Hamiltonian Perturbation Theory (And Transition to Chaos) H 4515 H

Hamiltonian Perturbation Theory (and Transition to Chaos) H 4515 H

Hamiltonian Perturbation Theory dle third from a closed interval) designates topological spaces that locally have the structure Rn Cantor dust (and Transition to Chaos) for some n 2 N. Cantor families are parametrized by HENK W. BROER1,HEINZ HANSSMANN2 such Cantor sets. On the real line R one can define 1 Instituut voor Wiskunde en Informatica, Cantor dust of positive measure by excluding around Rijksuniversiteit Groningen, Groningen, each rational number p/q an interval of size The Netherlands 2 2 Mathematisch Instituut, Universiteit Utrecht, ;>0 ;>2 : q Utrecht, The Netherlands Similar Diophantine conditions define Cantor sets Article Outline in Rn. Since these Cantor sets have positive measure their Hausdorff dimension is n.Wheretheunper- Glossary turbed system is stratified according to the co-dimen- Definition of the Subject sion of occurring (bifurcating) tori, this leads to a Can- Introduction tor stratification. One Degree of Freedom Chaos An evolution of a dynamical system is chaotic if its Perturbations of Periodic Orbits future is badly predictable from its past. Examples of Invariant Curves of Planar Diffeomorphisms non-chaotic evolutions are periodic or multi-periodic. KAM Theory: An Overview A system is called chaotic when many of its evolutions Splitting of Separatrices are. One criterion for chaoticity is the fact that one of Transition to Chaos and Turbulence the Lyapunov exponents is positive. Future Directions Diophantine condition, Diophantine frequency vector Bibliography A frequency vector ! 2 Rn is called Diophantine if there are constants >0and>n 1with Glossary jhk;!ij for all k 2 Znnf0g : Bifurcation In parametrized dynamical systems a bifur- jkj cation occurs when a qualitative change is invoked by a change of parameters. In models such a qualitative The Diophantine frequency vectors satisfying this con- change corresponds to transition between dynamical dition for fixed and form a Cantor set of half lines. regimes. In the generic theory a finite list of cases is ob- As the Diophantine parameter tends to zero (while tained, containing elements like ‘saddle-node’, ‘period remains fixed), these half lines extend to the origin. doubling’, ‘Hopf bifurcation’ and many others. The complement in any compact set of frequency vec- Cantor set, Cantor dust, Cantor family, Cantor strati- tors satisfying a Diophantine condition with fixed fication Cantor dust is a separable locally compact space has a measure of order O()as # 0. that is perfect, i. e. every point is in the closure of its Integrable system A Hamiltonian system with n degrees complement, and totally disconnected. This deter- of freedom is (Liouville)-integrable if it has n func- mines Cantor dust up to homeomorphisms. The term tionally independent commuting integrals of motion. Cantor set (originally reserved for the specific form of Locally this implies the existence of a torus action, Cantor dust obtained by repeatedly deleting the mid- a feature that can be generalized to dissipative sys- 4516 H Hamiltonian Perturbation Theory (and Transition to Chaos)

tems. In particular a mapping is integrable if it can be planes in !-space and, by means of the frequency interpolated to become the stroboscopic mapping of mapping, also in phase space. The smallest number aflow. jhjDjh1jCCjhk j is the order of the resonance. KAM theory Kolmogorov–Arnold–Moser theory is the Diophantine conditions describe a measure-theoret- perturbation theory of (Diophantine) quasi-periodic ically large complement of a neighborhood of the tori for nearly integrable Hamiltonian systems. In the (dense!) set of all resonances. format of quasi-periodic stability, the unperturbed and Separatrices Consider a hyperbolic equilibrium, fixed or perturbed system, restricted to a Diophantine Cantor periodic point or invariant torus. If the stable and un- set, are smoothly conjugated in the sense of Whitney. stable manifolds of such hyperbolic elements are codi- This theory extends to the world of reversible, volume- mension one immersed manifolds, then they are called preserving or general dissipative systems. In the latter separatrices, since they separate domains of phase KAM theory gives rise to families of quasi-periodic at- space, for instance, basins of attraction. tractors. KAM theory also applies to torus bundles, in Singularity theory AfunctionH : Rn ! R has a crit- which case a global Whitney smooth conjugation can ical point z 2 Rn where DH(z)vanishes.Inlo- be proven to exist, that keeps track of the geometry. In cal coordinates we may arrange z D 0 (and simi- an appropriate sense invariants like monodromy and larly that it is mapped to zero as well). Two germs Chern classes thus also can be defined in the nearly K :(Rn; 0) ! (R; 0) and N :(Rn ; 0) ! (R; 0) rep- integrable case. Also compare with Kolmogorov– resent the same function H locally around z if and only Arnold–Moser (KAM) Theory. if there is a diffeomorphism on Rn satisfying Nearly integrable system In the setting of perturbation theory, a nearly integrable system is a perturbation of N D K ı : an integrable one. The latter then is an integrable approximation of the former. See an above item. The corresponding equivalence class is called a singu- Normal form truncation Consider a dynamical system larity. in the neighborhood of an equilibrium point, a fixed or Structurally stable A system is structurally stable if it is periodic point, or a quasi-periodic torus, reducible to topologically equivalent to all nearby systems, where Floquet form. Then Taylor expansions (and their ana- ‘nearby’ is measured in an appropriate topology on the logues) can be changed gradually into normal forms, space of systems, like the Whitney Ck-topology [72]. that usually reflect the dynamics better. Often these A family is structurally stable if for every nearby family display a (formal) torus symmetry, such that the nor- there is a re-parametrization such that all correspond- mal form truncation becomes an integrable approxi- ing systems are topologically equivalent. mation, thus yielding a perturbation theory setting. See above items. Also compare with Normal Forms in Definition of the Subject Perturbation Theory. Persistent property In the setting of perturbation theory, The fundamental problem of mechanics is to study Hamil- a property is persistent whenever it is inherited from tonian systems that are small perturbations of integrable the unperturbed to the perturbed system. Often the systems. Also, perturbations that destroy the Hamiltonian perturbation is taken in an appropriate topology on the character are important, be it to study the effect of a small space of systems, like the Whitney Ck-topology [72]. amount of friction, or to further the theory of dissipa- Perturbation problem In perturbation theory the unper- tive systems themselves which surprisingly often revolves turbed systems usually are transparent regarding their around certain well-chosen Hamiltonian systems. Fur- dynamics. Examples are integrable systems or normal thermore there are approaches like KAM theory that his- form truncations. In a perturbation problem things are torically were first applied to Hamiltonian systems. Typi- arranged in such a way that the original system is well- cally perturbation theory explains only part of the dynam- approximated by such an unperturbed one. This ar- ics, and in the resulting gaps the orderly unperturbed mo- rangement usually involves both changes of variables tion is replaced by random or chaotic motion. and scalings. Resonance If the frequencies of an invariant torus with Introduction multi- or conditionally periodic flow are rationally dependent, this torus divides into invariant sub-tori. We outline perturbation theory from a general point of Such resonances hh;!iD0, h 2 Zk ,definehyper- view, illustrated by a few examples. Hamiltonian Perturbation Theory (and Transition to Chaos) H 4517

The Perturbation Problem a vector field, we obtain The aim of perturbation theory is to approximate a given t˙ D 1 dynamical system by a more familiar one, regarding the former as a perturbation of the latter. The problem then x˙ D y is to deduce certain dynamical properties from the unper- dV y˙ D (x) C " f (x; y; t) ; turbed to the perturbed case. dx What is familiar may or may not be a matter of taste, at R3 Df; ; g least it depends a lot on the dynamical properties of one’s now on the generalized phase space t x y .In interest. Still the most frequently used unperturbed sys- the case where the t-dependence is periodic, we can take S1 R2 tems are: for (generalized) phase space. Linear systems Remark Integrable Hamiltonian systems, compare with Dy- A small variation of the above driven system concerns namics of Hamiltonian Systems and references therein a parametrically forced oscillator like Normal form truncations, compare with Normal Forms in Perturbation Theory and references therein x¨ C (!2 C " cos t)sinx D 0 ; Etc. which happens to be entirely in the world of Hamilto- To some extent the second category can be seen as a special nian systems. case of the third. To avoid technicalities in this section we It may be useful to study the Poincaré or period map- assume all systems to be sufficiently smooth, say of class ping of such time periodic systems, which happens to C1 or real analytic. Moreover in our considerations " will be a mapping of the plane. We recall that in the Hamil- be a real parameter. The unperturbed case always corre- tonian cases this mapping preserves area. For general sponds to " D 0 and the perturbed one to " ¤ 0or">0. reference in this direction see, e. g., [6,7,27,66]. Examples of Perturbation Problems To begin with There are lots of variations and generalizations. One ex- consider the autonomous differential equation ampleisthesolarsystem,wheretheunperturbedcasecon- dV sists of a number of uncoupled two-body problems con- x¨ C "x˙ C (x) D 0 ; dx cerning the Sun and each of the planets, and where the in- teraction between the planets is considered as small [6,9, modeling an oscillator with small damping. Rewriting this 107,108]. equation of motion as a planar vector field Remark x˙ D y dV One variation is a restriction to fewer bodies, for y˙ D"y (x) ; example only three. Examples of this are systems dx like Sun–Jupiter–Saturn, Earth–Moon–Sun or Earth– 1 2 we consider the energy H(x; y) D 2 y C V(x). For " D 0 Moon–Satellite. the system is Hamiltonian with Hamiltonian function H. Often Sun, Moon and planets are considered as point Indeed, generally we have H˙ (x; y) D"y2,implyingthat masses, in which case the dynamics usually are mod- for ">0 there is dissipation of energy. Evidently for " ¤ 0 eled as a Hamiltonian system. It is also possible to ex- the system is no longer Hamiltonian. tend this approach taking tidal effects into account, The reader is invited to compare the phase portraits of which have a non-conservative nature. the cases " D 0and">0forV(x) Dcos x (the pendu- The Solar System is close to resonance, which makes 1 2 1 4 lum) or V(x) D 2 x C 24 bx (Duffing). application of KAM theory problematic. There exist, Another type of example is provided by the non-au- however, other integrable approximations that take tonomous equation resonance into account [3,63]. dV x¨ C (x) D " f (x; x˙; t) ; Quite another perturbation setting is local, e. g., near an dx equilibrium point. To fix thoughts consider which can be regarded as the equation of motion of an oscillator with small external forcing. Again rewriting as x˙ D Ax C f (x) ; x 2 Rn 4518 H Hamiltonian Perturbation Theory (and Transition to Chaos)

with A 2 gl(n; R), f (0) D 0andDx f (0) D 0. By the scal- point x0 2 ˙ \ 0. By transversality, for j"j small, a local ing x D "x¯ we rewrite the system to Poincaré map P" : ˙ ! ˙ is well-deﬁned for (1). Observe that ﬁxed points x" of P" correspond to periodic orbits " x¯˙ D Ax¯ C "g(x¯) : of (1). We now have, again as another direct consequence of the implicit function theorem. So, here we take the linear part as an unperturbed system. Observe that for small " the perturbation is small on Theorem 2 (Persistence of periodic orbits) In the above a compact neighborhood of x¯ D 0. assume that This setting also has many variations. In fact, any normal form approximation may be treated in this P0(x0) D x0 and Dx P0(x0) has no eigenvalue 1 : way Normal Forms in Perturbation Theory. Then the Then there exists a local arc " 7! x(") with x(0) D x such normalized truncation forms the unperturbed part and the 0 that higher order terms the perturbation.

Remark In the above we took the classical viewpoint P"(x(")) x" : which involves a perturbation parameter controlling the size of the perturbation. Often one can generalize this by Remark considering a suitable topology (like the Whitney topolo- gies) on the corresponding class of systems [72]. Also Often the conditions of Theorem 2 are not easy to ver- compare with Normal Forms in Perturbation Theory, ify.SometimesitisusefulheretouseFloquetTheory, Kolmogorov–Arnold–Moser (KAM) Theory and Dy- see [97]. In fact, if T0 is the period of 0 and ˝0 its namics of Hamiltonian Systems. Floquet matrix, then Dx P0(x0) D exp(T0˝0). The format of the Theorems 1 and 2 with the pertur- Questions of Persistence bation parameter " directly allows for algorithmic approaches. One way to proceed is by perturbation series, What are the kind of questions perturbation theory asks? leading to asymptotic formulae that in the real analytic A large class of questions concerns the persistence of cer- setting have positive radius of convergence. In the latter tain dynamical properties as known for the unperturbed case the names of Poincaré and Lindstedt are associated case. To ﬁx thoughts we give a few examples. with the method, cf. [10]. To begin with consider equilibria and periodic orbits. Also numerical continuation programmes exist based So we put on the Newton method. The Theorems 1 and 2 can be seen as special cases of x˙ D f (x;") ; x 2 Rn ;"2 R ; (1) aageneraltheoremfornormally hyperbolic invariant for a map f : RnC1 ! Rn. Recall that equilibria are given manifolds [73], Theorem 4.1. In all cases a contraction by the equation f (x;") D 0. The following theorem that principle on a suitable Banach space of graphs leads to continues equilibria in the unperturbed system for " ¤ 0, persistence of the invariant dynamical object. is a direct consequence of the implicit function theorem. This method in particular yields existence and persis- Theorem 1 (Persistence of equilibria) Suppose that tence of stable and unstable manifolds [53,54]. f (x0; 0) D 0 and that Another type of dynamics subject to perturbation theory Dx f (x0; 0) has maximal rank : is quasi-periodic. We emphasize that persistence of (Dio- phantine) quasi-periodic invariant tori occurs both in the Then there exists a local arc " 7! x(") with x(0) D x0 such conservative setting and in many others, like in the re- that versible and the general (dissipative) setting. In the latter case this leads to persistent occurrence of families of quasi- " ;" : f (x( ) ) 0 periodic attractors [125]. These results are in the domain of Kolmogorov–Arnold–Moser (KAM)theory.Fordetails Periodic orbits can be approximated in a similar way. In- we refer to Sect. “KAM Theory: An Overview” below or deed, let the system (1)for D 0 have a periodic orbit 0. to [24], Kolmogorov–Arnold–Moser (KAM) Theory, Let ˙ be a local transversal section of 0 and P0 : ˙ ! ˙ the former reference containing more than 400 references the corresponding Poincaré map. Then P0 has a ﬁxed in this area. Hamiltonian Perturbation Theory (and Transition to Chaos) H 4519

