Survey on dissipative KAM theory including quasi-periodic bifurcation theory

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1 I Survey on dissipative KAM theory including quasi-periodic bifurcation theory Maria-Cristina Ciocci, Anna Litvak-Hinenzon & Henk Broer Based on lectures by Henk Broer ABSTRACT Kolmogorov-Arnol d-moser Theory classically was mainly developed for conservative systems, establishing persistence results for quasi-periodic invariant tori in nearly integrable systems. In this survey we focus on dissipative systems, where similar results hold. In non-conservative settings often parameters are needed for the persistence of invariant tori. When considering families of such dynamical systems bifurcations of quasi-periodic tori may occur. As an example we discuss the quasi-periodic Hopf bifurcation. 1.1 Motivation 1 Introduction Kolmogorov-Arnold-Moser Theory is concerned with the occurrence of multior quasi-periodic invariant tori in nearly integrable systems. Integrable systems by definition have a toroidal symmetry which produces invariant tori as orbits under the corresponding torus action. The central problem of KAM Theory, is the continuation of such tori to nearly integrable perturbations of the system. Initially this part of perturbation theory was developed for conservative, i.e., Hamiltonian, systems that model the frictionless dynamics of classical mechanics. Related physical questions are concerned with the perpetual stability of the solar system, of tokamak accelerators, etc. Initiated by Poincaré at the end of the 19th century, the theory was further developed by Birkhoff and Siegel and later established by Kolmogorov, Arnold, Moser and others from the 1950s on. For a historical overview and further reference, see [36]. As pointed out in [66, 65] and later in [48, 24, 23], the conservative approach can be extended to many other settings, like to general dissipative systems, to volume preserving systems and to various equivariant or reversible settings. A unifying Lie algebra approach enables to reach all these results at once [66, 24].

2 2 I Survey on dissipative KAM theory In many cases the systems need to depend on parameters in order to ensure the persistent occurrence of quasi-periodicity. In the general dissipative formulation we therefore encounter families of quasi-periodic attractors, parametrized over a nowhere dense set of positive measure. The parametrization is smooth in the sense of Whitney [73, 24]. As an example of the physical relevance of this phenomenon we mention the Ruelle-Takens scenario for the onset of turbulence [76, 75]. In this scenario a fluid dynamical system depends on parameters, where upon changes of the parameters transitions from laminar to more complicated and even turbulent dynamics are described at a low dimensional level. Here chaotic dynamics plays a special role, related to the onset of turbulence and quasi-periodicity is an intermediate, pre-chaotic stage of this. The present material on KAM Theory focuses on results in this dissipative setting. In this way the mathematical difficulties are largely decoupled from the symplectic and Hamiltonian formalism. On the one hand, this clarifies the mathematics of KAM Theory, which is difficult enough to master by itself. On the other hand we illustrate the analogy of our approach in the conservative and other contexts by quite a few remarks and excercises, particularly in the Appendix. To this end we also include a number of references for further reading. Moreover, we included elements from quasi-periodic bifurcation theory, focussing on the Hopf case, where from -dimensional quasi-periodic attractors -dimensional quasi-periodic attractors branch off. This example fits very well in the Ruelle-Takens scenario for the onset of turbulence as sketched before, and its relationship to the more classical Landau-Hopf-Lifschitz scenario [57, 47, 58]. To a great extent the material of this course is contained in [23], to which we often shall refer for background material, for details and further reference. We thank Heinz Hanßmann, George Huitema, Jun Hoo, Vincent Naudot, Khairul Saleh, Floris Takens, Renato Vitolo and Florian Wagener for their help during the preparation of these notes. 1.2 Preliminaries We introduce a few basic concepts, for simplicity restricting to the world of smooth dynamical systems with continuous time. Such systems are generated by vector fields, locally given as systems of ordinary differential equation. For background information regarding see, e.g., Arnol d [1], Moser [66, 65], For example in a center manifold [45]. Similar considerations hold for systems with discrete time generated by diffeomorphisms.

