposition property noninvolutory time-reversal symmetries was long (3.11) generalizes to before acknowledged by Wigner [26]. Arnol'd and f = R o T, where R 2 o T 2 = id. (3.12) Sevryuk proposed to call systems with only noninvolutory reversing symmetries weakly reversible. In flows of nonantonomous vector fields (3.9) when They found that many results for reversible systems F(x, t) is periodic in time, i.e. F(x, t) = F(x, t + 1) actually also hold for weakly reversible systems, by (with period scaled to 1), then in a natural way the showing that in many problems the reversing symme- time-one return map of such a flow is autonomous. try enters the analysis with an effectively involutory Moreover, it is readily checked that when the nonan- action. tonomous system is invariant under Ra, then the time- one return map with respect to the surface of section 4 Despite the fact that Hale did not publish many papers on 3 Usually - but not always - one is interested in maps f time-reversal symmetry in dynamical systems, his interest has and R that are not just invertible, but also homeomorphisms or been a catalyst for further research, cf., e.g. the acknowledge- diffeomorphisms. ments in [Kirchg~issner, 1982a; Vanderbauwhede, 1990b]. J.S.W. Lamb, J.A.G. Roberts/Physica D 112 (1998) 1-39

In this respect it is interesting to note that if a lin- Example 3.2. All oscillation equations of the form ear system in R n possesses a reversing symmetry it also possesses a linear involutory reversing symmetry d 2 dt2q = F(q), F : ~m w-~ •m. (3.13) [Sevryuk, 1986]. (This follows directly from Jordan normal form theory.) However, one should be careful When (3.13) is rewritten as a first-order system in the interpreting this result. For instance, when considering variables qi and dqi/dt in R 2m, R is the involution continuous parameter families of matrices possessing that changes the signs of dqi/dt. a given noninvolutory reversing symmetry (such as in reversible linear normal form theory [Hoveijn, 1996; Example 3.3. In many partial differential equations, Lamb and Roberts, 1997]) this observation is not par- the equations goveming steady-state solutions are ticularly useful as, for instance, the form of the involu- reversible. tory reversing symmetry may change discontinuously. For example, Malomed and Tribelski [Malomed and Also, it should be stressed that this implication does Tribelski, 1984] considered a class of partial differen- not hold in general. For example, it does not hold for tial equations, one of which, nonlinear systems in R n [Lamb, 1994a, 1996a] and not even for linear diffeomorphisms of the 2-torus [Baake 3 0 4 3 2 and Roberts, 1997] (cf. Example 3.5). ~-7~ + ~x4~ + 2Ot~x2~ In the light of a more general approach towards symmetry properties of dynamical systems (see + ~ + ~ = 0, (3.14) Section 3.1), it turns out to be unnecessarily restric- tive to explicitly mention the nature of the reversing describes the evolution of a gas flame under certain symmetry in a definition of reversibility (and so dis- physical conditions. This equation is not reversible tinguish between reversible and weakly reversible with respect to the time variable. However, the steady- systems). Therefore, we define: state solutions are described by (3.14) with 0~ lot = O. The resulting steady-state ordinary differential equa- Definition 3.1 (Reversible Dynamical System). A dy- tion is reversible with respect to the space variable x. namical system is called reversible when it possesses a Namely, the fourth-order ordinary differential equa- reversing symmetry R satisfying (3.4), (3.8), or (3.10) tion can be written as a system of four first-order for autonomous flows, maps, or nonautonomous flows, equations in the variables ~, O~/Ox, 82~/0x 2, and respectively. 03~/0x 3, and this dynamical system is reversible with respect to the involution In the literature there is sometimes confusion about the use of terminology. Sometimes, a system is called R.@,0~,02 03 reversible when its inverse exists. This notion of 57 2 ' 57x invertibility differs from the notion of reversibility (~,_ 0 02 0 3 adopted here. In particular, note that all reversible -ff-Z2 - ) . % systems are invertible, but not all invertible systems are reversible (because not all invertible systems have In a similar way, one finds that all autonomous even- a reversing symmetry). order (odd-order) ordinary differential equations in We now list some examples of reversible dynamical which the odd (even) derivatives occur only in even systems: combinations are reversible when rewritten as a first- order system. The reversing symmetry R is the trans- Example 3.1. All Hamiltonian systems with Hamilto- formation of the even-dimensional (odd-dimensional) nian H(q, p) satisfying (2.2) with an anti-symplectic phase space that corresponds to changing the sign of R of the form (3.7). the variables corresponding to odd (even) derivatives. J.S.W. Lamb, J.A.G. Roberts/Physica D 112 (1998) 1-39 7

Example 3.4. Symmetric difference equations of the is reversible with the order-4 reversing symmetry R form given by

Xn+l Jr-Xn-1 = f(xn) (3.15) (;tt)---(01 ;1)(;)(modl). (3.18) may arise as a discretization of d2x /dt 2 = 2x - f (x), but may also arise due to spatial symmetry properties of physical models on a chain (n labels the position In fact, the map (3.17) has no involutory reversing on the chain). symmetry within the group of toral homeomorphisms. A well-known example of a mapping of the form We note that the reversibility of hyperbolic toral au- (3.15) arises in the study of stationary states of a chain tomorphisms has no obvious physical cause. The sym- of coupled oscillators with a convex nearest neighbour metry properties of hyperbolic toral automorphisms interaction potential. By Newton's action = reaction follow directly from the structure of the matrix group principle this interaction potential should be invariant Gl(2, ?7) [Baake and Roberts, 1997]. with respect to interchanging Xn and Xn+l. For instance, in the case of the Frenkel-Kontorova The above examples illustrate the fact that reversible model, the total (interaction + background) potential dynamical systems arising in the literature obtain their 1 is given by Y~n(g(xn - Xn+l) z --]- V COSXn), and the reversibility due to a variety of reasons. In particular, stationary states satisfy (3.15) with f(xn) = 2x~ - we observe: v sin(x~), which is equivalent to the area-preserving - Reversibility in time arising due to a natural assump- Chirikov-Taylor standard mapping. tion of time-reversibility of the equations of motion Introducing new variables, Pn := Xn and qn := (cf. Examples 3.1 and 3.2). x~-i the system (3.15) can be written as a mapping - Reversibility in space arising due to natural assump- of the plane tions of spatial symmetries of a physical model (cf. Examples 3.3 and 3.4). Pn+l = f(Pn) -- qn, qn+l = Pn. (3.16) - Reversibility arising due to the specific structure This mapping is reversible with respect to the in- of a mathematical problem under consideration volution R(p, q) = (q, p). This is a direct con- (cf. Example 3.5). sequence of the fact that (3.15) is invariant under For more examples in the above three categories, the transformation Xn-1 <--> Xn+l. In the context of see Section 5. the Frenkel-Kontorova model this is in turn a di- There are generally two perspectives from which rect consequence of the symmetry of the interaction reversible dynamical systems are considered. On the potential. one hand they can be treated from a symmetry per- Remarkably, (3.16) is not only reversible, but also spective, as reversible systems are defined in terms area-preserving. Many area-preserving (symplec- of a symmetry property. On the other hand, histori- tic) mappings studied in the literature are reversible cally the interest in reversible systems has often been (e.g. the well-studied area-preserving Hdnon map- in the context of Hamiltonian systems. Firstly this is ping, cf. [Roberts and Quispel, 1992] and references because many examples of reversible dynamical sys- therein). tems in applications are actually also Hamiltonian. Secondly there is the remarkable fact that reversible Example 3.5. The hyperbolic toral automorphism and Hamiltonian systems have many interesting dy- given by namical features in common. This duality may be the reason why so few systematic results on reversible dynamical systems have been derived, compared to (;'~)=(~ /0)(;)(modl) (3.17) the overwhelming machinery developed for symmetric J.S.W. Lamb, J.A.G. Roberts/Physica D 112 (1998) 1-39

under composition. We call a group of symmetries and reversing symmetries of a dynamical system a reversing symmetry group G [Lamb, 1992] and note that the symmetries (equivariances) form a normal subgroup H of G, i.e. H~ G. Moreover, when H 5~ G then H is a subgroup of index 2,

G/H ~-- 772. (3.21)

Note that G can be written as the semi-direct product G --~ H >~ 772 if and only if G \ H contains an involution. Fig. 1. Schematic diagram indicating the intersections of reversible dynamical systems (R), equivariant dynamical systems When a dynamical system possesses a reversing (E), and Hamiltonian dynamical systems (H). symmetry but no nontrivial symmetries - disregard- ing for the moment the trivial symmetries ~0t for flows (equivariant 5) and Hamiltonian dynamical systems. and fn for maps - it follows that H = {id} and G --~ The dual approach towards reversible systems will be 772, so that the dynamical system possesses precisely the dominant theme in this paper. A schematic view one involutory reversing symmetry. We will call such of the situation is depicted in Fig. 1, where it is shown a dynamical system purely reversible. how the Hamiltonian, reversible, and equivariant sys- The dynamical consequences of symmetries (equiv- tems overlap. ariance) and reversing symmetries differ substantially. Symmetries map trajectories to other trajectories pre- 3.1. Reversible versus equivariant dynamics serving the direction in which they are traversed in time. Reversing symmetries also map trajectories to When a reversible system possesses more than one trajectories, but now the time-direction of the two tra- reversing symmetry, one finds that the composition of jectories is reversed. an odd number of reversing symmetries yields again A very obvious difference resulting from this is the a reversing symmetry, but that the composition of an role of fixed point subspaces. The fixed point subspace even number of reversing symmetries yields a symme- of a map U : £2 ~ $2 is defined as Fix(U) :-----{x try. S is called a symmetry of the equations of motion [ U(x) = x}. Fixed point subspaces of symme- if, in the case of an autonomous or nonautonomous tries are setwise invariant under the dynamics. How- flow (3.1) or (3.9), we find ever, fixed point subspaces of reversing symmetries are usually not setwise invariant under the dynamics, dS(x) -- -- dSIx- F(x, t) = F(S(x), t), (3.19) but give rise to symmetric periodic orbits, homoclinics dt and heteroclinics (see Section 4.1 and Section 4.5). or, in the case of a map f, we have As a simple contrasting example, two phase portraits of flows of planar vector fields are sketched in S o f = f o S. (3.20) Fig. 2. Both flows are symmetric with respect to a re- Dynamical systems with a symmetry S are also called flection in a horizontal line, but in one portrait (a) the S-equivariant, and have attracted lots of attention in reflection is a symmetry and in the other (b) it is a recent years, cf. for instance [9,10,12]. reversing symmetry. In describing the symmetry properties of flows and Despite the dynamical differences between maps, it is natural to discuss symmetries and reversing reversible and equivariant dynamical systems, symmetries on an equal footing as they form a group techniques developed for the equivariant context sometimes carry over to the reversible one. For 5 A definition of eqnivariance will be given in Section 3.1. instance, local bifurcation problems in reversible J.S.W.Lamb, J.A.G. Roberts/Physica D 112 (1998) 1-39