Remark plies perpetual adiabatic stability of the action variables. In contrast, for n 3 the Lagrangian tori have codimen- Concerning the Solar System, KAM theory always has sion n 1 > 1 in the energy hypersurfaces and evolution aimed at proving that it contains many quasi-periodic curves may escape. motions, in the sense of positive Liouville measure. This actually occurs in the case of so-called Arnold This would imply that there is positive probability that diffusion. The literature on this subject is immense, and a given initial condition lies on such a stable quasi-pe- we here just quote [5,9,93,109], for many more references riodic motion [3,63], however, also see [85]. see [24]. Another type of result in this direction compares the Next we consider the motion in a neighborhood of distance of certain individual solutions of the perturbed the KAM tori, in the case where the systems are real ana- and the unperturbed system, with coinciding initial lytic or at least Gevrey smooth. For a definition of Gevrey conditions over time scales that are long in terms of ". regularity see [136]. First we mention that, measured in Compare with [24]. terms of the distance to the KAM torus, nearby evolution curves generically stay nearby over a superexponentially Apart from persistence properties related to invariant long time [102,103]. This property often is referred to as manifolds or individual solutions, the aim can also be superexponential stickiness of the KAM tori, see [24]for to obtain a more global persistence result. As an exam- more references. pleofthiswementiontheHartman–GrobmanTheorem, Second, nearly integrable Hamiltonian systems, in e. g., [7,116,123]. Here the setting once more is terms of the perturbation size, generically exhibit expo- nentially long adiabatic stability of the action variables, see x˙ D Ax C f (x) ; x 2 Rn ; e. g. [15,88,89,90,93,103,109,110,113,120], Nekhoroshev Theory and many others, for more references see [24]. with A 2 gl(n; R), f (0) D 0andDx f (0) D 0. Now we as- This property is referred to as the Nekhoroshev estimate sume A to be hyperbolic (i. e., with no purely imaginary or the Nekhoroshev theorem. For related work on pertur- eigenvalues). In that case the full system, near the origin, bations of so-called superintegrable systems, also see [24] is topologically conjugated to the linear system x˙ D Ax. and references therein. Therefore all global, qualitative properties of the unperturbed (linear) system are persistent under perturbation to the full system. For details on these notions see the above Chaos references, also compare with, e. g., [30]. In the previous subsection we discussed persistent and It is said that the hyperbolic linear system x˙ D Ax is some non-persistent features of dynamical systems un- (locally) structurally stable. This kind of thinking was in- der small perturbations. Here we discuss properties re- troduced to the dynamical systems area by Thom [133], lated to splitting of separatrices, caused by generic pertur- with a first, successful application to catastrophe theory. bations. For further details, see [7,30,69,116]. A first example was met earlier, when comparing the pendulum with and without (small) damping. The unper- General Dynamics turbed system is the undamped one and this is a Hamil- We give a few remarks on the general dynamics in a neigh- tonian system. The perturbation however no longer is borhood of Hamiltonian KAM tori. In particular this con- Hamiltonian. We see that the equilibria are persistent, as cerns so-called superexponential stickiness of the KAM tori they should be according to Theorem 1, but that none of and adiabatic stability of the action variables, involving the the periodic orbits survives the perturbation. Such quali- so-called Nekhoroshev estimate. tative changes go with perturbing away from the Hamilto- To begin with, emphasize the following difference be- nian setting. tween the cases n D 2andn 3 in the classical KAM the- Similar examples concern the breaking of a certain orem of Subsect. “Classical KAM Theory”. For n D 2the symmetry by the perturbation. The latter often occurs in level surfaces of the Hamiltonian are three-dimensional, the case of normal form approximations. Then the nor- while the Lagrangian tori have dimension two and hence malized truncation is viewed as the unperturbed system, codimension one in the energy hypersurfaces. This means which is perturbed by the higher order terms. The trunca- that for open sets of initial conditions, the evolution curves tion often displays a reasonable amount of symmetry (e. g., are forever trapped in between KAM tori, as these tori foli- toroidal symmetry), which generically is forbidden for the ate over nowhere dense sets of positive measure. This im- class of systems under consideration, e. g. see [25]. 4520 H Hamiltonian Perturbation Theory (and Transition to Chaos)

Hamiltonian Perturbation Theory (and Transition to Chaos), Figure 1 ! 1 "> Chaos in the parametrically forced pendulum. Left: Poincaré map P!;" near the 1 : 2 resonance D 2 and for 0 not too small. Right: A dissipative analogue

To fix thoughts we reconsider the conservative ex- more sophisticated tools see [65]. Often this leads to el- ample liptic (Abelian) integrals. In nearly integrable systems chaos can occur. This 2 x¨ C (! C " cos t)sinx D 0 fact is at the heart of the celebrated non-integrability of the three-body problem as addressed by Poincaré [12,59, of the previous section. The corresponding (time depen- 107,108,118]. A long standing open conjecture is that the dent, Hamiltonian [6]) vector field reads clouds of points as visible in Fig. 1, left, densely fill sets of positive area, thereby leading to ergodicity [9]. t˙ D 1 In the case of dissipation, see Fig. 1,right,wecon- x˙ D y jecture the occurrence of a Hénon-like strange attrac- y˙ D(!2 C " cos t)sinx : tor [14,22,126].

2 2 Remark Let P!;" : R ! R be the corresponding (area-preserving) Poincaré map. Let us consider the unperturbed map The persistent occurrence of periodic points of a given P!;0 which is just the flow over time 2 of the free rotation number follows from the Poincaré–Birkhoff pendulum x¨ C !2 sin x D 0. Such a map is called inte- fixed point theorem [74,96,107], i. e., on topological grable, since it is the stroboscopic map of a two-dimen- grounds. sional vector field, hence displaying the R-symmetry of The above arguments are not restricted to the conser- a flow. When perturbed to the nearly integrable case " ¤ 0, vative setting, although quite a number of unperturbed this symmetry generically is broken. We list a few of the systems come from this world. Again see Fig. 1. generic properties for such maps [123]:

The homoclinic and heteroclinic points occur at One Degree of Freedom transversal intersections of the corresponding stable Planar Hamiltonian systems are always integrable and the and unstable manifolds. orbits are given by the level sets of the Hamiltonian func- The periodic points of period less than a given bound tion. This still leaves room for a perturbation theory. The are isolated. recurrent dynamics consists of periodic orbits, equilibria This means generically that the separatrices split and that and asymptotic trajectories forming the (un)stable man- the resonant invariant circles filled with periodic points ifolds of unstable equilibria.Theequilibriaorganizethe with the same (rational) rotation number fall apart. In phase portrait, and generically all equilibria are elliptic any concrete example the issue remains whether or not (purely imaginary eigenvalues) or hyperbolic (real eigen- it satisfies appropriate genericity conditions. One method values), i. e. there is no equilibrium with a vanishing eigen- tocheckthisisduetoMelnikov,compare[66,137], for value. If the system depends on a parameter such van- Hamiltonian Perturbation Theory (and Transition to Chaos) H 4521 ishing eigenvalues may be unavoidable and it becomes where a; b; c 2 R are nonzero constants, for instance possible that the corresponding dynamics persist under a D b D c D 1. Note that this is a completely different un- perturbations. folding of the parabolic equilibrium at the origin. A closer Perturbations may also destroy the Hamiltonian char- look at the phase portraits and in particular at the Hamil- acter of the flow. This happens especially where the start- tonian function of the Hamiltonian pitchfork bifurcation ing point is a dissipative planar system and e. g. a scaling reveals the symmetry x 7!x of the Duffing oscillator. leads for " D 0 to a limiting Hamiltonian flow. The per- This suggests the addition of the non-symmetric term x. turbation problem then becomes twofold. Equilibria still The resulting two-parameter family persist by Theorem 1 and hyperbolic equilibria moreover persist as such, with the sum of eigenvalues of order O("). 1 2 1 4 1 2 H;(x; y) D y C bx C x C x Also for elliptic eigenvalues the sum of eigenvalues is of 2 24 2 O order (") after the perturbation, but here this number of Hamiltonian systems is indeed structurally stable. This measures the dissipation whence the equilibrium becomes implies not only that all equilibria of a Hamiltonian per- (weakly) attractive for negative values and (weakly) unsta- turbation of the Duffing oscillator have a local flow equiv- ble for positive values. The one-parameter families of pe- alent to the local flow near a suitable equilibrium in this riodic orbits of a Hamiltonian system do not persist under two-parameter family, but that every one-parameter fam- dissipative perturbations, the very fact that they form family of Z2-symmetric Hamiltonian systems that is a pertur- ilies imposes the corresponding fixed point of the Poincaré bation of (2) has equivalent dynamics. For more details mapping to have an eigenvalue one and Theorem 2 does see [36] and references therein. not apply. Typically only finitely many periodic orbits sur- This approach applies mutatis mutandis to every non- vive a dissipative perturbation and it is already a difficult degenerate planar singularity, cf. [69,130]. At an equilib- task to determine their number. rium all partial derivatives of the Hamiltonian vanish and the resulting singularity is called non-degenerate if it has Hamiltonian Perturbations finite multiplicity, which implies that it admits a versal unfolding H with finitely many parameters. The family of The Duffing oscillator has the Hamiltonian function Hamiltonian systems defined by this versal unfolding con- 1 1 1 tains all possible (local) dynamics that the initial equilib- H(x; y) D y2 C bx4 C x2 (2) 2 24 2 rium may be perturbed to. Imposing additional discrete symmetries is immediate, the necessary symmetric versal where b is a constant distinguishing the two cases b D˙1 unfolding is obtained by averaging and is a parameter. Under variation of the parameter the X 1 equations of motion HG D H ı g jGj g2G x˙ D y 1 along the orbits of the symmetry group G. y˙ D bx3 x 6 display a Hamiltonian pitchfork bifurcation, supercritical Dissipative Perturbations for positive b and subcritical in case b is negative. Corre- In a generic dissipative system all equilibria are hyperbolic. spondingly, the linearization at the equilibrium x D 0of Qualitatively, i. e. up to topological equivalence, the lo- the anharmonic oscillator D 0 is given by the matrix cal dynamics is completely determined by the number of eigenvalues with positive real part. Those hyperbolic equi- 01 libria that can appear in Hamiltonian systems (the eigen- 00 values forming pairs ˙) do not play an important role. Rather, planar Hamiltonian systems become important as whence this equilibrium is parabolic. a tool to understand certain bifurcations triggered off by The typical way in which a parabolic equilibrium bi- non-hyperbolic equilibria. Again this requires the system furcates is the center-saddle bifurcation. Here the Hamil- to depend on external parameters. tonian reads The simplest example is the Hopf bifurcation, a co-di- 1 1 mension one bifurcation where an equilibrium loses sta- H(x; y) D ay2 C bx3 C cx (3) bility as the pair of eigenvalues crosses the imaginary axis, 2 6 4522 H Hamiltonian Perturbation Theory (and Transition to Chaos)