3 P I. 1 Introduction 3 Broer et al. [7, 24, 23]. Also see Broer, Dumortier, Van Strien and Takens [14], Chs. 4 and 9. First we recall the notion of smooth conjugacy between vector fields. Let two vector fields and be given and a diffeomorphism of that takes solutions of the former to the latter vector field in a time preserving way. Then, if! and " # " $ are such solutions, it follows by the Chain Rule that %'&)(*,+ -'&%(*,+. where we took matrix products. Such a map is called a conjugacy between the two vector fields. It follows that the condition & ( / (1.1) where -4 3 is necessary for being a conjugacy. By the Existence and Uniqueness Theorem [39, 46] for solutions of ordinary differential equations, condition (1.1) also is sufficient for to be a conjugacy. In tensorial shorthand we often rewrite (1.1) as Remark There exists many variations on the definition of conjugacy, compare with [1, 72]. One variation relaxes the preservation of time-parametrization; the corresponding map then is called an equivalence. Another variation concerns the smoothness, which often is replaced by continuity. Indeed, if is just a homeomorphism we speak of a topological conjugacy or equivalence. With respect to all such equivalence relations the notion of structural stability can be introduced, meaning that with all nearby vector fields (in an appropriate topology) the corresponding relation holds. This is a strong way of formulating persistence of various dynamical properties, like the existence of equilibria, periodic solutions or invariant tori. It turns out that in the present setting of KAM Theory we can use smooth conjugacies or equivalences and corresponding forms of structural stability to express persistence results. Now we come to a central subject of KAM Theory, namely the quasi-periodic invariant torus. Let be a natural number and denote by :<; = A@CBED ;?> $; the standard -torus, equipped with angular coordinates ; 8 Definition 1.1 A vector field F on a smooth manifold GIH with an invariant submanifold J K5H is said to have parallel flow on JL if the restriction FNM O For all purposes one can take QSRUTWV for XZY\[ sufficiently large.

4 ; a a ; ; 8 4 I Survey on dissipative KAM theory ] PSfrag replacements ] PSfrag replacements ] ] Fig Evolution curve of a constant vector field on the 2-torus ^ `_ is smoothly conjugate to the constant vector field on : ; 8 a +.+.+b+.+.+b+.+.+ The conjugacy is a diffeomorphism between J and : ; and the evolution curves of a dcef parallel flow on JL in the coordinates associated to the conjugacy, are given by a ; a where the addition is mod, see Figure 1.1. We say that J is a parallel torus of F, with frequencies a a a The nature of the flow on J depends on the arithmetical properties of its frequencies, where we distinguish between dependence or independence over the rationals.

5 a : e a D D ; I. 1 Introduction 5 Definition 1.2 A parallel torus J is quasi-periodic (or non-resonant) if a a are rationally independent; that is, for all ; g D ;ih*j kml one has n a ogqpir k 8 n Here we abbreviate a `gsp 5t ;uwv a u g u 8 Quasi-periodic tori are densely filled by each of the evolution curves contained in them, see Exercise 1. The parallel torus is called resonant, n if its frequencies are rationally dependent, meaning that an integer relation a ogqp k exists for some g D ;<hxjykml 8 Resonant tori are foliated by lower dimensional tori [2]. Remarks - Consider the 8 For a given vector field F on :2 we can study the Poincaré return map z of the circle k 8 In the present setting where F is constant, this map is a well-defined diffeomorphism of the circle : provided that a r k 8 Indeed, the evolution curve of the vector field F starting at the point k A@CB passes through {z $ see Figure Note that when F is constant, the circle map z is a rigid rotation. We quote the following result on rigid rotations, which is quite well-known, e.g., see [35]: Lemma If } is irrational, then each orbit of the rigid rotation ~L} : " % mƒl@cb is dense in :. Exercise 1 vector fields on :< ) On :<C with coordinates 3 both counted mod consider the constant vector field F given by a 8 (1.2) (i) Suppose that a and a are rationally independent, then show that any integral curve of F is dense in : 8 (ii) Suppose that a > a >i with gcd { i 8 Show that : is foliated by periodic solutions, all of period 8 Exercise 2 (How intrinsic is the frequency vector?) If we define : ; ' ; > and count the angles modulo consider : ; -automorphisms of the affine form ce#ˆ ˆ where } ; and ŠmŒ the -Ž matrices of integers with determinant [2].