R0 to a reversing symmetry R for such a return map. The remaining time-shift symmetry properties arise (a) ~ (b)~~~ in a less obvious way. Namely, when a time-periodic flow admits a symmetry of the form S_l/q(x, t) = (S(x), t - 1/q) (for some q E N), then the time-one remm map with surface of section t = 0 can be writ- Fig. 2. Sketches of the phase portraits of planar flows that are ten as being decomposed into q pieces that are related symmetric with respect to a horizontal reflection, where (a) the to each other by time-shift symmetries. Consequently, reflection is a symmetry (eqnivariance), and (b) the reflection is the time-one return map can be written as S -q o fq a reversing symmetry. In (b) some typical reversible dynamical phenomena can be observed: the right-most equilibrium is a where f := So~o[0,1/q] and q0[0,1/q] denotes the first hit centre surrounded by a Liapunov centre family of periodic orbits map between surfaces of section at t = 0 and t = 1/q (Section 4.1). Going outward the periods of these orbits start [Lamb, 1995, 1997]. The map f conveniently char- tending to infinity (blue sky catastrophe) as they come close to a symmetric homoclinic orbit connecting the symmetric saddle acterizes the dynamics of the flow. Interestingly, the to itseff (Section 4.5). In the left side of the flow one observes space-time symmetry properties of the flow arise as an asymmetric attractor-repeller pair. (reversing) k-symmetries of the map f. k-Symmetry arises in a similar way in the study of local return maps systems can often be studied via equivariant singu- of symmetric periodic orbits in autonomous flows. For larity theory after performing a Liapunov-Schmidt a more detailed discussion, we refer to [Lamb, 1997]. reduction [Vanderbanwhede, 1982; Golubitsky et al., Interestingly, many results for reversible maps have 1995], cf. also Section 4.3 for more references. (nontrivial) extensions to the domain of k-reversible maps. We will give detailed references in relevant 3.1.1. k-Symmetry and space-time symmetry sections below. For a more extended introduction to k- It may happen that a map f possesses less sym- symmetric dynamical systems see [Lamb and Brands, metries and/or reversing symmetries than its kth iter- 1994; Lamb and Quispel, 1994; Lamb, 1994a, 1996b, ate fk. If k is the smallest positive integer for which 1997]. a transformation U is a (reversing) symmetry of fk, then U is called a (reversing) k-symmetry of f [Lamb 3.2. Reversible versus Hamiltonian dynamics and Quispel, 1994]. It turns out that k-symmetry naturally arises in the Reversibility is a symmetry property that most study of return maps of flows of time-periodic vec- prominently arises in Hamiltonian dynamical systems, tor fields with mixed space-time symmetries. In the in particular in the context of mechanical systems. In context of such systems we consider reversing sym- physics, the terms reversible and Hamiltonian might metries Ra :(x, t)~-~ (R(x),-t +a)and (time-shift) sometimes be thought to be nearly synonymous, since symmetries of the form Sa : (x, t) ~ (S(x), t + a), in practice the vast majority of reversible dynamical that leave the equations of motion invariant. In a nat- systems arising in applications so far appear to be ural way these space-time symmetries form a group Hamiltonian. under composition. In studies of classical mechanical systems re- Let us consider a nonantonomous system that is in- versibility has been gratefully welcomed as a tool variant with respect to the time-shift t --+ t + 1. The in studying periodic orbits, homoclinics and hete- symmetry properties of the time-one return map of roclinics, cf., e.g. [Devaney, 1976, 1977; Churchill such a system are related to the space-time symme- and Rod, 1980, 1986; Churchill et al., 1983; Meyer, tries of the flow under consideration. For instance, the 1981]. Reversibility began to be taken more seriously time-one return map with surface of section t = 0 in- as a symmetry property in Hamiltonian dynamical herits the space-time symmetries that fix the surface of systems after it turned out that many results origi- section setwise, i.e. So gives rise to a symmetry S and nally established for Hamiltonian dynamical systems 10 J.S.W. Lamb, J.A.G. Roberts/Physica D 112 (1998) 1-39

can also be obtained by assuming that the dynamical Hamiltonian systems has not really been developed. system is reversible (without taking into account the Also a systematic comparison between dynamical Hamiltonian structure). features of Hamiltonian systems, reversible sys- In two seminal papers, Devaney [Devaney, 1976, tems, and reversible Hamiltonian systems is far from 1977] established the reversible Liapunov centre established. theorem (Section 4.1) and the reversible blue sky In the literature there are not so many examples catastrophe theorem (Section 4.5) for periodic orbits of Hamiltonian dynamical systems that are not re- of reversible systems and noted the close analogy to versible. Arnol'd and Sevryuk [Arnol'd and Sevryuk, comparable results for Hamiltonian systems, noting 1986] and Roberts and Capel [Roberts and Capel, that "reversible systems, near symmetric periodic 1992, 1997] constructed examples of nonreversible orbits, behave qualitatively just like Hamiltonian Hamiltonian systems using local obstructions to re- systems". versibility. However, the nonreversibility in these ex- Along the same lines, various other "Hamiltonian" amples is not persistent under small (Hamiltonian) results have been extended to the reversible do- perturbations. Examples of persistently nonreversible main. Most notably, the Kolmogorov-Arnol'd-Moser Hamiltonian systems in R 2 were given by Mather (KAM) theory has a reversible analogue, as do some [Mackay, 1993] (global topological obstruction) and results in local bifurcation theory, cf. Sections 4.2 Lamb [Lamb, 1996a] (local topological obstructions). and 4.3. In the same spirit, the Aubry-Mather theory Recently, it was shown that the reversibility of hy- for area-preserving monotone twist maps has recently perbolic toral automorphisms can always be decided been extended to the domain of reversible mono- [Baake and Roberts, 1997]. Since such mappings are tone twist maps of the plane [Chow and Pei, 1995]. structurally stable, this yields persistent examples of The origin of these coincidences is still an area of both reversible and nonreversible area-preserving dif- investigation and not very well understood. feomorphisms of the toms. Many reversible systems in applications happen to Non-Hamiltonian reversible systems also do not be Hamiltonian at the same time. When studying re- appear frequently in the literature. A few examples versible Hamiltonian systems, it may be more con- are a laser model [Politi et al., 1986], the Stokeslet venient to prove results using the reversibility rather model describing sedimenting spheres [Caflisch et al., than the Hamiltonian structure. In dynamical systems 1988], and a model of coupled Josephson junctions obtained as reductions from partial differential equa- [Tsang et al., 1991a, 1991b]. Politi et al. and Tsang tions, reversibility is often more easily recognized than et al. observed that their reversible systems may pos- an underlying symplectic structure. 6 sess attractors and repellers (pair-wise), and at the Unfortunately, the fact that most reversible systems same time display Hamiltonian-like behaviour. The in applications are Hamiltonian and most Hamiltonian Stokeslet model has led to various interesting papers systems in applications are reversible, seems to on reversible equivariant systems, cf. [Golubitsky et have obscured a bit which properties of reversible al., 1991; Lim and McComb, 1995, 1998; McComb Hamiltonian systems are due to the reversibility and and Lim, 1993, 1995]. Roberts and Quispel [Roberts which are due to the Hamiltonian structure. and Quispel, 1992] studied scalings in non-area- Because of the overwhelming number of reversible preserving reversible mappings of the plane, and Hamiltonian systems of interest, it is somewhat also found a mixture of dissipative and conservative surprising that a systematic theory on reversible (Hamiltonian) behaviour, see also Section 4.4. In the following sections we will discuss in more 6 For instance, the authors of [Eckmann and Procaccia, 19911 detail certain aspects of the theory of reversible dy- did not realize that their dynamical system obtained by PDE namical systems. We will make more precise com- reduction is not only reversible but also Hamiltonian, until this was pointed out by R.S. MacKay (Woudtschoten Conference, ments on the reversible versus Hamiltonian dichotomy 1992). at various points. J.S.W. Lamb, J.A.G. Roberts/Physica D 112 (1998) 1-39 11