1 2 say at ˙i. At the bifurcation the linearization is a Hamil- All Hamiltonian functions H D 2 y C V(x) which have tonian system with an elliptic equilibrium (the co-dimen- an interpretation “kinetic C potential energy” are re- sion one bifurcations where a single eigenvalue crosses the versible, and in general the class of reversible systems is imaginary axis through 0 do not have a Hamiltonian lin- positioned between the class of Hamiltonian systems and earization). This limiting Hamiltonian system has a one- the class of dissipative systems. A guiding example is the parameter family of periodic orbits around the equilib- perturbed Duffing oscillator (with the roles of x and y ex- rium, and the non-linear terms determine the fate of these changed so that (4) remains the reversing symmetry) periodic orbits. The normal form of order three reads 1 x˙ D y3 y C "xy x˙ Dy 1 C b(x2 C y2) C x C a(x2 C y2) 6 y˙ D x x˙ D x 1 C b(x2 C y2) C y C a(x2 C y2) that combines the Hamiltonian character of the equilib- and is Hamiltonian if and only if (; a) D (0; 0). The sign rium at the origin with the dissipative character of the two of the coefficient distinguishes between the supercritical other equilibria. Note that all orbits outside the homoclinic case a > 0, in which there are no periodic orbits coexisting loop are periodic. with the attractive equilibria (i. e. when <0) and one at- There are two ways in which the reversing symme- tracting periodic orbit for each >0 (coexisting with the try (4) imposes a Hamiltonian character on the dynamics. unstable equilibrium), and the subcritical case a < 0, in An equilibrium that lies on the symmetry line fy D 0g has which the family of periodic orbits is unstable and coexists a linearization that is itself a reversible system and conse- with the attractive equilibria (with no periodic orbits for quently the eigenvalues are subject to the same constraints parameters >0). As ! 0 the family of periodic orbits as in the Hamiltonian case. (For equilibria z0 that do not lie shrinks down to the origin, so also this Hamiltonian fea- on the symmetry line the reflection R(z0)isalsoanequi- ture is preserved. librium, and it is to the union of their eigenvalues that Equilibria with a double eigenvalue 0 need two param- these constraints still apply.) Furthermore, every orbit that eters to persistently occur in families of dissipative sys- crosses fy D 0g more than once is automatically periodic, tems. The generic case is the Takens–Bogdanov bifurca- and these periodic orbits form one-parameter families. In tion. Here the linear part is too degenerate to be helpful, particular, elliptic equilibria are still surrounded by peri- but the nonlinear Hamiltonian system defined by (3)with odic orbits. a D 1 D c and b D3 provides the periodic and hetero- The dissipative character of a reversible system is most clinic orbit(s) that constitute the nontrivial part of the bi- obvious for orbits that do not cross the symmetry line. furcation diagram. Where discrete symmetries are present, Here R merely maps the orbit to a reflected counterpart. e. g. for equilibria in dissipative systems originating from The above perturbed Duffing oscillator exemplifies that other generic bifurcations, the limiting Hamiltonian sys- the character of an orbit crossing fy D 0g exactly once is tem exhibits that same discrete symmetry. For more details undetermined. While the homoclinic orbit of the saddle see [54,66,82] and references therein. at the origin has a Hamiltonian character, the heteroclinic The continuation of certain periodic orbits from an orbits between the other two equilibria behave like in a dis- unperturbed Hamiltonian system under dissipative per- sipative system. turbation can be based on Melnikov-like methods, again see [66,137]. As above, this often leads to Abelian integrals, Perturbations of Periodic Orbits for instance to count the number of periodic orbits that branch off. The perturbation of a one-degree-of-freedom system by a periodic forcing is a perturbation that changes the phase space. Treating the time variable t as a phase space vari- Reversible Perturbations able leads to the extended phase space S1 R2 and equi- A dynamical system that admits a reflection symmetry R libria of the unperturbed system become periodic orbits, mapping trajectories '(t; z0)totrajectories'(t; R(z0)) is inheriting the normal behavior. Furthermore introducing called reversible. In the planar case we may restrict to the an action conjugate to the “angle” t yields a Hamiltonian reversing reflection system in two degrees of freedom. While the one-parameter families of periodic orbits R : R2 ! R2 merely provide the typical recurrent motion in one degree (4) (x; y) 7! (x; y) : of freedom, they form special solutions in two or more de- Hamiltonian Perturbation Theory (and Transition to Chaos) H 4523 grees of freedom. Arcs of elliptic periodic orbits are partic- tonian. The arrangement consists of changes of vari- ularly instructive. Note that these occur generically in both ables and rescaling. An early example of this is the Bog- the Hamiltonian and the reversible context. danov–Takens bifurcation [131,132]. For other examples regarding nilpotent singularities, see [23,40] and refer- Conservative Perturbations ences therein. To fix thoughts, consider families of planar maps and Along the family of elliptic periodic orbits a pair e˙i˝ of let the unperturbed Hamiltonian part contain a center Floquet multipliers passes regularly through roots of unity. (possibly surrounded by a homoclinic loop). The question Generically this happens on a dense set of parameter val- then is which of these persist when adding the dissipative ues, but for fixed denominator q in e˙i˝ D e˙2ip/q the perturbation. corresponding energy values are isolated. The most im- Usually only a definite finite number persists. As in portant of such resonances are those with small denomi- Subsect. “Chaos”, a Melnikov function can be invoked nators q. here, possibly again leading to elliptic (Abelian) integrals, For q D 1 generically a periodic center-saddle bifurca- Picard Fuchs equations, etc. For details see [61,124]and tion takes place where an elliptic and a hyperbolic periodic references therein. orbit meet at a parabolic periodic orbit. No periodic orbit remains under further variation of a suitable parameter. The generic bifurcation for q D 2 is the period-dou- Invariant Curves of Planar Diffeomorphisms bling bifurcation where an elliptic periodic orbit turns hy- This section starts with general considerations on cir- perbolic (or vice versa) when passing through a parabolic cle diffeomorphisms, in particular focusing on persistence periodic orbit with Floquet multipliers 1. Furthermore, properties of quasi-periodic dynamics. Our main refer- a family of periodic orbits with twice the period emerges ences are [2,24,29,31,70,71,139,140]. For a definition of ro- from the parabolic periodic orbit, inheriting the normal tation number, see [58]. After this we turn to area preserv- linear behavior from the initial periodic orbit. ingmapsofanannuluswherewediscussMoser’stwist In case q D 3, and possibly also for q D 4, generically map theorem [104], also see [24,29,31]. The section is con- two arcs of hyperbolic periodic orbits emerge, both with cluded by a description of the holomorphic linearization of three (resp. four) times the period. One of these extends afixedpointinaplanarmap[7,101,141,142]. for lower and the other for higher parameter values. The Our main perspective will be perturbative, where we initial elliptic periodic orbit momentarily loses its stability consider circle maps near a rigid rotation. It turns out that duetotheseapproachingunstableorbits. generally parameters are needed for persistence of quasi- Denominators q 5 (and also the second possibility periodicity under perturbations. In the area preserving set- for q D 4) lead to a pair of subharmonic periodic orbits ting we consider perturbations of a pure twist map. of q times the period emerging either for lower or for higher parameter values. This is (especially for large q) Circle Maps comparable to the behavior at Diophantine e˙i˝ where a family of invariant tori emerges, cf. Sect. “Invariant We start with the following general problem. Given a two- Curves of Planar Diffeomorphisms”below. parameter family ˙i˝ For a single pair e of Floquet multipliers this 1 1 behavior is traditionally studied for the (iso-energetic) P˛;" : T ! T ; x 7! x C 2˛ C "a(x;˛;") Poincaré-mapping, cf. [92] and references therein. How- of circle maps of class C1. It turns out to be convenient to ever, the above description remains true in higher dimen- view this two-parameter family as a one-parameter family sions, where additionally multiple pairs of Floquet mul- of maps tipliers may interact. An instructive example is the La- grange top, the sleeping motion of which is gyroscopically 1 1 P" : T [0; 1] ! T [0; 1]; stabilized after a periodic Hamiltonian Hopf bifurcation; see [56]formoredetails. (x;˛) 7! (x C 2˛ C "a(x;˛;");˛)

of the cylinder. Note that the unperturbed system P0 is Dissipative Perturbations a family of rigid circle rotations, viewed as a cylinder map, There exists a large class of local bifurcations in the dis- where the individual map P˛;0 has rotation number ˛.The sipative setting, that can be arranged in a perturbation question now is what will be the fate of this rigid dynamics theory setting, where the unperturbed system is Hamil- for 0 ¤j"j1. 4524 H Hamiltonian Perturbation Theory (and Transition to Chaos)

The classical way to address this question is to look for a conjugation ˚", that makes the following diagram com- mute

P" T 1 [0; 1] ! T 1 [0; 1] " ˚" " ˚" P T 1 [0; 1] !0 T 1 [0; 1] ;

i. e., such that

P" ı ˚" D ˚" ı P0 :

Due to the format of P" we take ˚" as a skew map

˚"(x;˛) D (x C "U(x;˛;");˛C "(˛; ")) ;

which leads to the nonlinear equation

U(x C 2˛;˛;") U(x;˛;") D 2(˛; ") C a (x C "U(x;˛;");˛C "(˛; ");") Hamiltonian Perturbation Theory (and Transition to Chaos), Figure 2 in the unknown maps U and .Expandinginpowersof" Skew cylinder map, conjugating (Diophantine) quasi-periodic in- and comparing at lowest order yields the linear equation variant circles of P0 and P"

U0(x C 2˛;˛) U0(x;˛) D 20(˛) C a0(x;˛) Theorem 3 (Circle Map Theorem) For suﬃciently which can be directly solved by Fourier series. Indeed, small and for the perturbation "asuﬃcientlysmall writing in the C1-topology, there exists a C1 transformation 1 1 X ˚" : T [0; 1] ! T [0; 1], conjugating the restriction ;˛ D ˛ ikx ; a0(x ) a0k( )e P0j[0;1]; to a subsystem of P". k2Z X Theorem 3 in the present structural stability formula- U (x;˛) D U (˛)eikx 0 0k tion (compare with Fig. 2) is a special case of the results 2 k Z in [29,31]. We here speak of quasi-periodic stability.For

we ﬁnd 0 D1/(2)a00 and earlier versions see [2,9]. a (˛) Remark U (˛) D 0k : 0k 2ik˛ e 1 Rotation numbers are preserved by the map ˚" and ir- It follows that in general a formal solution exists if and rational rotation numbers correspond to quasi-period- only if ˛ 2 R n Q. Still, the accumulation of e2ik˛ 1 icity. Theorem 3 thus ensures that typically quasi-peri- on 0 leads to the celebrated small divisors [9,108], also odicity occurs with positive measure in the parameter see [24,29,31,55]. space. Note that since Cantor sets are perfect, quasi-pe- The classical solution considers the following Dio- riodicity typically has a non-isolated occurrence. phantine non-resonance conditions. Fixing >2and The map ˚" has no dynamical meaning inside the gaps. >0consider˛ 2 [0; 1] such that for all rationals p/q The gap dynamics in the case of circle maps can be il- ˇ ˇ lustrated by the Arnold family of circle maps [2,7,58], ˇ ˇ ˇ p ˇ given by ˇ˛ ˇ q : (5) q P˛;"(x) D x C 2˛ C " sin x This subset of such ˛sisdenotedby[0; 1]; and is well- known to be nowhere dense but of large measure as >0 which exhibits a countable union of open resonance gets small [115]. Note that Diophantine numbers are tongues where the dynamics is periodic, see Fig. 3.Note irrational. that this map only is a diﬀeomorphism for j"j < 1. Hamiltonian Perturbation Theory (and Transition to Chaos) H 4525

Remark The celebrated synchronization of Huygens’ clocks [77] is related to a 1 : 1 resonance, meaning that the corresponding Poincaré map would have its parameters in the main tongue with rotation number 0. Compare with Fig. 3. There exist direct generalizations to cases with n-oscillators (n 2 N), leading to families of invariant n-tori carrying quasi-periodic flow, forming a nowhere dense set of positive measure. An alteration with resonance occurs as roughly sketched in Fig. 3. In higher dimension the gap dynamics, apart from periodicity, also can contain strange attractors [112,126]. We shall come back to this subject in a later section. Hamiltonian Perturbation Theory (and Transition to Chaos), Figure 3 " Arnold resonance tongues; for 1 the maps are endomorphic Area-Preserving Maps The above setting historically was preceded by an area pre- We like to mention that non-perturbative versions of serving analogue [104] that has its origin in the Hamilto- Theorem 3 have been proven in [70,71,139]. nian dynamics of frictionless mechanics. 2 For simplicity we formulated Theorem 3 under Let R nf(0; 0)g be an annulus, with sympectic 1 C1-regularity, noting that there exist many ways to polar coordinates (';I) 2 T K,whereK is an interval. generalize this. On the one hand there exist Ck-versions Moreover, let D d' ^ dI be the area form on . for finite k and on the other hand there exist fine tun- We consider a -preserving smooth map P" : ! ings in terms of real-analytic and Gevrey regularity. For of the form details we refer to [24,31] and references therein. This P (';I) D (' C 2˛(I); I) C O(") ; same remark applies to other results in this section and " in Sect. “KAM Theory: An Overview”onKAM theory. whereweassumethatthemapI 7! ˛(I)isa(local)diffeomorphism. This assumption is known as the twist condi- A possible application of Theorem 3 runs as follows. Con- tion and P" is called a twist map. For the unperturbed case sider a system of weakly coupled Van der Pol oscillators " D 0 we are dealing with a pure twist map and its dynamics are comparable to the unperturbed family of cylinder y¨1 C c1 y˙1 C a1 y1 C f1(y1; y˙1) D "g1(y1; y2; y˙1; y˙2) maps as met in Subsect. “Circle Maps”. Indeed it is again y¨2 C c2 y˙2 C a2 y2 C f2(y2; y˙2) D "g2(y1; y2; y˙1; y˙2) : a family of rigid rotations, parametrized by I and where P0(:; I) has rotation number ˛(I). In this case the ques- Writing y˙j D z j; j D 1; 2, one obtains a vector field in tion is what will be the fate of this family of invariant cir- 2 2 the four-dimensional phase space R R Df(y1; z1); cles, as well as with the corresponding rigidly rotational (y2; z2)g.For" D 0 this vector field has an invariant dynamics. two-torus, which is the product of the periodic motions Regarding the rotation number we again introduce of the individual Van der Pol oscillations. This two- Diophantine conditions. Indeed, for >2and>0the torus is normally hyperbolic and therefore persistent for subset [0; 1]; is defined as in (5), i. e., it contains all j"j1[73]. In fact the torus is an attractor and we can de- ˛ 2 [0; 1], such that for all rationals p/q fine a Poincaré return map within this torus attractor. If we ˇ ˇ ˇ ˇ ˇ p ˇ include some of the coefficients of the equations as param- ˇ˛ ˇ q : eters, Theorem 3 is directly applicable. The above state- q ments on quasi-periodic circle maps then directly translate Pulling back [0; 1] along the map ˛ we obtain a subset to the case of quasi-periodic invariant two-tori. Concern- ; . ing the resonant cases, generically a tongue structure like ; in Fig. 3 occurs; for the dynamics corresponding to param- Theorem 4 (Twist Map Theorem [104]) For suffi- eter values inside such a tongue one speaks of phase lock. ciently small, and for the perturbation O(") sufficiently 4526 H Hamiltonian Perturbation Theory (and Transition to Chaos)