6 6 I Survey on dissipative KAM theory ] ] { ] PSfrag replacements ] PSfrag replacements ] o ] Fig Poincaré map associated to the section ]?, of ^ 3_ (i) Show that the frequency vector of a parallel torus is well-defined up to D the lattice ; 8 (ii) How does this translate to the present situation where : ; ; > ;? (iii) What can you say of the frequency vectors of F and 6 F when is sufficiently close to the identity map? (iv) Show that an individual vector field of the form (1.2) can never be structurally stable. 2 Quasi-periodic attractors In what dynamical systems does the phenomenon of quasi-periodicity occur? Our answer to this question should have the following two properties: - The occurrence is visible in the physical sense, meaning that the set of

7 I. 2 Quasi-periodic attractors 7 initial values showing quasi-periodicity or asymptotic quasi-periodicity has positive measure. - The occurrence is persistent for small perturbations of the system, which means that we are not interested in pathological examples of this phenomenon. Colloquially the above often is rephrased as for typical systems quasi-periodicity occurs in a visible way. Both in the conservative and in the dissipative setting many concrete examples can be given in this sense [24, 23]. Since we focus on the dissipative setting, we present a few examples with quasi-periodic attractors. For conservative analogues we refer to the Appendix. The first property of physical visibility is easily met: the attractors have an open basin of attraction (occupying almost the entire phase space). The second property of persistence is more problematic and belongs to the domain of KAM Theory, which is the subject of this course. As announced before, persistence requires that the systems depend on parameters, compare with the Exercises 1 and 2. The quasi-periodic attractors will be isolated in phase space and occur on parameter sets that are nowhere dense and of positive measure, compare with [75]. PSfrag replacements Fig Phase portrait of the free Van der Pol oscillator (2.1).

8 a 8 I Survey on dissipative KAM theory 2.1 Forcing or coupling of nonlinear oscillators Basic ingredient of our examples is a nonlinear oscillator with equation of motion where } and -ž Ÿ > š ˆ Nœ E k (2.1) which is assumed to have a hyperbolic periodic attractor, i.e., a periodic solution with ˆ a negative Floquet-exponent. For the moment we consider coefficients like and š as positive constants, but later on they also may be regarded as parameters. A classical example of such a system is the Van der Pol oscillator, where the nonlinearity is given by œ W 'o with a real constant. For a phase portrait see Figure A nonlinear oscillators with periodic forcing As a first example consider the oscillator (2.1) subject to a weak time-periodic forcing: /š ˆ œ y (2.2) where >Ÿ 7 1 E and where is a small perturbation parameter. As usual we take the time as an extra state-variable, intro- with coordinates ducing the -dimensional (generalized) phase space ŽU: and : ' A@CBED > 8 Here the non-autonomous oscillator (2.2) defines the vector field F% given by ª ª «ˆ «š ª «œ ª E y ª (2.3) 8 It is our aim to get this example in the format of Section 1.2, corresponding to and the definition of quasi-periodicity given there. From now on we set 8 We start considering the unperturbed case k 8 Here the oscillator is free and a decouples from the third equation 8 Combining the periodic attractor of the free oscillator (2.1) with this third equation gives rise to an invariant 2-torus to be denoted by J 8 Here we use certain elements of the theory of ordinary diffenential equations, compare [1, 39, 46]. The time-parameterization of the periodic attractor provides a (mod such that the corresponding evolution is generated by an equation Sa. It can be shown that the angular (mod and give a conjugacy

9 ˆ a I. 2 Quasi-periodic attractors 9 of the restriction F) v M Oy to the constant vector field a on the standard -torus :2 8 We conclude that for k the system (2.3) has an attracting parallel invariant 2-torus. What happens to such an invariant torus for r k? Note that by hyperbolicity of the periodic orbit, the unperturbed torus J is normally hyperbolic. Thus, according to the Center Manifold Theorem, compare [45, 81], J is persistent as an invariant This means that, for M M<±² the vector field F% has a smooth invariant -torus J, also depending smoothly on *8 Here smooth means finitely differentiable. In particular the degree of differentiability tends to ³ as e k 8 The remaining question then concerns the persistence of the dynamics inside J 8 In howfar is the parallellity or quasi-periodicity of the dynamics persistent under small perturbation? This question will be answered in the next subsection. Before that we give another example, which easily can be generalized to higher dimension Coupled nonlinear oscillators As a second example consider two nonlinear oscillators with a weak coupling /š ˆ Nœ W y /š Nœ W y 3 x8 This yields the following vector field F on the 4-dimensional phase space UŽ j ª 3 ª l : y µ ª. (2.5) ª «ˆ «š ª «œ ª E y ª ª Ÿ 8 Note that for k the system decouples to a system of two independent oscillators and has an attractor in the form of a two dimensional torus J 8 This torus arises as the product of two (topological) circles, along which each of the oscillators has its periodic solution. The circles lie in the two-dimensional planes given by ª Z k and ª ' k respectively. In the plane ª ' k we see the evolutions of the first oscillator only. The time-parameterization of its periodic attractor gives a coordinate " % such that the corresponding evolution is generated by 'a 8 Similarly, one finds a for the second oscillator in the plane ª ž k 8 We see that in this way the restriction F M J is conjugate to the constant vector field (2.4)