4. Aspects of reversible dynamics sults nowhere require that R is an involution [Lamb, 1992]. In this section we will discuss some dynamical consequences of reversibility in more detail. It is orga- Theorem 4.1 (Symmetric orbits for flows). Let o(x) nized as follows. In Section 4.1-4.5 we review results be an orbit of the flow of an autonomous vector field on symmetric periodic orbits (in particular in relation with time-reversal symmetry R. Then, to their natural occurrence in families), KAM theory, An orbit o(x) is symmetric with respect to R if and local bifurcations (including Birkhoff normal form only if o(x) intersects Fix(R), in which case the theory), scaling properties and renormalization, and orbit intersects Fix(R) in no more than two points homoclinic and heteroclinlc behaviour. Section 4;6 and is fully contained in Fix(R2). contains a brief discussion of some other topics. It An orbit o(x) intersects Fix(R) in precisely two should be noted that the length of our discussion of points if and only if the orbit is periodic (and not a the different topics in this section has been largely in- fixed point) and symmetric with respect to R. fluenced by the existence or absence of other recent relevant surveys. Theorem 4.2 (Symmetric orbits for maps). Let o(x) be an orbit of an invertible map f with reversing symmetry R. Then: 4.1. Symmetric periodic orbits - An orbit o(x) is symmetric with respect to R if and only if o(x) intersects Fix(R) ~ Fix(f e R), in Understanding a dynamical system on the basis of which case the orbit intersects Fix (R) U Fix(f e R) its periodic orbits has been a predominant theme in in no more than two points and is fully contained dynamical systems theory ever since the studies of in Fix(R2). Poincarr. - An orbit o(x) intersects Fix(R) UFix(f o R) in pre- It is therefore not surprising that a result on periodic cisely two points if and only if the orbit is sym- orbits is by far the most well known and used result metric with respect to R and periodic (but not a in reversible dynamical systems. In 1915, Birkhoff fixed point). Such an orbit intersects both Fix(R) [Birkhoff, 1915] described the use of reversibility and Fix(f o R) if and only if it has odd period. In to find periodic orbits of the restricted three-body particular: problem. In 1958 DeVogelaere [DeVogelaere, 1958] -o(x) is a periodic orbit of f with period described the method again, but now as a tool for 2p if and only if there exists a y 6 o(x) searching for symmetric periodic orbits of reversible such that y E Fix(R) n fPFix(R) or y systems (by computer). Fix(f o R) N fPFix(f o R). - o(x) is a periodic orbit of f with period 2p .a_ 1 Definition 4.1 (Symmetric orbits). Let o(x) be an or- if and only if there exists a y 6 o(x) such that bit of a dynamical system, i.e. o(x) = {~0t(x) I t ~ ~} y ~ Fix(R) N fPFix(f o R). in the case of flows and o(x) = {fn(x) [ n 6 2~} in the case of maps. Then o(x) is R-symmetric Theorem 4.1 or 4.2 is used in almost every paper dis- or symmetric with respect to R when the or- cussing reversible dynamical systems. In particular, bit is setwise invariant under R, i.e. R(o(x)) = these theorems imply efficient techniques for track- o(x). ing down R-symmetric periodic orbits, as it justifies searching for them in only a subset of the full phase The results on finding symmetric periodic orbits space, cf., e.g. [Greene et al., 1981; Kook and Meiss, in reversible systems are folklore and many have de- 1989], Section 4.4 and Appendix A for more exam- rived (and are still deriving!) these results apparently ples. In the special case of reversible maps in N2 with independently. We present here a general version an involutory reversing symmetry R fixing a one- of the results on periodic orbits. Note that the re- dimensional subspace, Fix(R), Fix(f o R) and their 12 J.S.W. Lamb, J.A.G. Roberts/Physica D 112 (1998) 1-39 iterates have been often referred to as the symmetry R-symmetric periodic orbits of a given period lines of the map, cf., e.g. [Mackay, 1993; Roberts and typically arise as (2 dim Fix(R) - dim Fix(R 2) + Quispel, 1992]. m)-parameter families. From the results on periodic orbits, we find that the (iii) In a continuous m-parameter family of au- fixed point subspaces of reversing symmetries are im- tonomous flows with reversing symmetry R, portant. The fixed sets of reversing symmetries Can R-symmetric periodic orbits typically arise as take various forms. For instance, when a reversing (2 dim Fix(R) -- dim Fix(R 2) +m + 1)-parameter symmetry acts freely then its fixed point subspace is families with smoothly varying period, in which empty. Examples of involutions with free actions in- the families with constant period mentioned in clude rotations on a 2-torus or the involutory action on (ii) are embedded. 7 a unit 2-sphere embedded in N 3 induced by the transformation -id (the latter example arose recently in Theorem 4.3 implies that under suitable conditions, a study of relative equilibria of molecules [Montaldi in reversible systems one may find n-parameter fam- and Roberts, 1997]). On more exotic manifolds, in- ilies (n > 0) of R-symmetric periodic orbits in phase volutions may even exist whose fixed point subspace space (whenever one of the formulas with m = 0 gives consists of several connected components of different a positive number). dimension, cf. [Quispel and Sevryuk, 1993]. Generalizations of Theorems 4.1 and 4.2 that apply 4.1.1. Stability properties of symmetric orbits to flows of time-periodic vector fields with space-time A well-known property of linear reversible systems symmetries were recently described in [Lamb, 1997]. is that their eigenvalue structure is similar to that of Generalizations that apply to maps with reversing k- Hamiltonian systems. symmetries can be found in [Lamb and Quispel, 1994; Brands et al.; 1995; Lamb, 1997]. Theorem 4.4 (Eigenvalues of linear reversible The above theorems imply that in reversible systems systems). periodic orbits generically arise in families. Under - Let )~ be an eigenvalue of a linear reversible vector some smoothness assumptions, the fixed point sub- field. Then so is -)~ and 2 (complex conjugate of spaces mentioned in the above theorems are man- )0. ifolds. Their intersection will again be a manifold - Let )~ be an eigenvalue of a linear reversible diffeo- and its generic dimension follows from elementary morphism. Then so is ;.-1 and 2. considerations. Hence, for linear flows the eigenvalues come in Theorem 4.3 (Families of symmetric periodic orbits). singlets {0}, doublets {)~, -)q with )~ c N or )~ c iR, or quadruplets {)~,-)~, 2,-2}. Also for finear re- (i) In a continuous m-parameter family of dif- versible maps the eigenvalues come in singlets {4-1}, feomorphisms fa with reversing symmetry R, doublets {)~,)-1} with )~ 6 N or )~ 6 S 1 := {z R-symmetric periodic orbits of a given even C I [z[ = 1}, or quadruplets {)~,)~-1,2,~-1}. Note period generically come in (2dimFix(R)- that when R is an involutory reversing symmetry dimFix(R 2) + m)-parameter families and (2 dim Fix ( fa o R) -- dim Fix(R 2) + m)-parameter 7 The observation in Theorem 4.3(iii) can be derived from families. R-symmetric periodic orbits of a given considering a local remm map for a symmetric periodic orbit of a flow. The dimension of the surface of section S is one lower odd period generically form (dimFix(R)+ than the dimension of the phase space and the periodic orbit is dimFix(fa o R) -- dim Fix(R 2) + m)-parameter a fixed point of the return map. Now, importantly dim Fix(R) = families. dimFix(R)ls. Hence, by Theorem 4.3(i) the symmetric fixed point is typically embedded in a (2 dim Fix(R) -- dim Fix(R 2) + (ii) In a continuous m-parameter family of au- m + 1)-parameter family of fixed points representing periodic tonomous flows with reversing symmetry R, orbits of the flow with nearby period. J.S.W. Lamb, J.A.G. Roberts/Physica D 112 (1998) 1-39 13 and dim Fix(R) ¢ 1 dim Fix(R2), then necessarily familiar planar picture of a centre type equilibrium the linearized vector field, respectively, diffeomor- surrounded by a family of periodic orbits (cf. the phism, at an R-symmetric fixed point is forced to rightmost equilibrium point in Fig. 2(b)). 8 have eigenvalues equal to 0, respectively, -t-1. In case Golubitsky et al. [Golubitsky et al., 1991] extended dimFix(R) > 1 dimFix(R 2) these eigenvalues pre- this result by allowing certain types of symmetry- cisely support the families of periodic orbits described induced resonances to occur, namely zero eigenvalues in Theorem 4.3. due to reversibility and 1 : 1 resonances due to equiv- From the characterization of eigenvalues of re- ariance. Contrasting Devaney's geometric approach, versible linear systems it can be seen that a range they used a Liapunov-Schmidt reduction, adapting of interesting phenomena might arise. For instance, an alternative proof of the reversible Liapunov centre note that the stability properties of fixed points and theorem given by Vanderbauwhede [Vanderbauwhede, periodic orbits are decided by linearized vector fields 1982]. (For a recent application of this result, see and return maps (via Floquet theory). The eigenvalue [Chang and Kazarinoff, 1996]). properties are consistent with confirm the fact that When resonances occur that are not a simple con- R-symmetric periodic solutions cannot be asymp- sequence of symmetry properties, then the families of totically (un)stable. (In fact, this comment applies periodic orbits of two pairs of purely imaginary eigen- to any R-symmetric w-limit set, cf., e.g. [Lamb and values interact. For a discussion, see Section 4.3. Nicol, 1998]). Indeed, precisely this generic occur- While considering the Liapunov centre family in rence of "balanced" stability characterizes reversible the right-hand side of Fig. 2(b), it is interesting to dynamical systems. It gives rise to complicated (and note that when we follow the one-parameter family interesting) dynamical behaviour which is partly sim- of periodic orbits going away from the centre point, ilar to dynamical features of volume preserving and the periods of the closed orbits tend to infinity as they Hamiltonian dynamical systems. approach a reversible homoclinic orbit (a closed orbit starting and ending at a symmetric saddle point). 4.1.2. Reversible Liapunov centre theorem Devaney called this a blue sky catastrophe [Devaney, Devaney [Devaney, 1976] showed that the Liapunov 1977], and proved that such families of symmetric pe- centre theorem for Hamiltonian systems has a re- riodic orbits always arise around reversible homoclinic versible analogue. The theorem describes the exis- orbits, cf. Section 4.5 for more details. tence of families of symmetric periodic orbits in the neighbourhood of an (partially) elliptic symmetric 4.2. KAm-theory fixed point 0 of a vector field F in R 2n with reversing symmetry R. The interest in reversible dynamical systems has Suppose that 4-ico are simple eigenvalues of the been boosted not simply because of the results on pe- linearized vector field dF]0, and that -t-ikco are not riodic orbits. Another important line of investigation eigenvalues for all k = 0, 2, 3, 4,... To avoid res- has been that of Kolmogorov-Arnol'd-Moser (KAM) onances, and assuming that none of the eigenvalues theory. KAM-theory deals with the persistence of in- of dFI0 are real, it then follows that R 2 = id and variant tori constituting quasiperiodic motion in nearly dimFix(R) = n. The reversible Liapunov centre integrable dynamical systems. theorem asserts that there exists a two-dimensional Originally, KAM-theory was developed in a invariant manifold containing 0 that, in a neighbour- Hamiltonian setting, i.e. only smooth perturbations hood of 0, contains a nested one-parameter family of were considered that preserve the symplectic structure R-symmetric periodic orbits. Moreover, the periods of these periodic orbits tend to 2~/co as the initial 8 Note that the Liapunov centre families of periodic orbits conditions of these orbits tend to 0. To fix the idea, in flows often appear in the associated return maps as one- note that the centre theorem precisely describes the parameter families of periodic orbits (with a fixed period). 14 J.S.W. Lamb, J.A.G. Roberts/Physica D 112 (1998) 1-39