small in C1-topology, there exists a C1 transformation Linearization of Complex Maps ˚ ! j " : , conjugating the restriction P0 #; to a sub- The Subsects. “Circle Maps”and“Area-Preserving Maps” system of P". both deal with smooth circle maps that are conjugated to As in the case of Theorem 3 again we chose the formula- rigid rotations. Presently the concern is with planar holo- tion of [29,31]. Largely the remarks following Theorem 3 morphic maps that are conjugated to a rigid rotation on an also apply here. open subset of the plane. Historically this is the first time that a small divisor problem was solved [7,101,141,142] Remark and Perturbative Expansions, Convergence of. Compare the format of the Theorems 3 and 4 and observe that in the latter case the role of the parameter ˛ Complex Linearization Given is a holomorphic germ has been taken by the action variable I.Theorem4im- F :(C; 0) ! (C; 0) of the form F(z) D z C f (z), with 0 plies that typically quasi-periodicity occurs with posi- f (0) D f (0) D 0. The problem is to find a biholomorphic tive measure in phase space. germ ˚ :(C; 0) ! (C; 0) such that In the gaps typically we have coexistence of periodicity, ˚ ı D ˚: quasi-periodicity and chaos [6,9,35,107,108,123,137]. F The latter follows from transversality of homo- and Such a diffeomorphism ˚ is called a linearization of F heteroclinic connections that give rise to positive topo- near 0. logical entropy. Open problems are whether the corre- We begin with the formal approach. Given the series sponding Lyapunov exponents also are positive, com- P P f (z) D f z j, we look for ˚(z) D z C z j.It pare with the discussion at the end of the intro- j2 j j2 j turns out that a solution always exists whenever ¤ 0is duction. not a root of unity. Indeed, direct computation reveals the following set of equations that can be solved recursively: Similar to the applications of Theorem 3 given at the end For j D 2 get the equation (1 )2 D f2 of Subsect. “Circle Maps”, here direct applications are pos- 2 For j D 3 get the equation (1 )3 D f3 C 2 f22 sible in the conservative setting. Indeed, consider a system n1 For j D n get the equation (1 )n D fn Cknown. of weakly coupled pendula The question now reduces to whether this formal solution has a positive radius of convergence. 2 @U y¨1 C ˛1 sin y1 D " (y1; y2) The hyperbolic case 0 < jj¤1 was already solved by @y1 Poincaré, for a description see [7]. The elliptic case jjD1 @U y¨ C ˛2 sin y D " (y ; y ) : again has small divisors and was solved by Siegel when for 2 2 2 @ 1 2 y2 some >0and>2wehavetheDiophantinenon-resonance condition Writing y˙j D z j, j D 1; 2 as before, we again get a vec- 2 2 p tor field in the four-dimensional phase space R R D 2i j e q jjqj : f(y1; y2); (z1; z2)g. In this case the energy The corresponding set of constitutes a set of full measure H"(y1; y2; z1; z2) in T 1 Dfg. 1 2 1 2 2 2 Yoccoz [141] completely solved the elliptic case using D z C z ˛ cos y1 ˛ cos y2 C "U(y1; y2) 2 1 2 2 1 2 the Bruno-condition. If

is a constant of motion. Restricting to a three-dimensional 2 i˛ pn 1 D e and when energy surface H" D const:, the iso-energetic Poincaré qn map P" is a twist map and application of Theorem 4 yields the conclusion of quasi-periodicity (on invariant two-tori) is the nth convergent in the continued fraction expansion occurring with positive measure in the energy surfaces of ˛ then the Bruno-condition reads of H . X " log(q ) nC1 < 1 : q Remark As in the dissipative case this example directly n n generalizes to cases with n oscillators (n 2 N), again leading to invariant n-tori with quasi-periodic ﬂow. We shall This condition turns out to be necessary and suﬃcient return to this subject in a later section. for ˚ having positive radius of convergence [141,142]. Hamiltonian Perturbation Theory (and Transition to Chaos) H 4527

Cremer’s Example in Herman’s Version As an example KAM Theory: An Overview consider the map In Sect. “Invariant Curves of Planar Diffeomorphisms”we described the persistent occurrence of quasi-periodicity in F(z) D z C z2 ; the setting of diffeomorphisms of the circle or the plane. The general perturbation theory of quasi-periodic motions where 2 T 1 is not a root of unity. is known under the name Kolmogorov–Arnold–Moser (or Observe that a point z 2 C is a periodic point of F with KAM) theory and discussed extensively elsewhere in this period q if and only if Fq(z) D z, where obviously encyclopedia Kolmogorov–Arnold–Moser (KAM) The- q ory. Presently we briefly summarize parts of this KAM Fq(z) D q z CCz2 : theory in broad terms, as this fits in our considerations, Writing thereby largely referring to [4,80,81,119,121,143,144], also see [20,24,55]. q In general quasi-periodicity is defined by a smooth Fq(z) z D z q 1 CCz2 1 ; conjugation. First on the n-torus T n D Rn/(2Z)n consider the vector field the period q periodic points exactly are the roots of the q right hand side polynomial. Abbreviating N D 2 1, it Xn @ directly follows that, if z ; z ;:::;z are the nontrivial X! D ! ; 1 2 N j @' roots, then for their product we have jD1 j

q z1 z2 ::: zN D 1 : where !1;!2;:::;!n are called frequencies [43,106]. Now, given a smooth (say, of class C1) vector field X It follows that there exists a nontrivial root within radius on a manifold M,withT M an invariant n-torus, we say that the restriction XjT is parallel if there exists jq 1j1/N ! 2 Rn and a smooth diffeomorphism ˚ : T ! T n, such that ˚(XjT ) D X! . We say that XjT is quasi-pe- of z D 0. riodic if the frequencies !1;!2;:::;!n are independent Now consider the set of T 1 defined as follows: over Q. 2 whenever A quasi-periodic vector field XjT leads to an integer affine structure on the torus T. In fact, since each orbit is lim inf jq 1j1/N D 0 : X q!1 dense, it follows that the self conjugations of ! exactly are the translations of T n, which completely determine n n It can be directly shown that is residual,againcompare the affine structure of T . Then, given ˚ : T ! T with with [115]. It also follows that for 2 linearization is ˚(XjT ) D X! , it follows that the self conjugations of XjT impossible. Indeed, since the rotation is irrational, the ex- determines a natural affine structure on the torus T.Note istence of periodic points in any neighborhood of z D 0 that the conjugation ˚ is unique modulo translations in T n implies zero radius of convergence. and T . Note that the composition of ˚ by a translation of Remark T n does not change the frequency vector !. However, the composition by a linear invertible map S 2 GL(n; Z) Notice that the residual set is in the complement of yields SX! D XS! .Weherespeakofaninteger affine the full measure set of all Diophantine numbers, again structure [43]. see [115]. Considering 2 T 1 as a parameter, we see a certain Remark analogy of these results on complex linearization with the Theorems 3 and 4. Indeed, in this case for a full The transition maps of an integer affine structure are measure set of s on a neighborhood of z D 0themap translations and elements of GL(n; Z). F D F is conjugated to a rigid irrational rotation. The current construction is compatible with the inte- Suchadomaininthez-plane often is referred to as grable affine structure on the Liouville tori of an inte- a Siegel disc. For a more general discussion of these and grable Hamiltonian system [6]. Note that in that case of Herman rings, see [101]. the structure extends to all parallel tori. 4528 H Hamiltonian Perturbation Theory (and Transition to Chaos)

Classical KAM Theory The classical KAM theory deals with smooth, nearly integrable Hamiltonian systems of the form

'˙ D !(I) C " f (I;';") (6) I˙ D "g(I;';") ;

where I varies over an open subset of Rn and ' over the standard torus T n. Note that for " D 0 the phase space as an open subset of Rn T n is foliated by invariant tori, parametrized by I. Each of the tori is parametrized by ' and the corresponding motion is parallel (or multi- peri- odic or conditionally periodic) with frequency vector !(I). Perturbation theory asks for persistence of the invariant n-tori and the parallelity of their motion for small values of j"j. The answer that KAM theory gives needs two essential ingredients. The first ingredient is that of Kolmogorov non-degeneracy which states that the map n n I 2 R 7! !(I) 2 R is a (local) diffeomorphism. Com- Hamiltonian Perturbation Theory (and Transition to Chaos), pare with the twist condition of Sect. “Invariant Curves of Figure 4 Rn Planar Diffeomorphisms”. The second ingredient general- The Diophantine set ; has the closed half line geometry Sn1 Rn izes the Diophantine conditions (5)ofthatsectionasfol- and the intersection \ ; is a Cantor set of measure Sn1 Rn lows: for >n 1and>0 consider the set n ; D O( )as # 0

n n n R; Df! 2 R jjh!;kij jkj ; k 2 Z nf0gg: (7) For regularity issues compare with a remark following The following properties are more or less direct. First Theorem 3. n R; has a closed half line geometry in the sense that For applications we largely refer to the introduction n n if ! 2 R; and s 1thenalsos! 2 R;. Moreover, and to [24,31] and references therein. n1 n the intersection S \ R; is a Cantor set of measure Continuing the discussion on affine structures at the n1 n S n R; D O()as # 0, see Fig. 4. beginning of this section, we mention that by means Completely in the spirit of Theorem 4, the classi- of the symplectic form, the domain of the I-variables in n cal KAM theorem roughly states that a Kolmogorov non- R inherits an affine structure [60], also see [91]and degenerate nearly integrable system (6)",forj"j1is references therein. smoothly conjugated to the unperturbed version (6)0,provided that the frequency map ! is co-restricted to the Dio- Statistical Mechanics deals with particle systems that n phantine set R;. In this formulation smoothness has to are large, often infinitely large. The Ergodic Hypothesis be taken in the sense of Whitney [119,136], also compare roughly says that in a bounded energy hypersurface, the with [20,24,29,31,55,121]. dynamics are ergodic, meaning that any evolution in the As a consequence we may say that in Hamiltonian sys- energy level set comes near every point of this set. tems of n degrees of freedom typically quasi-periodic in- The taking of limits as the number of particles tends variant (Lagrangian) n-tori occur with positive measure in to infinity is a notoriously difficult subject. Here we dis- phase space. It should be said that also an iso-energetic cuss a few direct consequences of classical KAM theory version of this classical result exists, implying a similar for many degrees of freedom. This discussion starts with conclusion restricted to energy hypersurfaces [6,9,21,24]. Kolmogorov’s papers [80,81], which we now present in The Twist Map Theorem 4 is closely related to the iso-en- a slightly rephrased form. First, we recall that for Hamil- ergetic KAM Theorem. tonian systems (say, with n degrees of freedom), typically the union of Diophantine quasi-periodic Lagrangian in- Remark variant n-tori fills up positive measure in the phase space We chose the quasi-periodic stability format as in and also in the energy hypersurfaces. Second, such a col- Sect. “Invariant Curves of Planar Diffeomorphisms”. lection of KAM tori immediately gives rise to non-ergodic- Hamiltonian Perturbation Theory (and Transition to Chaos) H 4529 ity, since it clearly implies the existence of distinct invari- taken in the sense of Whitney [29,119,136,143,144], also ant sets of positive measure. For background on Ergodic see [20,24,31,55]. Theory, see e. g. [9,27]and[24] for more references. Ap- It follows that the occurrence of normally hyperbolic parently the KAM tori form an obstruction to ergodicity, invariant tori carrying (Diophantine) quasi-periodic flow and a question is how bad this obstruction is as n !1. is typical for families of systems with sufficiently many Results in [5,78] indicate that this KAM theory obstruction parameters, where this occurrence has positive measure is not too bad as the size of the system tends to infinity. in parameter space. In fact, if the number of parameters In general the role of the Ergodic Hypothesis in Statistical equals the dimension of the tori, the geometry as sketched Mechanicshasturnedouttobemuchmoresubtlethan in Fig. 4 carries over in a diffeomorphic way. was expected, see e. g. [18,64]. Remark

Dissipative KAM Theory Many remarks following Subsect. “Classical KAM The- ory” and Theorem 3 also hold here. As already noted by Moser [105,106], KAM theory extends In cases where the system is degenerate, for instance outside the world of Hamiltonian systems, like to volume because there is a lack of parameters, a path formalism preserving systems, or to equivariant or reversible systems. can be invoked, where the parameter path is required to This also holds for the class of general smooth systems, n be a generic subfamily of the Diophantine set R;,see often called dissipative. In fact, the KAM theorem allows Fig. 4. This amounts to the Rüssmann non-degeneracy, for a Lie algebra proof, that can be used to cover all these that still gives positive measure of quasi-periodicity in special cases [24,29,31,45]. It turns out that in many cases the parameter space, compare with [24,31] and refer- parameters are needed for persistent occurrence of (Dio- ences therein. phantine) quasi-periodic tori. In the dissipative case the KAM theorem gives rise to As an example we now consider the dissipative setting, families of quasi-periodic attractors in a typical way. where we discuss a parametrized system with normally hy- This is of importance in center manifold reductions of perbolic invariant n-tori carrying quasi-periodic motion. infinite dimensional dynamics as, e. g., in fluid mechan- From [73] it follows that this is a persistent situation and ics [125,126]. In Sect. “Transition to Chaos and Turbu- that, up to a smooth (in this case of class Ck for large k) lence” we shall return to this subject. diffeomorphism, we can restrict to the case where T n is the phase space. To fix thoughts we consider the smooth system Lower Dimensional Tori '˙ D !() C " f (';;") We extend the above approach to the case of lower di- (8) ˙ D 0 ; mensional tori, i. e., where the dynamics transversal to the tori is also taken into account. We largely follow the set- where 2 Rn is a multi-parameter. The results of the up of [29,45] that follows Moser [106]. Also see [24,31] and references therein. Changing notation a little, we now classical KAM theorem regarding (6)" largely carry over consider the phase space T n Rm Dfx(mod 2); yg, to (8);". s Now, for " D 0 the product of phase space and param- as well a parameter space fgDP R .Weconsider 1 eter space as an open subset of T n Rn is completely fo- a C -family of vector fields X(x; y;) as before, having n n m liated by invariant n-tori and since the perturbation does T f0gT R as an invariant n-torus for D not concern the ˙-equation, this foliation is persistent. 0 2 P. The interest is with the dynamics on the resulting invariant x˙ D !() C f (y;) tori that remains parallel after the perturbation; compare with the setting of Theorem 3. As just stated, KAM the- y˙ D ˝() y C g(y;) (9) ory here gives a solution similar to the Hamiltonian case. ˙ D 0 ; The analogue of the Kolmogorov non-degeneracy condi- 2 tion here is that the frequency map 7! !()isa(local) with f (y;0) D O(jyj)andg(y;0) D O(jyj ), so we as- diffeomorphism. Then, in the spirit of Theorem 3, we state sume the invariant torus to be of Floquet type. that the system (8);" is smoothly conjugated to (8);0, The system X D X(x; y;) is integrable in the sense as before, provided that the map ! is co-restricted to the that it is T n-symmetric, i. e., x-independent [29]. The in- n n Diophantine set R;. Again the smoothness has to be terest is with the fate of the invariant torus T f0g and 4530 H Hamiltonian Perturbation Theory (and Transition to Chaos)