10 a 10 I Survey on dissipative KAM theory which, as before, lives on the -torus :< 8 The conclusion is that the system (2.5), for ¹ k has an attracting parallel invariant 2-torus. It may be clear that a similar coupling of nonlinear oscillators gives rise to an attracting parallel -torus inside ; 8 Regarding the fate of J for M Mº±» we can repeat the above discussion verbatim. This means that by the Center Manifold Theorem [45, 81] we find a smooth invariant 2-torus J C but that regarding the parallel and the quasiperiodic dynamics on J further discussion is needed. 2.2 Reduction to KAM Theory of circle maps We now discuss the persistence of the dynamics in the invariant 2-tori Jº as this came up in both of the above examples. After a first center manifold reduction to the standard 2-torus :<i we reduce to maps of the circle : by taking a Poincaré section Preliminaries As said before, the tori J are smooth (highly differentiable) center manifolds due to normal hyperbolicity [45, 81]. Indeed, for M M ±¼ Ÿ all J are diffeomorphic to the standard 2-torus :<C where the degree of differentiability increases to ³ as tends to k 8 To study the dynamics inside the torus, from now on we restrict to this center manifold and reduce the perturbation problem to :< 8 The KAM Theorem we are dealing with allows for formulations in the world of ½ ¾ -systems for g ½ sufficiently large, including compare [73, 24]. These versions are variations on the simpler setting where the systems are real analytic. For simplicity we therefore restrict to the case where our entire perturbation problem is real analytic. The present version of the KAM perturbation problem looks for smooth conjugacies or smooth equivalences between F% and F both living on the 2-torus :<C where M MŸ±¼ 8 Considering a simple example like a we observe that both quasi-periodicity and resonance have dense occurrence, compare Section 1.2. These two cases cannot be equivalent, since in the latter case all evolutions are compact, which they are not in the former. To be more precise, when classifying parallel 2-tori under (smooth) conjugacy the

11 @ e e e ˆ I. 2 Quasi-periodic attractors 11 frequencies a and a are invariants, while under (smooth) equivalence the frequency ratio a a is an invariant. Compare with the Exercises 1 and 2. So for a systematic study of the persistence problem it is necessary to introduce parameters. In the examples concerning the forced oscillator ˆ (2.2) or the coupled oscillators (2.5) we may consider the coefficients {š ˆ or {š {š respectively, as parameters. The frequencies a and a of the unperturbed vector field F), see (2.4), then are regarded as functions of these parameters. We claim that in our examples the following nondegeneracy condition holds, namely that the frequency ratio a > a varies as a function of the relevant multi-parameter: locally this correspondence is submersive. To simplify things further we consider a > a itself as a parameter. This leaves us with a family of vector fields F F qà on the standard -torus :, which is assumed real analytic in all the variables. We study this qà of the circle k considered as a : compare a remark following Definition 1.2. qà family by the Poincaré return map z circle diffeomorphism : The map z has the form ce Note that for - ˆ { < 8 (2.6) k the unperturbed map z qà ~ is just the rigid of the circle : 8 For a proper formulation of our problem it is convenient to regard the family of circle maps as a vertical map of the cylinder by considering z? : ŽÂÁ k..ã : ŽÂÁ k. Ã defined as z? $ - ˆ { < $? 8 The persistence problem now is further formalized as follows. We start looking for a diffeomorphism : Ž Á k 0 Ã : Ž Á k..ã conjugating the unperturbed family z? to the perturbed family z i.e., such that zw2å 5 2Å z 8 (2.7) The conjugacy equation (2.7) also is expressed by commutativity of the following diagram. : ŽÂÁ k. Ã «e È Æ Ç : ŽÂÁ k. Ã «Æ e : È ŽÂÁ k. Ã : ŽÂÁ k. Ã