of an integrable dynamical system. However, early text of equivariant dynamical systems, cf. Golubitsky on it was acknowledged that KAM-theory can also et al. [12]. be applied to the setting of reversible systems, cf. Bifurcation theory for reversible systems has been Moser [Moser, 1967] and Bibikov and Pliss [Bibikov developed in a less systematic way than for equivariant and Hiss, 1967]. Interestingly, in his expository paper systems. First of all, in most papers the analysis is [Moser, 1973], Moser chooses to present KAM- restricted to purely reversible systems (i.e. those with theory for reversible systems as he finds it "techni- no equivariance properties). Often, further assumption cally somewhat simpler" than for the Hamiltonian is made that the reversing symmetry R acts in E 2n as context. a linear involution with an n-dimensional fixed point Although integrable systems need not be reversible, subspace. 9 many integrable systems in the literature happen to be In papers on reversible equivariant systems there reversible. For examples of a large class of integrable are often additional hypotheses: e.g. the existence of reversible mappings of the plane that are the composi- an involutory reversing symmetry or some explicit as- tion of two involutions that also preserve the integrals, sumption on how a reversing symmetry R acts with see [Quispel et al., 1989; Roberts and Quispel, 1992]. respect to the action of the additional equivariances. For other examples of integrable reversible mappings, These hypotheses are usually motivated by properties cf. [Boukraa et al., 1994; Rerikh, 1995, 1996]. of the particular models under consideration, but un- Sevryuk has carried through a thorough program fortunately obscure the general applicability of some of studying reversible KAM-theory, starting with his of these results. lecture notes [Sevryuk, 1986] (for further references Nevertheless, many interesting results on local bi- see Appendix A). For a survey on the reversible KAM- furcations in reversible systems have been obtained theory see Sevryuk [Sevryuk, 1998] (in this volume) and their embedding in a systematic theory that ap- and Broer et al. [Broer et al., 1996c]. The latter treat plies to more general space-time symmetric systems KAM-theory from a general perspective, presenting is an interesting open problem. reversible systems as just one of the contexts to which Before we discuss reversible local bifurcation the- the KAM techniques apply. ory in more detail, we note that most results on With the recognition that KAM-theory applies to reversible systems are indeed of a local nature. In- reversible systems, the natural question has arisen terestingly, however, Fiedler and Heinze [Fiedler and whether there is a reversible analogue of (the problem Heinze, 1996a, 1996b] recently developed a topolog- of) Arnol'd-diffusion. Matveyev [Matveyev, 1995a, ical index theory for reversible periodic orbits. This 1995b, 1996] obtained some results that show that might well be a starting point for the use of global indeed there is diffusion related to the break-up of techniques in the study of reversible systems. Also invariant tori. However, the mechanism of diffusion is Fiedler and Turaev [Fiedler and Turaev, 1996] illus- not identical to the mechanism of diffusion observed trate how topological arguments can be used to prove in Hamiltonian systems. that elliptic periodic orbits arise at certain elementary homoclinic bifurcations. 4.3. Local bifurcation theory In discussing local bifurcations of reversible systems, it is important to note from Theorem 4.4 that It is well known that symmetry properties of a sys- instabilities may arise when eigenvalues are on the tem may influence the genericity of the occurrence imaginary axis or unit circle in the complex plane. In of local bifurcations. That is, bifurcations that typically occur in certain symmetric systems might only be rarely observed in nonsymmetric systems. The influ- 9 As many authors note, the assumption of a locally linear action of an involution is justified by the Montgomery-Bochner ence of symmetry on local bifurcations (steady-state theorem [19]. Actually this comment applies more generally to and Hopf) has been extensively studied in the con- local actions of compact Lie groups, cf. [3]. .I.S.W. Lamb, J.A.G. Roberts/Physica D 112 (1998) 1-39 15 particular, by analogy to the situation in Hamiltonian map into normal form, and then studying its (versal) systems, bifurcations may arise when such eigenval- unfoldings. ues pass through resonances. In the context of symmetric systems, the derivation This section is organized as follows. First, we briefly of Birkhoff normal forms is naturally done in a struc- discuss the Birkhoff normal form theory for reversible ture preserving (symmetry respecting) framework [5]. systems. Thereafter we survey the literature on several In the presence of a reversing symmetry group G this types of local bifurcations: steady-state bifurcations, means that only G-equivariant coordinate transforma- bifurcations at resonant centres, subharmonic branch- tions are to be considered. ing and reversible Krein crunch. For purely reversible systems, several papers on re- Steady-state bifurcations involve the collision at versible linear normal forms have appeared [Palmer, zero of eigenvalues of the linearized vector field at 1977; Wan, 1991; Sevryuk, 1992; Shih, 1993; an equilibrium point of a flow, or a collision at +1 of Hoveijn, 1996]. The paper by Hoveijn [Hoveijin, eigenvalues of the linearized map at a fixed point of 1996] is the most recent and complete reference, a diffeomorphism. More complicated bifurcations in- and includes a discussion of the Hamiltonian and volve the occurrence of resonances (rational relation- reversible Hamiltonian cases as well. For technical ships between the eigenvalues of a linearized vector details on the symmetry-respecting linear normal field or map on the imaginary axis or unit circle in form theory, see also Knobloch and Vanderbauwhede the complex plane). We distinguish between bifurca- [Knobloch and Vanderbauwhede, 1995]. Recently, tions at resonant equilibrium points of flows (when Lamb and Roberts [Lamb and Roberts, 1997] dis- eigenvalues on the imaginary axis pass through a cussed the normal form theory for finear reversible resonance), subharmonic branching (when the return equivariant systems on the basis of the representation map of a symmetric periodic orbit of a flow has (a theory of reversing symmetry groups. pair of) eigenvalues passing through a root of unity), After bringing the linear part of a system into nor- and the reversible Krein crunch (when two pairs of mal form, one can subsequently normalize higher- eigenvalues of a map collide on the unit circle). order terms in the Taylor expansion of a map or vector field. It turns out that most basic results for nonsymmetric systems (cf. [7,24]) carry over to the reversible 4.3.1. Birkhoff normal forms (and equivariant) context without further complica- Various authors have studied reversible bifurcation tions, cf., e.g. [Iooss and Adelmeyer, 1992; Lamb, problems using, in one way or another, a Birkhoff nor- 1996b; Shih, 1997; Vanderbauwhede, 1990a; van der mal form analysis. The starting point of Birkhoff nor- Meer et al., 1994]. mal form theory is to consider the Taylor expansion of The most common method for characterizing a diffeomorphism or vector field at a fixed point and Birkhoff normal forms uses the unique decomposition to find a local coordinate frame in which the Taylor of a matrix into its nilpotent and semisimple part. 10 expansion looks "simple", i.e. in normal form (with We first consider flows. Let dFI0 be the (Frechet) respect to a certain convention). The aim is then to derivative of a vector field F at a fixed point 0, and study certain aspects of the local dynamics around a dFI0 ---- As + An be the decomposition into its semi- fixed point (local bifurcations, stability properties) us- simple and nilpotent part. Then, when the flow is suf- ing the truncated normalized expansion of the diffeo- ficiently differentiable, a normal form to any desired morphism or vector field. However, it is important to order can be obtained that is equivariant with respect to keep in mind that this strategy should be applied with exp(tAs) for all t ~ N, while preserving the (reversing) care as some dynamical features of the truncated system may not arise in the original system. I°A linear operator is semi-simple whenever it is C- The Birkhoff normal form procedure about a fixed diagonalizable. A linear operator N is nilpotent if there exists point starts with bringing the linear part of the flow or an integer n such that N n is zero. 16 J.S.W. Lamb, J.A.G. Roberts/Physica D 112 (1998) 1-39