its parallel dynamics under small perturbation to a system us, for example, to deal with quasi-periodic versions X˜ D X˜(x; y;) that no longer needs to be integrable. of the Hamiltonian and the reversible Hopf bifurca- Consider the smooth mappings ! : P ! Rn and tion [38,41,42,44]. ˝ : P ! gl(m; R). To begin with we restrict to the case The Parameterized KAM Theory discussed here a pri- where all eigenvalues of ˝(0) are simple and nonzero. In ori needs many parameters. In many cases the pa- general for such a matrix ˝ 2 gl(m; R), let the eigenval- rameters are distinguished in the sense that they are

ues be given by ˛1 ˙ iˇ1;:::;˛N1 ˙ iˇN1 and ı1;:::;ıN2 , given by action variables, etc. For an example see Sub- where all ˛j;ˇj and ıj are real and hence m D 2N1 C N2. sect. “(n 1)-Tori” on Hamiltonian (n 1)-tori Also Also consider the map spec: gl(m; R) ! R2N1CN2 , given see [127]and[24,31] where the case of Rüssmann non- by ˝ 7! (˛; ˇ; ı). Next to the internal frequency vector degeneracy is included. This generalizes a remark at the ! 2 Rn, we also have the vector ˇ 2 RN1 of normal fre- end of Subsect. “Dissipative KAM Theory”. quencies. The present analogue of Kolmogorov non-degeneracy is the Broer–Huitema–Takens (BHT) non-degeneracy Global KAM Theory condition [29,127], which requires that the product map We stay in the Hamiltonian setting, considering La- n ! (spec) ı ˝ : P ! R gl(m; R)at D 0 has a sur- grangian invariant n-tori as these occur in a Liouville in- jective derivative and hence is a local submersion [72]. tegrable system with n degrees of freedom. The union of Furthermore, we need Diophantine conditions on these tori forms a smooth T n-bundle f : M ! B (where both the internal and the normal frequencies, generaliz- we leave out all singular fibers). It is known that this bun- ing (7). Given >n 1and>0, it is required for all dle can be non-trivial [56,60] as can be measured by mon- k 2 Zn nf0g and all ` 2 ZN1 with j`j2that odromy and Chern class. In this case global action angle variables are not defined. This non-triviality, among other jhk;!iCh`;ˇij jkj : (10) things, is of importance for semi-classical versions of the classical system at hand, in particular for certain spectrum n N Inside R R 1 Df!;ˇg this yields a Cantor set as be- defects [57,62,134,135], for more references also see [24]. fore (compare Fig. 4). This set has to be pulled back along Restricting to the classical case, the problem is what the submersion ! (spec) ı ˝, for examples see Sub- happens to the (non-trivial) T n-bundle f under small, sects. “(n 1)-Tori”and“Quasi-periodic Bifurcations” non-integrable perturbation. From the classical KAM the- below. ory, see Subsect. “Classical KAM Theory”wealreadyknow The KAM theorem for this setting is quasi-periodic sta- that on trivializing charts of f Diophantine quasi-periodic bility of the n-tori under consideration, as in Subsect. “Dis- n-tori persist. In fact, at this level, a Whitney smooth con- sipative KAM Theory”, yielding typical examples where jugation exists between the integrable system and its per- quasi-periodicity has positive measure in parameter space. turbation, which is even Gevrey regular [136]. It turns out In fact, we get a little more here, since the normal linear that these local KAM conjugations can be glued together behavior of the n-tori is preserved by the Whitney smooth so to obtain a global conjugation at the level of quasi-pe- conjugations. This is expressed as normal linear stability, riodic tori, thereby implying global quasi-periodic stabil- which is of importance for quasi-periodic bifurcations, see ity [43]. Here we need unicity of KAM tori, i. e., indepen- Subsect. “Quasi-periodic Bifurcations”below. dence of the action-angle chart used in the classical KAM Remark theorem [26]. The proof uses the integer affine structure on the quasi-periodic tori, which enables taking convex A more general set-up of the normal stability the- combinations of the local conjugations subjected to a suit- ory [45] adapts the above to the case of non-sim- able partition of unity [72,129]. In this way the geometry ple (multiple) eigenvalues. Here the BHT non-degen- of the integrable bundle can be carried over to the nearly- eracy condition is formulated in terms of versal un- integrable one. folding of the matrix ˝(0)[7]. For possible condi- The classical example of a Liouville integrable sys- tions under which vanishing eigenvalues are admissible tem with non-trivial monodromy [56,60] is the spher- see [29,42,69] and references therein. ical pendulum, which we now briefly revisit. The This general set-up allows for a structure preserv- configuration space is S2 Dfq 2 R3 jhq; qiD1g and ing formulation as mentioned earlier, thereby includ- the phase space TS2 Šf(q; p) 2 R6 jhq; qiD1and ing the Hamiltonian and volume preserving case, as hq; piD0g.ThetwointegralsI D q1 p2 q2 p1 (angu- 1 well as equivariant and reversible cases. This allows lar momentum) and E D 2 hp; piCq3 (energy) lead to Hamiltonian Perturbation Theory (and Transition to Chaos) H 4531

nearly integrable case, so where the axial symmetry is broken. Also compare [24] and many of its references. The global conjugations of [43] are Whitney smooth (even Gevrey regular [136]) and near the identity map in the C1-topology [72]. Geometrically speaking these diﬀeomorphisms also are T n-bundle isomorphisms between the unperturbed and the perturbed bundle, the basis of which is a Cantor set of positive measure.

Splitting of Separatrices KAM theory does not predict the fate of close-to-resonant tori under perturbations. For fully resonant tori the phe- Hamiltonian Perturbation Theory (and Transition to Chaos), nomenon of frequency locking leads to the destruction Figure 5 of the torus under (sufficiently rich) perturbations, and Range of the energy-momentum map of the spherical pendulum other resonant tori disintegrate as well. In the case of a single resonance between otherwise Diophantine frequencies the perturbation leads to quasi-periodic bifurcations, cf. an energy momentum map EM: TS2 ! R2, given by Sect. “Transition to Chaos and Turbulence”. (q; p) 7! (I; E) D q p q p ; 1 hp; piCq .InFig.5 1 2 2 1 2 3 WhileKAMtheoryconcernsthefateofmosttrajecto- we show the image of the map EM. The shaded area B ries and for all times, a complementary theorem has been consists of regular values, the fiber above which is a La- obtained in [93,109,110,113]. It concerns all trajectories grangian two-torus; the union of these gives rise to a bun- and states that they stay close to the unperturbed tori for dle f : M ! B as described before, where f D EMj . M long times that are exponential in the inverse of the per- The motion in the two-tori is a superposition of Huy- turbation strength. For trajectories starting close to sur- gens’ rotations and pendulum-like swinging, and the non- viving tori the diffusion is even superexponentially slow, existence of global action angle variables reflects that the cf. [102,103]. Here a form of smoothness exceeding the three interpretations of ‘rotating oscillation’, ‘oscillating mere existence of infinitely many derivatives of the Hamil- rotation’ and ‘rotating rotation’ cannot be reconciled in tonian is a necessary ingredient, for finitely differentiable a consistent way. The singularities of the fibration include Hamiltonians one only obtains polynomial times. the equilibria (q; p) D ((0; 0; ˙1); (0; 0; 0)) 7! (I; E) D Solenoids, which cannot be present in integrable sys- (0; ˙1). The boundary of this image also consists of sin- tems, are constructed for generic Hamiltonian systems gular points, where the fiber is a circle that corresponds in [16,94,98], yielding the simultaneous existence of rep- to Huygens’ horizontal rotations of the pendulum. The resentatives of all homeomorphy-classes of solenoids. Hy- fiber above the upper equilibrium point (I; E) D (0; 1) is perbolic tori form the core of a construction proposed apinchedtorus[56], leading to non-trivial monodromy, in [5] of trajectories that venture off to distant points of in a suitable bases of the period lattices, given by the phase space. In the unperturbed system the union of a family of hyperbolic tori, parametrized by the ac- 1 1 2 GL(2; R) : tions conjugate to the toral angles, form a normally hy- 01 perbolic manifold. The latter is persistent under pertur- Thequestionhereiswhatremainsofthebundlef when bations, cf. [73,100], and carries a Hamiltonian flow with the system is perturbed. Here we observe that locally Kol- fewer degrees of freedom. The main difference between in- mogorov non-degeneracy is implied by the non-trivial tegrable and non-integrable systems already occurs for pe- monodromy [114,122]. From [43,122] it follows that the riodic orbits. non-trivial monodromy can be extended in the perturbed case. Periodic Orbits Remark A sharp difference to dissipative systems is that it is generic for hyperbolic periodic orbits on compact energy shells in The case where this perturbation remains integrable is Hamiltonian systems to have homoclinic orbits, cf. [1]and covered in [95], but presently the interest is with the references therein. For integrable systems these form to- 4532 H Hamiltonian Perturbation Theory (and Transition to Chaos)

gether a pinched torus, but under generic perturbations to the internal resonances the necessary Diophantine con- the stable and unstable manifold of a hyperbolic periodic ditions (10) exclude the normal-internal resonances orbit intersect transversely. It is a nontrivial task to actually check this genericity condition for a given non-in- hk;!iD˛j (11) tegrable perturbation, a first-order condition going back to Poincaré requires the computation of the so-called hk;!iD2˛j (12) Mel’nikov integral, see [66,137] for more details. In two degrees of freedom normalization leads to approxima- hk;!iD˛i C ˛j (13) tions that are integrable to all orders, which implies that the Melnikov integral is a flat function. In the real ana- hk;!iD˛i ˛j : (14) lytic case the Melnikov criterion is still decisive in many The first three resonances lead to the quasi-periodic examples [65]. center-saddle bifurcation studied in Sect. “Transition to Genericity conditions are traditionally formulated in Chaos and Turbulence”, the frequency-halving (or quasi- the universe of smooth vector fields, and this makes the periodic period doubling) bifurcation and the quasi-peri- whole class of analytic vector fields appear to be non- odic Hamiltonian Hopf bifurcation, respectively. The res- generic. This is an overly pessimistic view as the conditions onance (14) generalizes an equilibrium in 1 : 1 resonance defining a certain class of generic vector fields may cer- whence T n1 persists and remains elliptic, cf. [78]. When tainly be satisfied by a given analytic system. In this respect passing through resonances (12)and(13)thelower-di- it is interesting that the generic properties may also be for- mensional tori lose ellipticity and acquire hyperbolic Flo- mulated in the universe of analytic vector fields, see [28] quet exponents. Elliptic (n 1)-tori have a single normal for more details. frequency whence (11)and(12)aretheonlynormal-internal resonances. See [35] for a thorough treatment of the (n 1)-Tori ensuing possibilities. The (n 1)-parameter families of invariant (n 1)-tori The restriction to a single normal-internal resonance organize the dynamics of an integrable Hamiltonian sys- is dictated by our present possibilities. Indeed, already the tem in n degrees of freedom, and under small pertur- bifurcation of equilibria with a fourfold zero eigenvalue bations the parameter space of persisting analytic tori is leads to unfoldings that simultaneously contain all possi- Cantorized. This still allows for a global understanding of ble normal resonances. Thus, a satisfactory study of such a substantial part of the dynamics, but also leads to addi- tori which already may form one-parameter families in in- tional questions. tegrable Hamiltonian systems with five degrees of freedom A hyperbolic invariant torus T n1 has its Floquet ex- has to await further progress in local bifurcation theory. ponents off the imaginary axis. Note that T n1 is not a normally hyperbolic manifold. Indeed, the normal linear Transition to Chaos and Turbulence behavior involves the n 1 zero eigenvalues in the direc- One of the main interests over the second half of the twen- tion of the parametrizing actions as well; similar to (9)the tieth century has been the transition between orderly and format complicated forms of dynamics upon variation of either initial states or of system parameters. By ‘orderly’ we here x˙ D !(y) C O(y) C O(z2) mean equilibrium and periodic dynamics and by compli- 3 y˙ D O(y) C O(z ) cated quasi-periodic and chaotic dynamics, although we z˙ D ˝(y)z C O(z2) note that only chaotic dynamics is associated to unpre- dictability, e. g. see [27]. As already discussed in the in- in Floquet coordinates yields an x-independent matrix ˝ troduction systems like a forced nonlinear oscillator or the that describes the symplectic normal linear behavior, planar three-body problem exhibit coexistence of periodic, cf. [29]. The union fz D 0g over the family of (n 1)-tori quasi-periodic and chaotic dynamics, also compare with is a normally hyperbolic manifold and constitutes the cen- Fig. 1. ter manifold of T n1. Separatrices splitting yields the di- Similar remarks go for the onset of turbulence in fluid viding surfaces in the sense of Wiggins et al. [138]. dynamics. Around 1950 this led to the scenario of Hopf– The persistence of elliptic tori under perturbation Landau–Lifschitz [75,76,83,84], which roughly amounts from an integrable system involves not only the internal to the following. Stationary fluid motion corresponds to frequencies of T n1, but also the normal frequencies. Next an equilibrium point in an 1-dimensional state space Hamiltonian Perturbation Theory (and Transition to Chaos) H 4533 of velocity fields. The first transition is a Hopf bifurca- Overview”, for the ensuing quasi-periodic bifurcation the- tion [66,75,82], where a periodic solution branches off. ory the same holds. In a second transition of similar nature a quasi-periodic two-torus branches off, then a quasi-periodic three-torus, Hamiltonian Cases To fix thoughts we start with an ex- etc. The idea is that the motion picks up more and more ample in the Hamiltonian setting, where a robust model frequencies and thus obtains an increasingly complicated for the quasi-periodic center-saddle bifurcation is given by power spectrum. In the early 1970s this idea was modified in the Ruelle–Takens route to turbulence, based on the ob- H!1;!2;;"(I;';p; q) servation that, for flows, a three-torus can carry chaotic 1 2 D ! I C ! I C p C V(q) C " f (I;';p; q) (or strange) attractors [112,126], giving rise to a broad 1 1 2 2 2 band power spectrum. By the quasi-periodic bifurcation (15) theory [24,29,31] as sketched below these two approaches are unified in a generic way, keeping track of measure the- 1 3 with V(q) D 3 q q,comparewith[67,69]. The un- oretic aspects. For general background in dynamical sys- perturbed (or integrable) case " D 0, by factoring out the tems theory we refer to [27,79]. T 2-symmetry, boils down to a standard center-saddle bi- Another transition to chaos was detected in the furcation, involving the fold catastrophe [133]inthepo- quadratic family of interval maps tential function V D V(q). This results in the existence of two invariant two-tori, one elliptic and the other hyper- f(x) D x(1 x) ; bolic. For 0 ¤j"j1 the dense set of resonances compli- cates this scenario, as sketched in Fig. 6, determined by the see [58,99,101], also for a holomorphic version. This tran- Diophantine conditions sition consists of an infinite sequence of period doubling bifurcations ending up in chaos; it has several universal jhk;!ij jkj ; for q < 0 ; (16) aspects and occurs persistently in families of dynamical jhk;!iC`ˇ(q)jjkj ; for q > 0 systems. In many of these cases also homoclinic bifurca- n tions show up, where sometimes the transition to chaos for all kp2 Z nf0g and for all ` 2 Z with j`j2. Here is immediate when parameters cross a certain boundary, ˇ(q) D 2q is the normal frequency of the elliptic torus p for general theory see [13,14,30,117]. There exist quite given by q D for >0.Asbefore,(cf.Sects.“Invari- anumberofcasestudieswhereallthreeoftheabovesce- ant Curves of Planar Diffeomorphisms”, “KAM Theory: narios play a role, e. g., see [32,33,46] and many of their An Overview”), this gives a Cantor set of positive mea- references. sure [24,29,31,45,69,105,106]. For 0 < j"j1Fig.6 will be distorted by a near- identity diffeomorphism; compare with the formulations Quasi-periodic Bifurcations of the Theorems 3 and 4. On the Diophantine Cantor set For the classical bifurcations of equilibria and periodic the dynamics is quasi-periodic, while in the gaps generi- orbits, the bifurcation sets and diagrams are generally cally there is coexistence of periodicity and chaos, roughly determined by a classical geometry in the product of comparable with Fig. 1, at left. The gaps at the border fur- phase space and parameter space as already established thermore lead to the phenomenon of parabolic resonance, by, e. g., [8,133], often using singularity theory. Quasi-pe- cf. [86]. riodic bifurcation theory concerns the extension of these Similar programs exist for all cuspoid and umbilic bifurcations to invariant tori in nearly-integrable systems, catastrophes [37,39,68]aswellasfortheHamiltonian e. g., when the tori lose their normal hyperbolicity or when Hopf bifurcation [38,44]. For applications of this approach certain (strong) resonances occur. In that case the dense see [35]. For a reversible analogue see [41]. As so often set of resonances, also responsible for the small divisors, within the gaps generically there is an infinite regress of leads to a Cantorization of the classical geometries ob- smaller gaps [11,35]. For theoretical background we refer tained from Singularity Theory [29,35,37,38,39,41,44,45, to [29,45,106], for more references also see [24]. 48,49,67,68,69], also see [24,31,52,55]. Broadly speaking, one could say that in these cases the Preparation Theo- Dissipative Cases In the general dissipative case we ba- rem [133]ispartlyreplacedbyKAM theory. Since the KAM sically follow the same strategy. Given the standard bifur- theory has been developed in several settings with or with- cations of equilibria and periodic orbits, we get more com- out preservation of structure, see Sect. “KAM Theory: An plex situations when invariant tori are involved as well. 4534 H Hamiltonian Perturbation Theory (and Transition to Chaos)