12 Î ˆ e 12 I Survey on dissipative KAM theory Remarks - Conjugacies between return maps directly translate to equivalences between the corresponding vector fields, see [1, 72]. In the case of the first example (2.2) these equivalences can even be made conjugacies. - For orientation preserving circle homeomorphisms, the rotation number is an invariant under (topological) conjugacy, e.g., compare with [1, 35]. The rotation number of such homeomorphism is the average amount of rotation effected by the homeomorphism, which for the unperturbed map z qà exactly coincides with the frequency ratio 8 - The Denjoy Theorem [1, 35] asserts that for irrational < whenever the rotation number of z É"À equals < a topological conjugacy exists between z qà and z É À Formal considerations and small divisors We now study equation (2.7) for the conjugacy to some detail. To simplify the notation first set -'. Assuming that : Ž}Á k 0 Ã : ŽÁ k 0 Ã has the general form {? Ê $ < $ - ië < { 3 we get the following nonlinear equation for the function Ê and the parameter shift Ë <{ < «Ê $ < Ë < ˆ\ W Ê $ 2 3{ - ië As is common in classical perturbation theory,ì ˆ we expand Ê and Ë as formal power series in and solve (2.8) by comparing coefficients. We only consider the coefficients of power zero in not only because asymptotically these coefficients have the strongest effect, but also since the coefficients of higher -powers satisfy similar equations. So, writing ô { < ˆ $? Í Ê $ 2 5Ê $? NÍ 3 Ë 2 Ë? W NÍ substitution in equation (2.8) leads to the following, so-called homological, equation <$ «,Ê Compare with Poincaré-Lindstedt series. Ë?? {? 3 (2.9)

13 í î ˆ À ˆ Û Û Û Û ˆ I. 2 Quasi-periodic attractors 13 which has to be solved for Ê, and Ë 8 Equation (2.9) is linear and therefore can be directly solved by Fourier series. Indeed, introducing Ï $ 5Ð ¾yÑiÒ ¾ u ¾ (? $Ó and Ê $? 5Ð ¾ ÑiÒ and comparing coefficients in (2.9), directly yields that Ê ¾ u ¾ (? ÔÓ Ë «@ib ` Ê ¾? ¾? Õ u `Ö ¾ «og D h*j0kml (2.10) while Ê ` which corresponds to the position of the origin k on JL remains arbitrary. We conclude that in general a formal solution exists if and only if is irrational. Even then one meets the problem of small divisors, caused by the accumulation of the denominators in (2.10) on k which makes the convergence of the Fourier series of Ê problematic. This problem can be solved by a further restriction of by so-called Diophantine conditions. and ÙÚØ k be given. We say that Á k 0 à is Diophantine if for all { D with )Ø k we have that ÛÛ «Ù 8 ÛqÜ (2.11) C Let us denote the set of satisfying (2.11) by Á k 0 à À Þ KßÁ k..ã 8 It is easily seen that Á k 0 à À Þ is a closed set. From this, by the Cantor-Bendixson Theorem [42] it follows that Á k. à À Þ is the union of a perfect set and a countable set. The perfect set, for sufficiently small Ù,Ø k has to be a Cantor set, since it is compact and totally disconnected.à The latter means that every point of Á k..ã À Þ has arbitrarily small neighbourhoods with empty boundary, which directly follows from the fact that the dense set of rationals is in its complement. Note that, since Á k 0 à À Þ KÚÁ k..ãa so as a subset of the real line, the property of being totally disconnected is equivalent to being nowhere dense. Anyhow, the set Á k 0 à À Þ is small in the topological sense. In this case, however, the Lebesgue measure of Á k..ã À Þ is not small, since measure á`á k..ã h Á k..ã À Þ â Ù Ðäoå Ôé çæ?è æ Í Ù (2.12) Definition 2.1 Let =Ø, e.g., compare [1, 24, 23], also for further reference. Note that the estimate (2.12) implies that the union as ÙZê k, by our assumption that Ø ë Þçì Á k. à À Þ In [23] this is called a 1 bite small divisor problem. Or zero-dimensional.