symmetry properties of the flow. 11 The additional a Hamiltonian vector field with purely imaginary symmetry properties of the normal form are usually eigenvalues -4-ico has a normal form that is, up to referred to as the formal normal form symmetry. any desired order, rotationally (SO (2)) equivariant. For smooth diffeomorphisms f : R n ~ R n an Consequently, it can be written in polar coordinates analogous scheme applies. After decomposing the as [Birkhoff, 1927] (Frechet) derivative df[0 of f at 0 as d f[0 = As -k An, up to any desired order of the Taylor expan- dr 0, dO g(r 2), with g(0) co, (4.1) dt dt sion a normal form can be obtained that is (formally) A s-equivariant. which is reversible with respect to the involution Above we described normalizations with respect to (r, v~) ~-~ (r, -0). For counterexamples of such coin- the semi-simple part As. Further normalization with cidences, see [Roberts and Capel, 1992, 1997; Lamb respect to the nilpotent part An can also be imple- et al., 1993]. The reason for the formal reversibifity of mented. Note however, that in the case of maps, when Hamiltonian normal forms is not well understood, but df[0 is not semi-simple the explicit calculations for it foreshadows some of the similarities in the (local) obtaining reversible normal forms for diffeomor- dynamics of reversible and Hamiltonian systems. phisms are potentially cumbersome, cf. [Lamb et al., Despite the inherent problem of the convergence 1993; Lamb, 1996b]. properties of Birkhoff normal forms, they are very Jacquemard and Teixeira [Jacquemard and Teixeira, helpful in bifurcation analysis and in understanding 1996b, 1996c, 1997] have developed an alternative certain aspects of local dynamics, cf., e.g. [Iooss and method for calculating reversible normal forms of Kirchg~issner, 1992; Iooss and Ptroubme, 1993; Iooss, diffeomorphisms. They successfully implement this 1995a, 1997] and [Broer et al., 1998b; Hangmann, method within a computer algebra program. Their 1998] in this volume. method might be particularly useful, as an alternative to the method described above, when calculating 4.3.2. Steady-state bifurcations in reversible systems normal forms for maps with non-semi-simple linear To our knowledge, Rimmer [Rimmer, 1978, 1983], parts. was one of the first to discuss a reversible bifurcation The normal form results for flows and diffeomor- problem. He considered reversible symplectic diffeo- phisms extend in a natural way to the reversible morphisms of the plane with a reversing reflection Hamiltonian setting under natural additional assump- symmetry R. He showed that symmetry-breaking tions on the (anti-)symplecticity of the representation pitchfork bifurcations from an R-symmetric fixed of G. point are generic (codimension one). He also showed The structure preserving Birkhoff normal form that such pitchfork bifurcations cease to be generic strategy also extends naturally to diffeomorphisms when the fixed point under consideration is not R- with a reversing k-symmetry group G in which case symmetric (e.g. when the symplectic diffeomorphism the structure preserving transformations are again is not reversible). This result nicely illustrates the im- G-equivariant [Lamb, 1996b]. portance of acknowledging the presence of a reversing In relation to the reversible-Hamiltonian duality symmetry, also in Hamiltonian systems. discussed in Section 3.2, it is interesting to note More recently, steady-state bifurcations in reversible- that Hamiltonian normal forms are almost always Hamiltonian systems in R 2 have been studied in formally reversible. For example, an equilibrium of [Broer et al., 1998a, 1998b; Hangmann, 1998] (in this volume). It is interesting to note that (anti)symplectic 11 For local actions of compact Lie groups, we army assume (reversing) symmetries of Hamiltonian vector fields without loss of generality that a reversing symmetry group G arise as invariance (or anti-invariance) properties acts locally linear and orthogonal. However, for a normal form approach allowing for nonlinear symmetry actions, see Gaeta of the Hamiltonian, posing the problem of steady- [Gaeta, 1994]. state bifurcations automatically as a singularity J.S.W. Lamb, J.A.G. Roberts/Physica D 112 (1998) 1-39 17

theory problem. We note, in this respect, that in the classification of planar polynomial vector fields reversibility is a recurrent theme, either in connection with the study of centres [Zoladek, 1994; Berthier and Moussu, 1994; Teixeira, 1997b] or as a symmetry Fig. 3. Schematic diagram illustrating the eigenvaluemovements assumption restricting the class of vector fields under at the reversible-Hopfbifi.trcation (1 : 1 resonance). consideration [Guimond and Rousseau, 1996]. Surprisingly, steady-state bifurcations in reversible systems have not been studied very intensively. Mo- of purely imaginary eigenvalues on the imaginary axis. tivated by a laser model, Politi et al. [Politi et al., In the absence of eqnivariance, these eigenvalues are 1986] observed a symmetry breaking pitchfork bifur- typically simple (no two eigenvalues are the same). cation in non-area-preserving planar maps and flows Also, typically, they are nonresonant (no eigenvalues of the plane and noted the birth of an attractor-repeller are positive integer multiples of others). However, such pair in such a bifurcation, cf. also [Post et al., 1990; situations may arise in generic one-parameter families Roberts and Qnispel, 1992]. Local steady-state bifur- for suitable values of the parameter. cations of certain planar reversible equivariant vector We consider the situation where two pairs of purely fields were discussed by Lamb and Capel [Lamb and imaginary (nonzero) eigenvalues +io)1, -t-io92 collide Capel, 1995]. Most of this analysis was done on the when a parameter is varied. Then, typically, after such basis of a Birkhoff normal form approach, a collision they branch off as a quadruplet in the com- A different approach towards steady-state bifur- plex plane, cf. Fig. 3. In Hamiltonian systems, such cations of reversible flows has been pursued by a bifurcation is known as a Hamiltonian Hopf bifur- Teixeira [Teixeira, 1997a; da Rocha Medrado and cation; in reversible systems, by analogy, it is called Teixeira, 1998]: purely reversible flows in Nn where a reversible-Hopf bifurcation or reversible 1 : 1 res- dimFix(R) = n - 1 are treated as a half-infinite onance. By the centre theorem, before the collision system with a boundary Fix(R). In this approach, each pair of purely imaginary eigenvalues has a one- symmetric local bifurcations (and indeed the local parameter family of periodic orbits associated with it. flow around Fix(R)) are characterized by the contact However, after the collision the quadruplet of eigen- of the vector field with Fix(R). values is in the complex plane and does not give rise Jacquemard and Teixiera [Jacquemard and Teixeira, anymore to families of periodic solutions around the 1996a, 1996c, 1997] study local bifurcations of fixed origin. points of reversible diffeomorphisms in R 2 and N 3 Amol'd and Sevryuk [Amol'd and Sevryuk, 1986; using normal forms to describe the contact between Sevryuk, 1986] studied this bifurcation and found the fixed sets of the two involutions that constitute a that either both families simultaneously disappear, (purely) reversible map. shrinking as a unit at the origin (the so-called elliptic Yet another approach has been pursued by Lim and regime), or they disappear only around the origin but McComb [Lim and McComb, 1998] (in this volume). persist outside (the hyperbolic regime). The approach They use a Liapunov-Schmidt reduction to prove the by Arnol'd and Sevryuk is geometrical and based genericity of symmetry-breaking pitchfork bifurca- on an analysis of the curves along which the fami- tions in reversible systems. Their technique allows lies of periodic orbits intersect the fixed point sub- for the occurrence of resonances and zero eigenvalues space of the reversing involution. Vanderbauwhede that arise due to equivariance and reversibility. [Vanderbauwhede, 1990a] also studied this bifurcation but from a different (analytical) perspective, 4.3.3. Bifurcations at resonant centres using Liapunov-Schmidt reduction. This allows him The linearized flow at an R-symmetric equilibrium to include generalizations to some situations in which point of a reversible autonomous flow may have pairs the system is not only reversible but also equivariant. 18 J.S.W. Lamb, J.A.G. Roberts/Physica D 112 (1998) 1-39