Hamiltonian Perturbation Theory (and Transition to Chaos), Figure 6 Hamiltonian Perturbation Theory (and Transition to Chaos), Sketch of the Cantorized Fold, as the bifurcation set of the quasi- Figure 7 periodic center-saddle bifurcation for n D 2[67], where the hor- Bifurcation diagram of the Hopf bifurcation izontal axis indicates the frequency ratio !2 : !1,cf.(15). The lower part of the figure corresponds to hyperbolic tori and the upper part to elliptic ones. See the text for further interpretations than five, see [7,132], and where O(jyj2)shouldcontain generic third order terms. As before, we let ˛ D serve as a bifurcation parameter, varying near 0. On one side of The simplest examples are the quasi-periodic saddle-node the bifurcation value D 0, this system has by normal hy- and quasi-periodic period doubling [29]alsosee[24,31]. perbolicity and [73], an invariant circle. Here, due to the To illustrate the whole approach let us start from invariance of the rotation numbers of the invariant circles, the Hopf bifurcation of an equilibrium point of a vec- no topological stability can be obtained [111]. Still this bi- tor field [66,75,82,116] where a hyperbolic point attrac- furcation can be characterized by many persistent prop- tor loses stability and branches off a periodic solution, erties. Indeed, in a generic two-parameter family (18), say cf. Subsect. “Dissipative Perturbations”. A topological nor- with both ˛ and ˇ as parameters, the periodicity in the pa- mal form is given by rameter plane is organized in resonance tongues [7,34,82]. (The tongue structure is hardly visible when only one pa- y˙1 ˛ ˇ y1 2 2 y1 rameter, like ˛, is used.) If the diffeomorphism is the re- D y1 C y2 y˙2 ˇ˛ y2 y2 turn map of a periodic orbit for flows, this bifurcation pro- (17) duces an invariant two-torus. Usually this counterpart for flows is called Ne˘ımark–Sacker bifurcation. The period- where y D (y ; y ) 2 R2, ranging near (0; 0). In this rep- 1 2 icity as it occurs in the resonance tongues, for the vector resentation usually one fixes ˇ D 1andlets˛ D (near field is related to phase lock. The tongues are contained in 0) serve as a (bifurcation) parameter, classifying modulo gaps of a Cantor set of quasi-periodic tori with Diophan- topological equivalence. In polar coordinates (17)sogets tine frequencies. Compare the discussion in Subsect. “Cir- the form cle Maps”, in particular also regarding the Arnold family '˙ D 1 ; and Fig. 3.AlsoseeSect.“KAM Theory: An Overview”and r˙ D r r3 : again compare with [115]. Quasi-periodic versions exist for the saddle-node, the Figure 7 shows an amplitude response diagram (often period doubling and the Hopf bifurcation. Returning to called the bifurcation diagram). Observe the occurrence the setting with T n Rm as the phase space, we remark of the attracting periodic solution for >0ofamplitude p that the quasi-periodic saddle-node and period doubling . already occur for m D 1, or in an analogous center man- Let us briefly consider the Hopf bifurcation for fixed ifold. The quasi-periodic Hopf bifurcation needs m 2. points of diffeomorphisms. A simple example has the form We shall illustrate our results on the latter of these cases, P(y) D e2(˛Ciˇ) y C O(jyj2) ; (18) compare with [19,31]. For earlier results in this direction see [52]. Our phase space is T n R2 Dfx(mod 2); yg, y 2 C Š R2,near0.Tostartwithˇ is considered a con- where we are dealing with the parallel invariant torus stant, such that ˇ is not rational with denominator less T n f0g.Intheintegrablecase,byT n-symmetry we can Hamiltonian Perturbation Theory (and Transition to Chaos) H 4535 reduce to R2 Dfyg and consider the bifurcations of relative equilibria. The present interest is with small non-integrable perturbations of such integrable models. We now discuss the quasi-periodic Hopf bifurcation [17,29], largely following [55]. The unperturbed, in- n 2 tegrable family X D X(x; y)onT R has the form

X(x; y)

D [!() C f (y;)]@x C [˝()y C g(y;)]@y ; (19) were f D O(jyj)andg D O(jyj2) as before. Moreover 2 P is a multi-parameter and ! : P ! Rn and ˝ : P ! gl(2; R) are smooth maps. Here we take ˛() ˇ() Hamiltonian Perturbation Theory (and Transition to Chaos), ˝() D ; Figure 8 ˇ() ˛() (2) Planar section of the Cantor set ; which makes the @y component of (19)compatiblewith the planar Hopf family (17). The present form of Kol- mogorov non-degeneracy is Broer–Huitema–Takens sta- ! R2 bility [29,42,45], requiring that there is a subset P on with a plane f g for a Diophantine (internal) fre- ! which the map quency vector ,cf.(7). From [17,29] it now follows that for any family X˜ 2 P 7! (!();˝()) 2 Rn gl(2; R) on T n R2 P, sufficiently near X in the C1-topology a near-identity C1-diffeomorphism ˚ : T n R2 ! is a submersion. For simplicity we even assume that is T n R2 exists, defined near T n f0g ,thatconju- replaced by n (2) gates X to X˜ when further restricting to T f0g;. (!;(˛; ˇ)) 2 Rn R2 : So this means that the Diophantine quasi-periodic invariant n-tori are persistent on a diffeomorphic image of the (2) Observe that if the non-linearity g satisfies the well-known Cantor set ;, compare with the formulations of the Hopf non-degeneracy conditions, e. g., compare [66,82], Theorems 3 and 4. then the relative equilibrium y D 0 undergoes a standard Similarly we can find invariant (n C 1)-tori. We first planar Hopf bifurcation as described before. Here ˛ again have to develop a T nC1 symmetric normal form approxi- plays the role of bifurcation parameter and a closed or- mation [17,29]and Normal Forms in Perturbation The- bit branches off at ˛ D 0. To fix thoughts we assume that ory. For this purpose we extend the Diophantine condi- y D 0 is attracting for ˛<0. and that the closed orbit oc- tions (20) by requiring that the inequality holds for all curs for ˛>0, and is attracting as well. For the integrable j`jN for N D 7. We thus find another large Cantor set, family X, qualitatively we have to multiply this planar sce- again see Fig. 8, where Diophantine quasi-periodic invari- nario with T n, by which all equilibria turn into invariant ant (n C 1)-tori are persistent. Here we have to restrict to attracting or repelling n-tori and the periodic attractor into ˛>0 for our choice of the sign of the normal form coeffi- an attracting invariant (n C 1)-torus. Presently the ques- cient, compare with Fig. 7. tion is what happens to both the n-andthe(n C 1)-tori, In both the cases of n-tori and of (n C 1)-tori, the when we apply a small near-integrable perturbation. nowhere dense subset of the parameter space containing The story runs much like before. Apart from the BHT the tori can be fattened by normal hyperbolicity to open non-degeneracy condition we require Diophantine condi- subsets. Indeed, the quasi-periodic n-and(n C 1)-tori are tions (10), defining the Cantor set infinitely normally hyperbolic [73]. Exploiting the normal form theory [17,29]and Normal Forms in Perturbation (2) ; Df(!;(˛; ˇ)) 2 jjhk;!iC`ˇjjkj ; Theory to the utmost and using a more or less standard 8k 2 Zn nf0g ; 8` 2 Z with j`j2g ; (20) contraction argument [17,53], a fattening of the parameter domain with invariant tori can be obtained that leaves out (2) n 2 In Fig. 8 we sketch the intersection of ; R R only small ‘bubbles’ around the resonances, as sketched 4536 H Hamiltonian Perturbation Theory (and Transition to Chaos)

As an example consider a family of maps that undergoes a generic quasi-periodic Hopf bifurcation from circle to two-torus. It turns out that here the Cantorized fold of Fig. 6 is relevant, where now the vertical coordinate is a bifurcation parameter. Moreover compare with Fig. 3, where also variation of " is taken into account. The Cantor set contains the quasi-periodic dynamics, while in the gaps we can have chaos, e. g., in the form of Hénon like strange attractors [46,112]. A fattening process as explained above, also can be carried out here.