14 ] ] ò ò e e 14 I Survey on dissipative KAM theory ð ïð ñ PSfrag replacements ó iôoõ`öw 3ø ù Fig Conjugacy between the Poincaré maps ñ and ð on ^ ú ó iô3õ`ö øù _ for any fixed =Ø is of full measure. For a thorough discussion of the discrepancy between the measure theoretical and the topological notion of the size of number sets, see Oxtoby [71]. Returning to (2.10), we conclude the following on the convergence of the Fourier series. ˆ ˆ First we recall that for a real analytic function the Fourier coefficients ¾ decay exponentially as M gem ³ 8 This is implied by the Paley- Wiener estimate, which, for completeness, is included in the Appendix. Also À Þ it see Exercise 15. Second, a brief calculation reveals that for Á k 0 à follows that for all g D hçj k*l we have M Õ u `Ö ¾ «*M Üüû Ù<M gemýæ 8 We conclude that the coefficients Ê À ¾ still have exponential decay as M gem ³Z which implies that the sum Ê again is a real analytic function. Of course this does not yet prove that the function Ê==Ê $ < exists in one way or another for M MŸ±¼ 8 Yet we do have the following.

15 e I. 2 Quasi-periodic attractors 15 Theorem 2.2 For M M and Ù ½ sufficiently small, there exists a -diffeomorphism : ŽÁ k..ã : ŽÁ k..ãa conjugating the restriction z Mýþ À ÿ to a subsystem of z 8 Observe that Theorem 2.2 is independent of the oscillator background provided by the above examples. Indeed, it is a general result for circle diffeomorphisms and also it forms the first KAM Theorem of this course. There exist several more or less independent proofs of Theorem 2.2. We refer to [24, 23] for further discussion and a large bibliography. The proofs of the KAM Theorems in these notes usually are not based on extending the above formal argument by showing that the power series in converges. Indeed, an appropriate Newtonian iteration process is set up, based on a linearization related to the above homological equation (2.9). For an earlier version also see [1]. Note that by more global methods of Herman-Yoccoz [43, 88] versions of Theorem 2.2 have been obtained for large values of M M 8 The present formulation is related to the conservative analogue by Pöschel [73], who follows Zehnder [92, 93] and to Moser [66]. This method uses the concept of Whitney differentiable functions defined on Diophantine Cantor sets. Several aspects of this theory will be treated in more detail below, also see the Appendix. Figure 2.2 illustrates Theorem 2.2. The invariant subsystem of the perturbed family z mentioned in Theorem 2.2 consists of a collection of parallel (quasiperiodic) circles, smoothly parametrized over a Cantor set of positive measure. In the context of our examples (2.3) and (2.5) this corresponds to a similar family of quasi-periodic invariant and attracting 2-tori. By a straightforward construction, the conjugacy can be extended as an equivalence between the corresponding families of vector fields, restricted to the 2-tori. Compare with [72, 24]. These results all can be phrased in terms of an appropriate form of structural stability, for the occasion called quasi-periodic stability, cf. [24, 23]. Compare with the notion of -stability where structural stability is restricted to the nonwandering set or -set. Remarks - It is known that in the gaps of the Cantor set Á k. Ã À Þ in general we meet periodicity, which in the oscillator context often is called phase lock. An example of this is given by the Arnold family of circle maps, where z ) 7 compare [1, 35]. In these examples the periodicity in the < -plane is organized in a open dense family of resonance tongues. For results and an overview on circle maps see [90, 91, 63], for references also see [23].