Recently, Knobloch and Vanderbauwhede [Knobloch multipliers), we may have a pair of eigenvalues on the and Vanderbauwhede, 1995] extended the analysis unit circle in the complex plane. When the eigenval- to reversible-Hopf bifurcations at k-fold resonances ues pass through a resonance ()~± = exp(-t-2i top~q), (where k pairs of purely imaginary eigenvalues collide, with gcd(p, q) = 1) then, in the absence of other res- involving the merging of k one-parameter families onances, Vanderbauwhede [Vanderbauwhede, 1986, of symmetric periodic orbits), using general results 1990b, 1992a] showed that at such resonances gener- on periodic solutions obtained by the same authors ically Liapunov-centre families of periodic solutions in [Knobloch and Vanderbanwhede, 1996]. Iooss and meet, cf. also Gervais [Gervais, 1988] and Sevryuk P6roubme [Iooss and P6roubme, 1993] used a normal [Sevryuk, 1986]. The cases q = 1, q = 2 (period dou- form approach to analyze homoclinic solutions in bling) and q > 3 (subharmonic bifurcations) require the reversible 1 : 1 resonance. For peculiarities about separate discussions. the normal form at the reversible 1 : 1 resonance, see For a detailed bifurcation analysis in the strongly van der Meer et at. [van der Meer et al., 1994] and resonant case q = 1, see also P6roubme [P6rou~me, Bridges [Bridges, 1998] (in this volume). 1993]. The situation where there are additional reso- Higher 1 : N resonances (where icol = Nico2) were nances due to reversibility in the form of eigenvalues also studied by Arnol'd and Sevryuk [Arnol'd and locked at +1 is discussed by Furter [Furter, 1991] on Sevryuk, 1986; Sevryuk, 1986]. An important differ- the basis of a singularity theoretic approach of the re- ence with the reversible Hopf-bifurcation is that now, duced bifurcation equations. before and after passing through the resonance, the Reversible subharmonic branching is similar to purely imaginary eigenvalues remain on the imaginary the corresponding phenomenon of subharmonic axis. The two corresponding Liapunov centre families branching in Hamiltonian systems, and gives rise of (short and long) periodic orbits interact at the bifur- to a subharmonic branching tree in its phase space cation points. There are again two regimes (elliptic and [Vanderbauwhede, 1990b]. In the Hamiltonian case, hyperbolic) and the cases N = 2, N = 3, and N > 4 this tree is normally foliated by the level sets of are treated separately. McComb and Lim [McComb the Hamiltonian. However, in the non-Hamiltonian and Lim, 1995] extended these results by allowing for reversible case there need not be any conserved zero eigenvalues and resonances due to reversibility quantities. The tree is embedded in phase space and and equivariance along the lines of [Gohibitsky et al., longitudinal drifting motion parallel to such a tree 1991]. Sevryuk [Sevryuk, 1986] further describes p : may arise [Arnol'd, 1984]. For a picture of part q resonances (with gcd(p, q) = 1) in which case the of a branching tree in R 3 see [Roberts and Lamb, scenarios involve short and long periodic orbits (asso- 1995]. Associated scaling behaviour is discussed in ciated with p and q) and very long periodic orbits (as- Section 4.4. sociated with gcd(p, q)). Recently, Shih [Shih, 1997] extended the analysis to case studies of resonances in- 4.3.5. Reversible Krein crunch volving three frequencies. In the study of diffeomorphisms in R n (n > 4), resonances may arise when a quadruplet of eigenvalues 4.3.4. Subharmonic branching on the unit circle collides at exp(+i co) and the four The dynamics of periodic orbits in flows can be eigenvalues branch off the unit circle as a quadruplet studied by a (local) Poincar6 return map such that into the complex plane (the eigenvalues thus behaving the periodic orbit corresponds to a fixed point of as the exponents of eigenvalues in the reversible-Hopf this map. When the flow is reversible and the peri- bifurcation depicted in Fig. 3). By analogy to a similar odic orbit is symmetric with respect to a reversing bifurcation in Hamiltonian systems, this is sometimes symmetry, the return map is reversible too and the called the reversible Krein crunch. The normal form fixed point is symmetric. Hence, among the eigenval- theory is formally similar to the reversible 1 : 1 reso- ues of the linearized map at the fixed point (Floquet nance, which is used in Sevryuk and Lahiri [Sevryuk J.S.W. Lamb, J.A.G. Roberts/Physica D 112 (1998) 1-39 19

and Lahiri, 1991] to conjecture a description in case has been employed widely to study break-up of toil co/(2Jr) is sufficiently irrational, cf. also [Bridges et in area-preserving reversible mappings and universal al., 1995]. In a number ofpapers, Lahiri et al. have fur- scalings associated with the break-up have been iden- ther numerically studied bifurcations near resonant re- tified [Mackay, 1983b, 1988, 1993]. Renormalization versible Krein crunches in four-dimensional reversible group explanations of the scalings have been advanced maps [Bhowal et al., 1993b, 1993a; Lahiri et al., 1993, by MacKay [Mackay, 1988, 1992]. Numerical results 1995], see also [Lahiri et al., 1998] (in this volume) suggest that the same scalings characterize reversible for a recent account. mappings that are not area-preserving [Roberts and Quispel, 1992]. Khanin and Sinai [Khanin and Sinai, 4.4. Renormalization and scaling 1986] have given a renormalization group theory that works in the space of reversible (not necessarily area- Universal scaling in dissipative dynamical systems, preserving) maps. and accompanying explanations using renormalization group theory, were introduced by Feigenbaum. 4.4.2. Period-multifurcation cascades Such investigations have also been made in low- In the early 1980s, various authors discovered dimensional conservative/reversible mappings in two universal scalings in parameter and phase space in main areas: (i) break-up of KAM-tori; and (ii) period- period-doubling trees in area-preserving reversible multifurcation cascades. Historically, both areas were mappings [Greene et al., 1981; Bountis, 1981; first explored numerically in area-preserving map- Benettin et al., 1980b, 1980a]. Only the paper [Greene pings. As both investigations require finding many et al., 1981] identified two phase-space scalings, as- long periodic orbits, it was natural to study reversible sociating them with scaling along and scaling across area-preserving mappings in which Theorem 4.2 the so-called dominant symmetry line containing above could be used to find symmetric periodic orbits. two points of each even cycle (cf. Theorem 4.2). In fact, it seems prohibitive to conduct such studies Renormalization group explanations in the space of without the benefits of reversibility. This means that area-preserving reversible mappings were advanced the universal results obtained pertain to mappings by various authors [Greene et al., 1981; Collet et that are both area-preserving and reversible. Various al., 1981; Mackay, 1993], the fixed point of the authors have investigated whether the results are dif- doubling operator being assumed reversible. There ferent if one of the properties is relaxed. This seems was limited discussion as to how important it was to be another area where the similarities induced by to have both properties : area-preservation and re- symplecticity and by reversibility are quite striking. versibility. Again, Roberts and Quispel [Roberts and Although a complete explanation is yet to be given, Quispel, 1992] show that symmetric period-doubling it seems analysis should focus on how dependent the cascades in reversible mappings that are not neces- spectrum of appropriate renormalization operators is sarily area-preserving appear to be governed by the on the properties of the space of maps on which the scalings found earlier. Meanwhile, the analysis in operators act. [Davie and Dutta, 1993] would seem to explain this by highlighting the significance of the spectrum of the 4.4.1. Break-up of KAM-tori doubling operator when restricted to area-preserving Greene's residue criterion [Greene, 1979] uses the maps. stability of a sequence of nearby symmetric periodic Roberts and Lamb [Roberts and Lamb, 1995] orbits to suggest the existence or nonexistence of a showed that the self-same scalings found in 2D (re- given KAM-torus. The sequence of periodic orbits versible) period-doubling describe self-similarity of analysed has rotation number converging to the irra- period-doubling branching trees in 3D reversible tional winding number of the torus. Since its illus- mappings (cf. also [Komineas et al., 1994]). Here tration for the standard mapping [Greene, 1979], it the symmetric periodic orbits in the tree form 20 J.S.W. Lamb, J.A.G. Roberts/Physica D 112 (1998) 1-39 one-parameter families (because of Theorem 4.3) and Homoclinic (and heteroclinic) orbits are of great branch in phase space (rather than parameter space) interest in dynamical systems because in their according to the subharmonic branching theory of neighbourhood one usually finds chaotic behaviour. Vanderbauwhede described in Section 4.3.4. Importantly, the above characterization yields the Self-similar universal scalings also govern multifur- persistent occurrence of symmetric homoclinic and cations in (area-preserving) reversible maps [Meiss, heteroclinic solutions in many reversible systems of 1986; Turner and Quispel, 1994]. interest. In this respect it is interesting to note that in the prototype model for homoclinic dynamics, the 4.5. Reversible homoclinics and heteroclinics Smale-horseshoe map, the symbolic dynamics on the nonwandering set is reversible in a very natural way, Homoclinics and heteroclinics form connec- with the reversing symmetry interchanging the stable tions between saddle points and thereby usually and unstable leaves [Devaney, 1989]. constitute recurrent transport through a dynamical Devaney [Devaney, 1984, 1988] Used reversibil- system. ity to prove the existence of transversal homoclinics Let x0 be an equilibrium point of a dynamical sys- (horse-shoes) in the area-preserving reversible H6non tems with reversing symmetry R. We denote the stable map (cf. also [Brown, 1995]). Others have used and unstable manifolds of x0 by WxS0 and WU0. Recall normal forms to prove the existence of homoclinic that stable, respectively, unstable manifolds, contain and/or heteroclinic solutions, cf., e.g. Churchill and all the points that tend to x0 for t --+ +~x~, respec- Rod [Churchill and Rod, 1986] in the context of the tively, t --+ -cx~. A point y is a homoclinicpoint of x0 H6non-Heiles system and Iooss and P6roubme [Iooss if y lies in the intersection of WxS0 and WU0. A point y and P6rou~me, 1993] in the context of the reversible is a heteroclinic point of two points Xl and xz when 1 : 1 resonance. Y6WxSlNWU2. The dynamics around reversible homoclinics enjoys In general, it is not easy to locate homoclinic or special properties. For instance, Devaney [Devaney, heteroclinic points and orbits. However, in reversible 1977] showed that an R-symmetric homoclinic orbit systems, symmetric homoclinic and heteroclinic orbits in a reversible vector field invokes a "blue sky catas- can be found relatively easily, because of the character- trophe", i.e. a family of periodic orbits with periods izations of symmetric orbits in Theorems 4.1 and 4.2. tending to infinity. More precisely, he found that in In fact, if a dynamical system has a reversing symme- the neighbourhood of a nondegenerate R-symmetric try R, it follows immediately that the intersection of homoclinic orbit of an R-reversible vector field there Fix(R) and the stable or unstable manifold of a hyper- exists a one-parameter family of R-symmetric pe- bolic point yields homoclinic or heteroclinic points. riodic orbits whose periods tend to infinity as the More precisely, let y 7~ x0, then y c Fix(R) N wS;u periodic orbits approach the homoclinic orbit. De- is a homoclinic point of x0 if and only if x0 is R- vaney's approach is mainly geometrical and uses symmetric. Alternatively, y ~ Fix(R) f? W~ u is a the classical properties of symmetric periodic or- heteroclinic point of x0 if and only if x0 is not R- bits. Vanderbauwhede and Fiedler [Vanderbauwhede symmetric. and Fiedler, 1992] proved this theorem with a dif- For flows, this gives a full description of R- ferent (analytical) method which works for both re- symmetric homoclinics and heteroclinics. For maps versible and Hamiltonian systems. Interestingly, the f, one should also consider the above statements not latter paper extends a result from the reversible cat- only with Fix(R) but also with Fix(f o R), by anal- egory into the Hamiltonian setting, rather than vice ogy to the characterization of R-symmetric orbits in versa. Section 4.1. Note in this respect that R-symmetric ho- In case the equilibrium of the homoclinic orbit moclinic and heteroclinic orbits are always contained is of saddle-focus type, the dynamics around the in Fix(R2). homoclinic orbit tends to become very intricate, £S.W. Lamb, J.A.G. Roberts/Physica D 112 (1998) 1-39 21 cf. [Devaney, 1977] and [H~rterich, 1998] (in this a recent series of publications, Goodson et al. have volume). begun a study of reversible dynamical systems from The interest in reversible homoclinics arises not an ergodic theory point of view. We refer the reader only from their relation to complicated dynamics, to [Goodson et al., 1996; Goodson and Lemaficzyk, but also for their practical relevance. Namely, in the 1996; Goodson, 1996c; Goodson, 1996b] for further context of travelling waves of certain partial dif- details. ferential equations, homoclinic solutions represent solitary waves. Such waves are of interest in various 4.6.3. Reversible integrators applications, e.g. in optical communication systems The numerical study of a dynamical system [Sandstede et al., 1997]. For a survey on the the- with continuous time (i.e. a flow) often involves ory and applications of reversible homoclinics (in a discretization method to integrate the equations particular in the context of partial differential equa- of motion. In recent years there has been much tions), see Champneys [Champneys, 1998] in this interest in studying the properties of such inte- volume. grators. In particular, when a dynamical system Reversible heteroclinics have attracted consid- possesses certain structures (symmetry, reversibil- erably less attention than reversible homoclin- ity, Hamiltonian, gradient) one would fike to find a ics. A few exceptions are[Churchill and Rod, discretized approximant that preserves such a struc- 1986; Vanderbauwhede, 1992b; Rabinowitz, 1994a, ture. This is done to prevent the numerical calcula- 1994b; Maxwell, 1997]. Following up the recent tion of evidently erroneous global phenomena, such interest in heteroclinic cycles in equivariant sys- as asymptotically stable attractors in Hamiltonian tems, reversible heteroclinic cycles are also be- systems. ginning to be studied in more detail, cf. [Reil3ner, Especially in the context of reversible Hamiltonian 1998]. systems, it turns out that preserving reversibility in integrators has lots of advantages (in the context of 4.6. Miscellaneous topics such systems it is sometimes argued that preserving reversibility in an integration method is more important 4.6.1. Admissible symmetry properties of periodic than preserving the symplectic structure), cf. [Scovel, orbits and attractors 1991; Stoffer, 1995; McLachlan et al., 1995; Hairer One may address the following question: what and Stoffer, 1997; Leimkuhler, 1997], and McLachlan symmetry properties of periodic orbits, attractors, and Quispel [McLachlan and Qulspel, 1998] in this or other types of co-limit sets are admissible in dy- volume. namical systems with a given reversing symmetry group? In the equlvariant setting (without reversibility) there is a fairly complete understanding [10]. 5. Reversible dynamical systems in physics and However, in reversible equivariant systems many el- mathematics ementary questions are still open. For a discussion, see Lamb and Nicol [Lamb and Nicol, 1998] in this In this section we will briefly describe some ar- volume. eas of physics and mathematics in the context of which reversible dynamical systems have occurred 4.6.2. Ergodic theory in the literature. In particular, we follow up our ob- It is somewhat surprising that in the field of er- servation in Section 3 that reversible systems have godic theory, reversibility has, until recently, received appeared in the literature not only in relation to con- very little attention. This is despite the success of er- ventional reversibility with respect to time, but also godic theory in many areas of dynamical systems the- due to spatial symmetries in the context of partial ory and its obvious relevance to thermodynamics. In difference and differential equations, or due to the 22 J.S.W. Lamb, J.A.G. Roberts/Physica D 112 (1998) 1-39