Future Directions One important general issue is the mathematical charac- terization of chaos and ergodicity in dynamical systems, in conservative, dissipative and in other settings. This is a tough problem as can already be seen when considering two-dimensional diffeomorphisms. In particular we refer to the still unproven ergodicity conjecture of [9] and to the conjectures around Hénon like attractors and the principle ‘Hénon everywhere’, compare with [22,32]. For a discussion see Subsect. “A Scenario for the Onset of Turbu- Hamiltonian Perturbation Theory (and Transition to Chaos), lence”. In higher dimension this problem is even harder to Figure 9 handle, e. g., compare with [46,47] and references therein. Fattening by normal hyperbolicity of a nowhere dense parame- In the conservative case a related problem concerns a better set with invariant n-tori in the perturbed system. The curve ter understanding of Arnold diffusion. H is the Whitney smooth (even Gevrey regular [136]) image Somewhat related to this is the analysis of dynamical of the ˇ-axis in Fig. 8. H interpolates the Cantor set Hc that contains the non-hyperbolic Diophantine quasi-periodic in- systems without an explicit perturbation setting. Here nu- (2) variant n-tori, corresponding to ; ,see(20). To the points merical and symbolic tools are expected to become useful H A 1;2 2 c discs 1;2 are attached where we find attracting to develop computer assisted proofs in extended perturba- normally hyperbolic n-tori and similarly in the discs R re- 1;2 tion settings, diagrams of Lyapunov exponents, symbolic pelling ones. The contact between the disc boundaries and H is infinitely flat [17,29] dynamics, etc. Compare with [128]. Also see [46,47]for applications and further reference. This part of the theory is important for understanding concrete models, that of- and explained in Fig. 9 for the n-tori. For earlier results in ten are not given in perturbation format. the same spirit in a case study of the quasi-periodic saddle- Regarding nearly-integrable Hamiltonian systems, node bifurcation see [49,50,51], also compare with [11]. several problems have to be considered. Continuing the above line of thought, one interest is the development of Hamiltonian bifurcation theory without integrable normal A Scenario for the Onset of Turbulence form and, likewise, of KAM theory without action angle co- Generally speaking, in many settings quasi-periodicity ordinates [87]. One big related issue also is to develop KAM constitutes the order in between chaos [31]. In the Hopf– theory outside the perturbation format. Landau–Lifschitz–Ruelle–Takens scenario [76,83,84,126] The previous section addressed persistence of Dio- we may consider a sequence of typical transitions as phantine tori involved in a bifurcation. Similar to Cre- given by quasi-periodic Hopf bifurcations, starting with mer’s example in Subsect. “Cremer’s Example in Her- the standard Hopf or Hopf–Ne˘ımark–Sacker bifurcation man’s Version” the dynamics in the gaps between persis- as described before. In the gaps of the Diophantine Can- tent tori displays new phenomena. A first step has been tor sets generically there will be coexistence of periodic- made in [86] where internally resonant parabolic tori in- ity, quasi-periodicity and chaos in infinite regress. As said volved in a quasi-periodic Hamiltonian pitchfork bifur- earlier, period doubling sequences and homoclinic bifur- cationareconsidered.Theresultinglargedynamicalin- cations may accompany this. stabilities may be further amplified for tangent (or flat) Hamiltonian Perturbation Theory (and Transition to Chaos) H 4537 parabolic resonances, which fail to satisfy the iso-energetic In: History of Mathematics, vol 11. Am Math Soc, Providence; non-degeneracy condition. London Math Soc, London 2 The construction of solenoids in [16,94] uses ellip- 13. Benedicks M, Carleson L (1985) On iterations of 1 ax on (1; 1). Ann Math 122:1–25 tic periodic orbits as starting points, the simplest exam- 14. Benedicks M, Carleson L (1991) The dynamics of the Hénon ple being the result of a period-doubling sequence. This map. Ann Math 133:73–169 construction should carry over to elliptic tori, where nor- 15. Benettin G (2005) Physical applications of Nekhoroshev the- mal-internal resonances lead to encircling tori of the same orem and exponential estimates. In: Giorgilli A (ed) Hamilto- dimension, while internal resonances lead to elliptic tori nian dynamics theory and applications, Cetraro 1999, Lecture of smaller dimension and excitation of normal modes in- Notes in Mathematics, vol 1861. Springer, pp 1–76 16. Birkhoff BD (1935) Nouvelles recherches sur les systemes dy- creases the torus dimension. In this way one might be able namiques. Mem Pont Acad Sci Novi Lyncaei 1(3):85–216 to construct solenoid-type invariant sets that are limits of 17. Braaksma BLJ, Broer HW (1987) On a quasi-periodic Hopf bi- tori with varying dimension. furcation. Ann Inst Henri Poincaré, Anal non linéaire 4(2): Concerning the global theory of nearly-integrable 115–168 torus bundles [43], it is of interest to understand the ef- 18. Bricmont J (1996) Science of chaos or chaos in science? In: Gross PR, Levitt N, Lewis MW (eds) The Flight from Science fects of quasi-periodic bifurcations on the geometry and and Reason (New York, 1995), Ann New York Academy of Sci- its invariants. Also it is of interest to extend the results ences, vol 775. New York Academy of Sciences, New York, of [134] when passing to semi-classical approximations. In pp 131–175; Also appeared in: Phys Mag 17:159–208 (1995) that case two small parameters play a role, namely Planck’s 19. Broer HW (2003) Coupled Hopf-bifurcations: Persistent exam- constant as well as the distance away from integrability. ples of n-quasiperiodicity determined by families of 3-jets. Astérisque 286:223–229 20. Broer HW (2004) KAM theory: the legacy of Kolmogorov’s 1954 paper. Bull Am Math Soc (New Series) 41(4):507–521 Bibliography 21. Broer HW, Huitema GB (1991) A proof of the isoenergetic KAM-theorem from the “ordinary” one. J Differ Equ 90:52–60 1. Abraham R, Marsden JE (1978) Foundations of Mechanics, 22. Broer HW, Krauskopf B (2000) Chaos in periodically driven sys- 2nd edn. Benjamin tems. In Krauskopf B, Lenstra D (eds) Fundamental Issues of 2. Arnold VI (1961) Small divisors I: On mappings of the circle Nonlinear Laser Dynamics. American Institute of Physics Con- onto itself. Izv Akad Nauk SSSR Ser Mat 25:21–86 (in Rus- ference Proceedings 548:31–53 sian); English translation: Am Math Soc Transl Ser 2(46):213– 23. Broer HW, Roussarie R (2001) Exponential confinement of 284 (1965); Erratum: Izv Akad Nauk SSSR Ser Mat 28:479–480 chaos in the bifurcation set of real analytic diffeomorphisms. (1964, in Russian) In: Broer HW, Krauskopf B, Vegter G (eds) Global Analysis of 3. Arnold VI (1962) On the classical perturbation theory and the Dynamical Systems, Festschrift dedicated to Floris Takens for stability problem of the planetary system. Dokl Akad Nauk his 60th birthday. Bristol and Philadelphia IOP, pp 167–210 SSSR 145:487–490 4. Arnold VI (1963) Proof of a theorem by A.N. Kolmogorov 24. Broer HW, Sevryuk MB (2007) KAM Theory: quasi-periodicity on the persistence of conditionally periodic motions under in dynamical systems. In: Broer HW, Hasselblatt B, Takens F a small change of the Hamilton function. Russ Math Surv (eds) Handbook of Dynamical Systems, vol 3. North-Holland 18(5):9–36 (English; Russian original) (to appear) 5. Arnold VI (1964) Instability of dynamical systems with several 25. Broer HW, Takens F (1989) Formally symmetric normal forms degrees of freedom. Sov Math Dokl 5:581–585 and genericity. Dyn Rep 2:36–60 6. Arnold VI (1978) Mathematical Methods of Classical Mechan- 26. Broer HW, Takens F (2007) Unicity of KAM tori. Ergod Theory ics, GTM 60. Springer, New York Dyn Syst 27:713–724 7. Arnold VI (1983) Geometrical Methods in the Theory of Ordi- 27. Broer HW, Takens F (2008) Dynamical Systems and Chaos. To nary Differential Equations. Springer be published by Epsilon Uitgaven 8. Arnold VI (ed) (1994) Dynamical Systems V: Bifurcation The- 28. Broer HW, Tangerman FM (1986) From a differentiable to ory and Catastrophe Theory. Encyclopedia of Mathematical a real analytic perturbation theory, applications to the Kupka Sciences, vol 5. Springer Smale theorems. Ergod Theory Dyn Syst 6:345–362 9. Arnold VI, Avez A (1967) Problèmes Ergodiques de la Mé- 29. Broer HW, Huitema GB, Takens F, Braaksma BLJ (1990) Unfold- canique classique, Gauthier-Villars; English edition: Arnold ings and bifurcations of quasi-periodic tori. In: Memoir AMS, VI, Avez A (1968) Ergodic problems of classical mechanics. vol 421. Amer Math Soc, Providence Benjamin 30. Broer HW, Dumortier F, van Strien SJ, Takens F (1991) Struc- 10. Arnol’d VI, Kozlov VV, Neishtadt AI (1988) Mathematical As- tures in dynamics, finite dimensional deterministic studies. In: pects of Classical and Celestial Mechanics. In: Arnold VI (ed) de Jager EM, van Groesen EWC (eds) Studies in Mathematical Dynamical Systems, vol III. Springer Physics, vol II. North-Holland 11. Baesens C, Guckenheimer J, Kim S, MacKay RS (1991) Three 31. Broer HW, Huitema GB, Sevryuk MB (1996) Quasi-Periodic Mo- coupled oscillators: Mode-locking, global bifurcation and tions in Families of Dynamical Systems: Order amidst Chaos. toroidal chaos. Phys D 49(3):387–475 In: Lecture Notes in Mathematics, vol 1645. Springer 12. Barrow-Green J (1997) Poincaré and the Three Body Problem. 32. Broer HW, Simó C, Tatjer JC (1998) Towards global models 4538 H Hamiltonian Perturbation Theory (and Transition to Chaos)

near homoclinic tangencies of dissipative diffeomorphisms. 53. Chow S-N, Hale JK (1982) Methods of Bifurcation Theory. Nonlinearity 11(3):667–770 Springer 33. Broer HW, Simó C, Vitolo R (2002) Bifurcations and strange at- 54. Chow S-N, Li C, Wang D (1994) Normal Forms and Bifur- tractors in the Lorenz-84 climate model with seasonal forcing. cation of Planar Vector Fields. Cambridge University Press, Nonlinearity 15(4):1205–1267 Cambridge 34. Broer HW, Golubitsky M, Vegter G (2003) The geometry of res- 55. Ciocci MC, Litvak-Hinenzon A, Broer HW (2005) Survey on dis- onance tongues: a singularity theory approach. Nonlinearity sipative KAM theory including quasi-periodic bifurcation the- 16:1511–1538 ory based on lectures by Henk Broer. In: Montaldi J, Ratiu T 35. Broer HW, Hanßmann H, Jorba À, Villanueva J, Wagener (eds) Geometric Mechanics and Symmetry: the Peyresq Lec- FOO (2003) Normal-internal resonances in quasi-periodi- tures, LMS Lecture Notes Series, vol 306. Cambridge Univer- cally forced oscillators: a conservative approach. Nonlinearity sity Press, Cambridge, pp 303–355 16:1751–1791 56. Cushman RH, Bates LM (1997) Global Aspects of Classical In- 36. Broer HW, Hoveijn I, Lunter G, Vegter G (2003) Bifurcations tegrable Systems. Birkhäuser, Basel in Hamiltonian systems: Computing Singularities by Gröbner 57. Cushman RH, Dullin HR, Giacobbe A, Holm DD, Joyeux M, Bases. In: Lecture Notes in Mathematics, vol 1806. Springer Lynch P, Sadovskií DA and Zhilinskií BI (2004) CO2 molecule as 37. Broer HW, Hanßmann H, You J (2005) Bifurcations of normally a quantum realization of the 1 : 1 : 2 resonant swing-spring parabolic tori in Hamiltonian systems. Nonlinearity 18:1735– with monodromy. Phys Rev Lett 93:024302 1769 58. Devaney RL (1989) An Introduction to Chaotic Dynamical Sys- 38. Broer HW, Hanßmann H, Hoo J, Naudot V (2006) Nearly-inte- tems, 2nd edn. Addison-Wesley, Redwood City grable perturbations of the Lagrange top: applications of KAM 59. Diacu F, Holmes P (1996) Celestial Encounters. The Origins of theory. In: Denteneer D, den Hollander F, Verbitskiy E (eds) Chaos and Stability. Princeton University Press, Princeton Dynamics & Stochastics: Festschrift in Honor of MS Keane Lec- 60. Duistermaat JJ (1980) On global action-angle coordinates. ture Notes, vol 48. Inst. of Math. Statistics, pp 286–303 Commun Pure Appl Math 33:687–706 39. Broer HW, Hanßmann H, You J (2006) Umbilical torus bifurca- 61. Dumortier F, Roussarie R, Sotomayor J (1991) Generic 3-pa- tions in Hamiltonian systems. J Differ Equ 222:233–262 rameter families of vector fields, unfoldings of saddle, focus 40. Broer HW, Naudot V, Roussarie R (2006) Catastrophe theory in and elliptic singularities with nilpotent linear parts. In: Du- Dulac unfoldings. Ergod Theory Dyn Syst 26:1–35 mortier F, Roussarie R. Sotomayor J, Zoladek H (eds) Bifur- 41. Broer HW, Ciocci MC, Hanßmann H (2007) The quasi-periodic cations of Planar Vector Fields: Nilpotent Singularities and reversible Hopf bifurcation. In: Doedel E, Krauskopf B, Sanders Abelian Integrals. LNM 1480, pp 1–164 J (eds) Recent Advances in Nonlinear Dynamics: Theme sec- 62. Efstafhiou K (2005) Metamorphoses of Hamiltonian systems tion dedicated to André Vanderbauwhede. Intern J Bifurc with symmetries. LNM, vol 1864. Springer, Heidelberg Chaos 17:2605–2623 63. Féjoz J (2004) Démonstration du “théorème d’Arnold” sur la 42. Broer HW, Ciocci MC, Hanßmann H, Vanderbauwhede A stabilité du système planétaire (d’après Herman). Ergod The- (2009) Quasi-periodic stability of normally resonant tori. Phys ory Dyn Syst 24:1–62 D 238:309–318 64. Gallavotti G, Bonetto F, Gentile G (2004) Aspects of Ergodic, 43. Broer HW, Cushman RH, Fassò F, Takens F (2007) Geometry Qualitative and Statistical Theory of Motion. Springer of KAM tori for nearly integrable Hamiltonian systems. Ergod 65. Gelfreich VG, Lazutkin VF (2001) Splitting of Separatrices: per- Theory Dyn Syst 27(3):725–741 turbation theory and exponential smallness. Russ Math Surv 44. Broer HW, Hanßmann H, Hoo J (2007) The quasi-periodic 56:499–558 Hamiltonian Hopf bifurcation. Nonlinearity 20:417–460 66. Guckenheimer J, Holmes P (1983) Nonlinear Oscillations, Dy- 45. Broer HW, Hoo J, Naudot V (2007) Normal linear stability of namical Systems, and Bifurcations of Vector Fields. Springer quasi-periodic tori. J Differ Equ 232:355–418 67. Hanßmann H (1988) The quasi-periodic centre-saddle bifur- 46. Broer HW, Simó C, Vitolo R (2008) The Hopf–Saddle-Node bi- cation. J Differ Equ 142:305–370 furcation for fixed points of 3D-diffeomorphisms, the Arnol’d 68. Hanßmann H (2004) Hamiltonian Torus Bifurcations Related resonance web. Bull Belg Math Soc Simon Stevin 15:769–787 to Simple Singularities. In: Ladde GS, Medhin NG, Samband- 47. Broer HW, Simó C, Vitolo R (2008) The Hopf–Saddle-Node bi- ham M (eds) Dynamic Systems and Applications, Atlanta furcation for fixed points of 3D-diffeomorphisms, analysis of 2003. Dynamic Publishers, pp 679–685 a resonance ‘bubble’. Phys D Nonlinear Phenom (to appear) 69. Hanßmann H (2007) Local and Semi-Local Bifurcations in 48. Broer HW, Hanßmann H, You J (in preparation) On the de- Hamiltonian Dynamical Systems – Results and Examples. In: struction of resonant Lagrangean tori in Hamiltonian systems Lecture Notes in Mathematics, vol 1893. Springer, Berlin 49. Chenciner A (1985) Bifurcations de points fixes elliptiques I, 70. Herman M (1977) Mesure de Lebesgue et nombre de rota- courbes invariantes. Publ Math IHÉS 61:67–127 tion. In: Palis J, do Carmo M (eds) Geometry and Topology. In: 50. Chenciner A (1985) Bifurcations de points fixes elliptiques II, Lecture Notes in Mathematics, vol 597. Springer, pp 271–293 orbites périodiques et ensembles de Cantor invariants. Invent 71. Herman MR (1979) Sur la conjugaison différentiable des dif- Math 80:81–106 féomorphismes du cercle à des rotations. Publ Math IHÉS 51. Chenciner A (1988) Bifurcations de points fixes elliptiques III, 49:5–233 orbites périodiques de “petites” périodes et élimination ré- 72. Hirsch MW (1976) Differential Topology. Springer sonnante des couples de courbes invariantes. Publ Math IHÉS 73. Hirsch MW, Pugh CC, Shub M (1977) Invariant Manifolds. In: 66:5–91 Lecture Notes in Mathematics, vol 583. Springer 52. Chenciner A, Iooss G (1979) Bifurcations de tores invariants. 74. Hofer H, Zehnder E (1994) Symplectic invariants and Hamilto- Arch Ration Mech Anal 69(2):109–198; 71(4):301–306 nian dynamics. Birkhäuser Hamiltonian Perturbation Theory (and Transition to Chaos) H 4539