16 16 I Survey on dissipative KAM theory - The smoothness of the conjugacies implies the following: Corollary For typical families of dynamical systems, -tori occur on a set of positive measure in the parameter space. By the perfectness of the Cantor sets, typically quasi-periodicity is not isolated in the parameter space. One can even show that typically each parameter point of quasi-periodicity is a Lebesgue density point of quasiperiodicity, in the sense that the relative Lebesgue measure tends to full measure as the volume of the neighborhood tends to zero, [23] pp Note that such measure theoretic results could not be obtained if were only known to be continuous, again see Oxtoby [71]. - A result similar to Theorem 2.2, including the above remarks on the measure theoretical and topological consequences, generally holds for the existence of quasi-periodic -tori, again we refer to the set-up of the next section. In the seventies of the last century a paradox arose in this respect, involving the names of R. Thom and V.I. Arnol d, among others. However, while quasi-periodicity is not persistent for individual systems, for families of systems it is generally persistent on a set of positive measure in the parameter space. Again see [5, 24, 23]. - For the measure theoretic aspect it would also have sufficed to use the somewhat weaker concept of Lipschitz continuity for the conjugacy 8 However, the Whitney smoothness beyond this keeps a fine track of the geometry of the foliations that are generated by the Diophantine conditions in the product of phase space and parameter space. This is especially of importance when studying quasi-periodic bifurcations. Exercise 3 (An equivalence turned into a conjugacy) Show that in the case of the forced nonlinear oscillator (2.3) the conjugacy between the return maps z? and z can be extended to a (smooth) conjugacy between the corresponding vector fields. 3 Towards a KAM Theory of vector fields One of the main aims of this course is to sketch set-up and proof of the general KAM Theorem as developed in [24, 23]. We like to point out here that a completely similar theory exists for diffeomorphisms, compare the examples in Section 2.2 and the Appendix. For simplicity we stay in the context of quasi-periodic attractors, so with the standard -torus : ; as phase-space, the center manifold. This set-up is very close to that of the KAM Theorem for Lagrangean invariant tori in Hamiltonian mechanics, compare [73].

17 F z e œ I. 3 Towards a KAM Theory of vector fields Formulation of the Main Theorem Let z K be an open set of parameters and consider families of vector fields F with F : ; ; > ; and z 8 Often such a family is considered as a vertical vector field on the product : ; Ž z 8 Throughout we assume a real analytic dependence of all vector fields in both and 8 Also we often use vector field notation, writing œ W ( instead of œ W 8 Starting point is an integrable family F Za W ( (3.1) :2; z9 where integrability amounts to -independence, which expresses symmetry (equivariance) with respect to a natural : ; -action. Our interest is with the family of F -invariant -tori : ; Ž j l where z 8 For obvious reasons, the analytic map a ; is called the frequency map. is said to be nondegenerate at z if the derivative & a The family F is surjective. As before, our interest is with the fate of the F -invariant tori : ; Ž j l z9 under real analytic perturbations F F? W ( (3.2) where the size of F «F is small in the compact-open topology on holomorphic extensions. The main question of KAM Theory concerns the fate of the tori : ; Ž j l z9 when perturbing from F to F 8 Again as before, we shall use Diophantine conditions. Indeed, for a given,ø «and ÙNØ k we define the set of Diophantine frequency vectors as follows. ; À Þ j an n MqM a `gsp.m Ü Ù M gemýæ Eg D ; hçjykml l 8 (3.3) Let us describe its structure, compare with Figure 3.1. First of all it directly follows that ; À Þ is a closed set. Secondly, note that if a 5 ; À Þ then also is a union of closed half lines. š a ; À Þ for any š Ü 8 Therefore ; À Þ Thirdly, if ; æ ; is the unit «-sphere, then the intersection ; À Þ ; æ is another closed set, which again is the union of a Cantor set and a countable set, compare the arguments of Section For this Ù Ø k has to be sufficiently small. Finally, the complement of this Cantor set in ; æ has a measure n of order Ù as Ù,ê k 8 Note that the resonance hyperplanes with equations a `gqp k Cg D ;9hdjykml densely fill the complement of ; À Þ 8 À Þ a æ á ; À Þ â If z is any open subset then we define restriction of the map a À Þ to is a submersion, then The topology of uniform convergence on compact sets.. If the is a Whitney-smooth