(group) structure in certain abstract mathematical 5.1.2. Reversible models for nonequilibrium problems. thermodynamics The study of the behaviour of the dynamics of 5.1. Reversibility in time: Mechanics and many particles is naturally the domain of statistical nonequilibrium thermodynamics physics. Recently, in the study of nonequilibrium systems, a reversible dynamical systems point of 5.1.1. Conventional time-reversibility in mechanical view towards such problems has received a lot of systems interest. Needless to say, mechanical (Hamiltonian) systems In 1984, Nos6 showed that a molecular system con- contain a large class of examples of reversible dynam- nected to a heat reservoir can be described as an iso- ical systems with the conventional anti-symplectic lated system after the introduction of additional bath reversing symmetry (3.7). The bibliography in variables. These bath variables can be chosen in such Appendix A contains only a limited number of ref- a way that the thermodynamic properties of the sys- erences in this direction, trying to include those that tem can be derived using microcanonical rather than use reversibility in a systematic, rather than ad hoc, canonical ensembles. Importantly, after adopting the way. ergodic hypothesis, in the latter formulation the rel- Reversibility is certainly an important symme- evant thermodynamic variables of a system can be try property, even (or especially) in the context of found by averaging over an ergodic trajectory of the Hamiltonian systems. A nice illustration of this point system (computationally this is very advantageous). was made by Montaldi [Montaldi, 1991]. He showed Interestingly, Nos6's additional bath variables keep that in configuration (q-) space, projections of toil of the dynamical system reversible. The reversibility in reversible-Hamiltonian systems have different caus- the extended nonequilibrium system is such that the tics than projections of tori in nonreversible systems. reversing symmetry maps sources to sinks and vice The configuration-space point of view was also taken versa. After Nos6's discovery, various modifications by Golubitsky et al. [Golubitsky et al., 1996] in a to his initial ideas have been made. We refer to the recent study of the admissible types of symmetric review paper of Hoover [Hoover, 1998] in this vol- periodic orbits in configuration space for reversible ume for a discussion and more references. Despite equivariant potential systems. the large number of experimental (numerical) studies As hardly any mechanical system in engineering of the Nos6-Hoover type dynamical systems (cf., e.g. is perfectly reversible, it is interesting to study how Dellago and Posch [Dellago and Posch, 1998] in this a reversible symmetric system behaves when small volume), as yet not many theoretical studies have been additional nonreversible perturbations are taken into devoted to revealing their properties. account. O'Reilly et al. [O'Reilly et al., 1995, 1996] Gallavotti and Cohen [Gallavotti and Cohen, have considered the consequences of nonreversible 1995] propose a point of view in which the ergodic dissipative perturbations to reversible mechanical hypothesis common in equilibrium statistical me- systems. We emphasize that this approach is differ- chanics is replaced by a similar chaotic hypothesis ent to the approach in which generic phenomena of for reversible nonequilibrium systems (e.g. of the symmetric systems are studied. In the latter stud- Nos6-Hoover type). The latter approach assumes the ies, from a practical point of view the underlying existence of a transitive reversible Anosov system. idea is that in many physically relevant situations For some studies of the properties of such systems, idealized symmetric models represent less ideally see, e.g., [Gallavotti, 1995; Tasaki and Gaspard, 1995] symmetric experiments quite well (thus quietly and [Gallavotti, 1998] in this volume. The property assuming that small symmetry-breaking perturba- of reversibility in the theory is important as it ensures tions do not cause drastic changes in the dynamical the pairing of negative and positive Liapunov expo- behaviour). nents. See also [Biferale et al., 1997] for a recent J.S.W. Lamb, J.A.G. Roberts/Physica D 112 (1998) 1-39 23 discussion on applying these ideas in models for reversibility surprisingly appears. Examples are found turbulence. in the work of Moser and Webster [Moser and Webster, 1983] who study reversible maps in the con- 5.2. Reversibility in space: Reductions of partial text of normal forms for real surfaces in C e, or in the differential and difference equations work of Teixeira [Teixeira, 1981] on discontinuous ordinary differential equations. In both these cases, in Many physically significant examples of reversible a natural way the problem comes down to the study flows arise by considering partial differential equations of compositions of two involutions (i.e. a reversible (PDEs) involving space and time, and looking for map). These are not isolated examples. A more re- their steady-state or travelling wave solutions. The cent example is provided in the works of [Boukraa resulting ordinary differential equations can then be et al., 1994; Maillard and Rollet, 1994; Meyer et reversible with the independent "time" variable now al,, 1994] which studies birational representations of being played by a spatial coordinate (the reversibility discrete groups generated by involutions: This work here is thus equivalent to a spatial symmetry). An ex- has connections with hyperbolic Coxeter groups (but, ample was presented earlier in Example 3.3 (in fact, interestingly enough, also has physical connections this example was an important motivation for the stud- to statistical mechanical models). Another example ies by Arnol'd and Sevryuk [Arnol'd and Sevryuk, is the study of holomorphic correspondences, which 1986]). we mention next. We follow this with some alge- Once the spatial reversibility is noted, the full braic aspects of the study of reversible dynamical force of the reversible theory can be applied to yield systems. information about the steady-states and/or travelling waves, e.g. their appearance in one-parameter families, their stability and their bifurcations. Also, 5.3.1. Holomorphic correspondences reversible homoclinics describe physically relevant Bullett and co-workers have studied the dynamics solutions such as defects or solitary waves, cf. of complex polynomial correspondences z w-~ z ~ de- Section 4.5. fined implicitly by g(z, z r) = 0 with g polynomial There is a wide range of applications in which re- in both arguments and having complex coefficients versibility arises in the study of steady-states and trav- [Bullett, 1988; Bullett, 1991; Bullett et al., 1986; elling waves in partial differential equations. Examples Bullett and Penrose, 1994b, 1994a] (cf. also [Webster, range from steady-states of reaction diffusion equa- 1996] for related work). In particular, they have tions [Kazarinoff and Yan, 1991; Yan, 1992, 1993b, studied the case with g quadratic in both variables, 1995; Yan and Hwang, 1996b; Yan et al., 1995], to whereby we have a 2-valued map of the Riemann water waves (see [Iooss, 1995a] for a recent survey). sphere with 2-valued inverse. Under suitable con- Sleeman [Sleeman, 1996a, 1996b] recently stressed ditions on g, involutory reversing symmetries arise the importance of reversibility in these kind of mod- naturally in the correspondence dynamics. For exam- els arising in the context of mathematical biology. ple, if g(z, z') = 0 if and only if g(~', ~) = 0, then We note, though, that in many of these applications complex conjugation reverses time so that z ~+ z ~ if the reversible differential equations obtained by re- and only if ~ ~ ~. In the ensuing dynamics induced duction from a PDE are not only reversible, but also by the correspondence, one observes Hamiltonian- Hamiltonian. like behaviour in the form of Siegel discs around symmetric periodic orbits, together with attracting 5.3. Reversibility in abstract mathematical settings and repelling asymmetric periodic orbits (we remark that reversible polynomial mappings of C n to itself Arnol'd [Arnol'd, 1984] already noted that there are necessarily volume-preserving and cannot have are several interesting mathematical contexts in which attractors/repellers [Roberts, 1997]). 24 J.S.W. Lamb, J.A.G. Roberts/Physica D 112 (1998) 1-39