75. Hopf E (1942) Abzweigung einer periodischen Lösung von 92. MacKay RS (1993) Renormalisation in area-preserving maps. einer stationären Lösung eines Differentialsystems. Ber Math- World Scientific Phys Kl Sächs Akad Wiss Leipzig 94:1–22 93. Marco J-P, Sauzin D (2003) Stability and instability for 76. Hopf E (1948) A mathematical example displaying features of Gevrey quasi-convex near-integrable Hamiltonian systems. turbulence. Commun Appl Math 1:303–322 Publ Math Inst Hautes Etud Sci 96:199–275 77. Huygens C Œvres complètes de Christiaan Huygens, (1888– 94. Markus L, Meyer KR (1980) Periodic orbits and solenoids 1950), vol 5, pp 241–263 and vol 17, pp 156–189. Martinus in generic Hamiltonian dynamical systems. Am J Math 102: Nijhoff, The Hague 25–92 78. de Jong HH (1999) Quasiperiodic breathers in systems of 95. Matveev VS (1996) Integrable Hamiltonian systems with weakly coupled pendulums: Applications of KAM theory two degrees of freedom. Topological structure of saturated to classical and statistical mechanics. Ph D Thesis, Univ. neighborhoods of points of focus-focus and saddle-saddle Groningen types. Sb Math 187:495–524 79. Katok A, Hasselblatt B (1995) Introduction to the Modern 96. McDuff D, Salamon D (1995) Introduction to Symplectic Ge- Theory of Dynamical Systems. Cambridge University Press, ometry. Clarendon/Oxford University Press Cambridge 97. Meyer KR, Hall GR (1992) Introduction to Hamiltonian Dynam- 80. Kolmogorov AN (1954) On the persistence of conditionally ical Systems and the N-Body Problem. In: Applied Mathemat- periodic motions under a small change of the Hamilton func- ical Sciences, vol 90. Springer tion. Dokl Akad Nauk SSSR 98:527–530 (in Russian); English 98. Meiss JD (1986) Class renormalization: Islands around islands. translation: Stochastic Behavior in Classical and Quantum Phys Rev A 34:2375–2383 Hamiltonian Systems, Volta Memorial Conference (Como, 99. de Melo W, van Strien SJ (1991) One-Dimensional Dynamics. 1977). In: Casati G, Ford J (eds) Lecture Notes in Physics, Springer vol 93. Springer, Berlin pp 51–56 (1979); Reprinted in: Bai Lin 100. Mielke A (1991) Hamiltonian and Lagrangian Flows on Center Hao (ed) Chaos. World Scientific, Singapore, pp 81–86 (1984) Manifolds – with Applications to Elliptic Variational Problems. 81. Kolmogorov AN (1957) The general theory of dynamical sys- In: Lecture Notes in Mathematics, vol 1489. Springer tems and classical mechanics. In: Gerretsen JCH, de Groot 101. Milnor JW (2006) Dynamics in One Complex Variable, 3rd edn. J (eds) Proceedings of the International Congress of Math- In: Ann. Math. Studies, vol 160. Princeton University Press, ematicians, vol 1 (Amsterdam, 1954), North-Holland, Ams- Princeton terdam, pp 315–333 (in Russian); Reprinted in: International 102. Morbidelli A, Giorgilli A (1995) Superexponential Stability of Mathematical Congress in Amsterdam, (1954) (Plenary Lec- KAM Tori. J Stat Phys 78:1607–1617 tures). Fizmatgiz, Moscow, pp 187–208 (1961); English trans- 103. Morbidelli A, Giorgilli A (1995) On a connection between KAM lation as Appendix D in: Abraham RH (1967) Foundations of and Nekhoroshev’s theorems. Physica D 86:514–516 Mechanics. Benjamin, New York, pp 263–279; Reprinted as 104. Moser JK (1962) On invariant curves of area-preserving map- Appendix in [1], pp 741–757 pings of an annulus. Nachr Akad Wiss Göttingen II, Math-Phys 82. Kuznetsov YA (2004) Elements of Applied Bifurcation Theory, Kl 1:1–20 3rd edn. In: Applied Mathematical Sciences, vol 112. Springer, 105. Moser JK (1966) On the theory of quasiperiodic motions. SIAM New York Rev 8(2):145–172 83. Landau LD (1944) On the problem of turbulence. Akad Nauk 106. Moser JK (1967) Convergent series expansions for quasi-peri- 44:339 odic motions. Math Ann 169:136–176 84. Landau LD, Lifschitz EM (1959) Fluid Mechanics. Pergamon, 107. Moser JK (1968) Lectures on Hamiltonian systems. Mem Am Oxford Math Soc 81:1–60 85. Laskar J (1995) Large scale chaos and marginal stability in 108. Moser JK (1973) Stable and random motions in dynamical sys- the Solar System, XIth International Congress of Mathemat- tems, with special emphasis to celestial mechanics. In: Ann. ical Physics (Paris, 1994). In: Iagolnitzer D (ed) Internat Press, Math. Studies, vol 77. Princeton University Press, Princeton Cambridge, pp 75–120 109. Nekhoroshev NN (1977) An exponential estimate of the time 86. Litvak-Hinenzon A, Rom-Kedar V (2002) Parabolic resonances of stability of nearly-integrable Hamiltonian systems. Russ in 3 degree of freedom near-integrable Hamiltonian systems. Math Surv 32:1–65 Phys D 164:213–250 110. Nekhoroshev NN (1985) An exponential estimate of the time 87. de la Llave R, González A, Jorba À, Villanueva J (2005) KAMthe- of stability of nearly integrable Hamiltonian systems II. In: ory without action-angle variables. Nonlinearity 18:855–895 Oleinik OA (ed) Topics in Modern Mathematics, Petrovskii 88. Lochak P (1999) Arnold diffusion; a compendium of remarks Seminar No.5. Consultants Bureau, pp 1–58 and questions. In: Simó C (ed) Hamiltonian systems with three 111. Newhouse SE, Palis J, Takens F (1983) Bifurcations and stabil- or more degrees of freedom (S’Agaró, 1995), NATO ASI Se- ity of families of diffeomorphisms. Publ Math IHÉS 57:5–71 ries C: Math Phys Sci, vol 533. Kluwer, Dordrecht, pp 168–183 112. Newhouse SE, Ruelle D, Takens F (1978) Occurrence of 89. Lochak P, Marco J-P (2005) Diffusion times and stability ex- strange Axiom A attractors near quasi-periodic flows on T m, ponents for nearly integrable analytic systems, Central Eur J m 3. Commun Math Phys 64:35–40 Math 3:342–397 113. Niederman L (2004) Prevalence of exponential stability 90. Lochak P, Ne˘ıshtadt AI (1992) Estimates of stability time for among nearly-integrable Hamiltonian systems. Ergod Theory nearly integrable systems with a quasiconvex Hamiltonian. Dyn Syst 24(2):593–608 Chaos 2:495–499 114. Nguyen Tien Zung (1996) Kolmogorov condition for inte- 91. Lukina O (2008) Geometry of torus bundles in Hamiltonian grable systems with focus-focus singularities. Phys Lett A systems, Ph D Thesis, Univ. Groningen 215(1/2):40–44 4540 H

115. Oxtoby J (1971) Measure and Category. Springer 141. Yoccoz J-C (1995) Théorème de Siegel, nombres de Bruno et 116. Palis J, de Melo M (1982) Geometric Theory of Dynamical Sys- polynômes quadratiques. Astérisque 231:3–88 tems. Springer 142. Yoccoz J-C (2002) Analytic linearization of circle diffeomor- 117. Palis J, Takens F (1993) Hyperbolicity & Sensitive Chaotic phisms.In:MarmiS,YoccozJ-C(eds)DynamicalSystems Dynamics at Homoclinic Bifurcations. Cambridge University and Small Divisors, Lecture Notes in Mathematics, vol 1784. Press, Cambridge Springer, pp 125–174 118. Poincaré H (1980) Sur le problème des trois corps et les équa- 143. Zehnder E (1974) An implicit function theorem for small divi- tions de la dynamique. Acta Math 13:1–270 sor problems. Bull Am Math Soc 80(1):174–179 119. Pöschel J (1982) Integrability of Hamiltonian systems on Can- 144. Zehnder E (1975) Generalized implicit function theorems tor sets. Commun Pure Appl Math 35(5):653–696 with applications to some small divisor problems, I and II. 120. Pöschel J (1993) Nekhoroshev estimates for quasi-convex Commun Pure Appl Math 28(1):91–140; (1976) 29(1):49–111 Hamiltonian systems. Math Z 213:187–216 121. Pöschel J (2001) A lecture on the classical KAM Theorem. In: Proc Symp Pure Math 69:707–732 122. Rink BW (2004) A Cantor set of tori with monodromy near a focus-focus singularity. Nonlinearity 17:347–356 123. Robinson C (1995) Dynamical Systems. CRC Press 124. Roussarie R (1997) Smoothness properties of bifurcation diagrams. Publ Mat 41:243–268 125. Ruelle D (1989) Elements of Differentiable Dynamics and Bi- furcation Theory. Academic Press 126. Ruelle D, Takens F (1971) On the nature of turbulence. Com- mun Math Phys 20:167–192; 23:343–344 127. Sevryuk MB (2007) Invariant tori in quasi-periodic non-autonomous dynamical systems via Herman’s method. DCDS-A 18(2/3):569–595 128. Simó C (2001) Global dynamics and fast indicators. In: Broer HW, Krauskopf B, Vegter G (eds) Global Analysis of Dynami- cal Systems, Festschrift dedicated to Floris Takens for his 60th birthday. IOP, Bristol and Philadelphia, pp 373–390 129. Spivak M (1970) Differential Geometry, vol I. Publish or Perish 130. Takens F (1973) Introduction to Global Analysis. Comm. 2 of the Math. Inst. Rijksuniversiteit Utrecht 131. Takens F (1974) Singularities of vector fields. Publ Math IHÉS 43:47–100 132. Takens F (1974) Forced oscillations and bifurcations. In: Appli- cations of Global Analysis I, Comm 3 of the Math Inst Rijksuni- versiteit Utrecht (1974); In: Broer HW, Krauskopf B, Vegter G (eds) Global Analysis of Dynamical Systems, Festschrift dedicated to Floris Takens for his 60th birthday. IOP, Bristol and Philadelphia, pp 1–62 133. Thom R (1989) Structural Stability and Morphogenesis. An Outline of a General Theory of Models, 2nd edn. Addison- Wesley, Redwood City (English; French original) 134. VuNgo˜ . c San (1999) Quantum monodromy in integrable systems. Commun Math Phys 203:465–479 135. Waalkens H, Junge A, Dullin HR (2003) Quantum monodromy in the two-centre problem. J Phys A Math Gen 36:L307-L314 136. Wagener FOO (2003) A note on Gevrey regular KAM theory and the inverse approximation lemma. Dyn Syst 18:159–163 137. Wiggins S (1990) Introduction to Applied Nonlinear Dynami- cal Systems and Chaos. Springer 138. Wiggins S, Wiesenfeld L, Jaffe C, Uzer T (2001) Impenetrable barriers in phase-space. Phys Rev Lett 86(24):5478–5481 139. Yoccoz J-C (1983) C1-conjugaisons des difféomorphismes du cercle. In: Palis J (ed) Geometric Dynamics, Proceedings, Rio de Janeiro (1981) Lecture Notes in Mathematics, vol 1007, pp 814–827 140. Yoccoz J-C (1992) Travaux de Herman sur les tores invariants. In: Séminaire Bourbaki, vol 754, 1991–1992. Astérisque 206:311–344