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 ap- proximation 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.