5.3.2. Reversing symmetry groups and algebraic 6. Discussion structures The reversing symmetry group G obtained from In this paper we have presented a compact survey combining the symmetries and reversing symme- of the literature on reversible dynamical systems. Dy- tries of a map f has been previously mentioned in namical systems with time-reversal symmetry have Section 3. From the algebraic structure of revers- certainly received a lot of interest in recent years, cf. ing symmetry groups, interesting consequences can the bibliography in Appendix A, and a lot of inter- be drawn. For instance, Goodson [Goodson, 1996a] esting results have been obtained. However, given the shows that if f2 7~ id, f has a reversing symmetry R importance and relevance of reversible dynamical sys- and the group of symmetries H of f is precisely the tems, there is still a range of problems to be tackled. trivial centralizer {fn:n E 2~}, then it follows from A theme throughout this survey has been the re- algebraic considerations that R 4 = id. This is an ex- lation of reversible dynamical systems to equivariant ample where the nature of f and its symmetry group dynamical systems, on the one hand, and Hamiltonian impose the nature of any reversing symmetry R, cf. dynamical systems on the other hand. The main task also [Goodson, 1996a, 1997]. for the future seems to be bringing the theory of re- It might also be the case that the dynamical systems versible systems to a similar maturity as that of equiv- under consideration form a group with a known struc- ariant and Hamiltonian systems (e.g. many results on ture. Then it may be possible to deduce the structure of reversible systems are obtained in specific problem- possible reversing symmetry groups within this group related contexts). In so doing, the interconnections be- (and so decide, for instance, if a particular system has tween the three classes of systems will also be better any reversing symmetry). An example in which this understood. We conclude by making some further re- can be done is the group of hyperbolic toral automor- marks along these lines. phisms [Baake and Roberts, 1997], which belong to Because reversibility is a symmetry property, and the integer matrix group Gl(2, Z) (in a related prob- the present theory for equivariant dynamical systems lem, the reversing symmetry group can be calculated is powerful and successful, it seems most desirable to for a group of 3D polynomial maps arising in solid adopt an approach that smoothly connects to the the- state physics [Baake and Roberts, 1997; Roberts and ory for equivariant dynamical systems. In particular a Baake, 1994]). More generally, since, e.g. the set of theory for reversible systems could be developed as invertible polynomial maps of C n also form a group, an extension of the equivariant one, in a similar way there may be possibilities to also understand the preva- as reversing symmetry groups are extensions of sym- lence of reversibility in this situation. metry groups. In this way equivariance and reversibil- Another example in which algebraic considerations ity can be studied on an equal footing, as particular arise is in the work of McLachlan et al. [McLachlan cases of space-time symmetry properties. In order to et al., 1995]. They recently pointed out, in the context achieve this, the introduction of a more systematic use of reversible integration methods, that large classes of of group (representation) theory for reversing symme- reversible maps can be viewed as fixed points of anti- try groups would be useful. (A first step in this direc- automorphisms. tion has recently been made in [Lamb and Roberts, In k-symmetric systems analogous algebraic con- 1997]). As we mentioned in Section 3, a historical siderations arise very naturally and nontrivially in the distinction in reversible dynamical systems has been description of the interaction between a map and its among systems with involutory reversing symmetries (reversing) k-symmetry group, cf., e.g. [Lamb and and ones with noninvolutory reversing symmetries. It Quispel, 1995]. The algebraic structures arising in the seems, from a unified symmetry approach, more nat- context of reversing symmetry groups generalize in a ural to distinguish between purely reversible systems nontrivial way to the context of reversing k-symmetry (with only one involutory reversing symmetry) and re- groups. versible equivariant systems. J.S.W. Lamb, J.A.G. Roberts/Physica D 112 (1998) 1-39 25

The relationship between reversible and Hamilto- - will provide encouragement for further studies into nian dynamical systems is a very intriguing one and reversible dynamical systems. deserves further attention. Till now, most Hamiltonian systems of interest in the literature are reversible and most reversible systems of interest are Hamiltonian. Acknowledgements Reversibility is often used as a tool in reversible- Hamiltonian systems to study a particular dynam- We are very grateful to all colleagues and ical phenomenon. Although it is often noted that friends who supplied us with valuable remarks on reversible systems have many features in common the survey and bibliography. Appendix A has its with Hamiltonian systems, this is by no means a roots in the bibliographies of Sevryuk [Sevryuk, guarantee of no differences, cf. e.g. [Rimmer, 1978; 1991b] and Roberts and Quispel [Roberts and Champneys, 1994; Matveyev, 1995a, 1996]. Hence, Quispel, 1992], and was updated and extended in order to understand the dynamics of reversible with the use of Mathematical Reviews (Math- Hamiltonian systems, it will be essential to take the SciNet: http://www.ams .org/) and Science full structure (reversibility as well as the Hamiltonian Citation Index (Bath Information Services (BIDS): properties) of such systems into account. For a http : //www. bids. ac. uk/) "on-line". Without deeper understanding of the similarities of reversible, the electronic availability and search tools of these Hamiltonian and reversible-Hamiltonian dynamical databases, our bibliography would certainly be more systems, more comparative studies of these three incomplete than it is in its present form. categories will be needed. JSWL was supported in part by an EC Human From the literature it appears that KAM-theory, Capital and Mobility Research Fellowship (ERBCH- local bifurcations and homoclinics have been focus BICT941533). JAGR's visit to the workshop on points for the research in reversible dynamical sys- Time-Reversal Symmetry in Dynamical Systems was tems. However, many basic problems in these fields financially supported by the EPSRC (GR/L21051). of research are still open and deserve prompt attention. In particular, we think of the embedding of results for purely reversible systems (e.g. on local Appendix A. A Bibliography on time-reversal bifurcations) into the context of reversible equivari- symmetry in dynamical systems ant (Hamiltonian) systems. Also, investigations into reversible homoclinic bifurcations and reversible het- Aharonov, D., Devaney, R.L., Elias, U., 1997. The eroclinic networks have began only recently [Fiedler dynamics of a piecewise linear map and its smooth and Turaev, 1996; Knobloch, 1997; Reigner, 1998]. approximation, Inter. J. Bifurc. Chaos 7, 351-372. Other future directions of research might include Arnol'd, V.I., 1984. Reversible systems. In: Sagdeev, the study of more general space-time symmetries of R.Z. (Ed.), Nonfinear and Turbulent Processes in ordinary differential equations, and symmetry proper- Physics, vol. 3. Harwood, Chur, pp. 1161-1174. ties of partial differential and difference equations that Arnol'd, V.I., Sevryuk, M.B., 1986. Oscillations and involve transformations of both the dependent and in- bifurcations in reversible systems. In: Sagdeev, R.Z. dependent variables. We note in this respect that the (Ed.), Nonlinear Phenomena in Plasma Physics and possibility for PDEs to be reversible in both space and Hydrodynamics, Mir, Moscow, pp. 31-64. time seems an interesting starting point for such in- Baake, M., Roberts, J.A.G., 1997. Reversing symme- vestigations [4]. try group of Gl(2, Z) and PGI(2, Z) matrices with We feel that our survey illustrates that the field of connections to cat maps and trace maps, J. Phys. A reversible dynamical systems is still in its adolescense, 30, 1549-1573. but enjoying growing interest. We hope that this paper Baesens, C., MacKay, R.S., 1992. Uniformly trav- - and indeed this entire special volume of Physica D elling water waves from a dynamical systems 26 J.S.W. Lamb, J.A.G. Roberts/Physica D 112 (1998) 1-39

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