18 n e Ð 18 I Survey on dissipative KAM theory foliation of smooth manifolds (with boundary) parameterized over a Cantor set. We colloquially call such a foliation a Cantor set. According to the Inverse Function Theorem, these considerations apply for a sufficiently small is nondegenerate at the neighborhood of in z whenever the family F torus : ; Ž j l 8 We now are ready to formulate the Main Theorem of these Theorem 3.1 [24, 23] Let 8 Ü Consider the integrable real analytic family F F of vector fields (3.1), : ; z 8 For z, let F be nondegenerate at the torus : ; Ž j l 8 Then, for Ù=Ø k sufficiently small, there exist a neighborhood of in z and a neighborhood of F in the compact-open topology, such that for any perturbed family F as in (3.2), there exists a mapping : ; Ž! : ; Ž}z with the following properties. (i) ½ ½ is a diffeomorphism onto its image which is -close to the identity map. Also, preserves the projection to z and is real analytic in #" (ii) The restriction of to : ; À Þ Ž! conjugates F to F 8 This is the second KAM Theorem mentioned in this course and its content forms a paradigm for a great many similar results; also the circle map KAM Theorem 2.2 is just a small variation of this. Many remarks following Theorem 2.2 also hold here. In particular we recall the notion of quasi-periodic stability for the integrable family F and the Corollary that typically quasi-periodicity occurs with positive measure in parameter space. Also we like to recall the remarks in Section regarding ½ ¾ -versions of Theorems 2.2 and 3.1, that can be obtained in a straightforward way [73, 24]. Here it is sufficient that g Ø ) 8 For details, compare with Exercise 15 for the case and [23] for the multi-frequency case. Remark For Ü the situation in the gaps of the Cantor sets in between the quasi-periodic tori can be a little different from that in Section 2.2, where we mostly dealt with 2-tori. One reason is that for flows 3-tori can contain strange attractors [70]. Exercise 4 (On Diophantine conditions) In the literature there exist many versions of the Diophantine conditions. Our present interest is in howfar these are equivalent. The Euclidean inner product of two vectors $ $ $ ; and a/ a a ; in ; is denoted by $?%*p ; $ u % u 8 u v

19 & 1 J Þ n Û & g Û Û a & Ù g a Û Û Û & Ù g ; Ù a a ; Þ Þ l l I. 3 Towards a KAM Theory of vector fields 19 For such vectors, we sometimes use the maximum norm and for integer vectors g D & a & ('*) u v À,+,+,+ M a u M3 ; the norm (or length)- & ; Ð uwv M g u M 8 Given ÙØ k and }Ø, consider the sets À Þ j } ; M M g $?p «/. M Ü & 0g D ; hçj kql.i D À Þ j } ; M ÛÛÕ u32 `Ö ¾ À p «Û Û Û Ü & Eg D ; hçjykml l À Þ j } ; n{ M M g «4. 3 <. y $p0m Ü &i g. & 0g D ; h*j kql.\ D À Þ j aü} n36 ;*5 MqM a p.m Ù 6 Ü &768& 0 D ;*5 hçj kqlÿl Show that: (i) Given ÙØ k there exists Ù Ø k such that & À Þ 1 À:9 ; (ii) Given ÙØ k there exists Ù Ø k 1 À Þ such that & À:9 ; (iii) & À Þ À Þ ; (iv) Given À Þ there existsùø k such that & À;9 Þ ; (v) If À Þ then there exists Ù Ø k such that a <. y 4 ;*5 belongs to À;9 ; (vi) If a, a a ;<5 } À:9 Þ then ÙØ k exists such that >= ;*5 ;*5 belongs to À:9 Þ. (vii) Fix. Sketch a geometrical picture of the set À Þ 8 For inspiration see Figure 3.1. How can you interpret (iii) in terms of this picture? What is the relation between & À Þ and the set AB À Þ DC } s.t. ÛÛ «Ù B ÛqÜ M qm E D h j k*l DGFIH In the text the subscripts K and L usually are omitted.

20 20 I Survey on dissipative KAM theory M M PSfrag replacements Fig Sketch of the set N 3øù (and a horizontal line _o_o_ ). 3.2 On the proof of the Main Theorem The map conjugating the unperturbed family F with its perturbation F/ will be obtained from a nonlinear conjugacy equation, compare with Section 2.2. Here we present the set-up of a proof based on a Newtonian iteration pro- cedure that solves this nonlinear equation. At iteration step number D 5 the map is approximated by an analytic map < where < ž< 5 Å#O and where O is determined by a linearized conjugacy equation, also called homological equation. This is similar to Section 2.2. The limit QP R < is taken by the Inverse Approximation Lemma, see the Appendix, in such a way that ½ is a Whitney- map. Here the domains of the < have shrunk À Þ to the nowhere dense union ( Cantor set ) of Diophantine tori : ; ŽS appropriate way. in an Introductory remarks We need some preparations for the set-up.

21 ^ F ã ã F I. 3 Towards a KAM Theory of vector fields 21 Reparameterization. By the Inverse Function Theorem, near z there exists an analytic ce diffeomorphism a W T W $ E such that in the new parameters a Ts we get the simplification À V F(U Za ( 8 So now the frequency vector parametrizes the F -invariant tori. Observe that the parameter T does not show up in the unperturbed system F 8 We shall drop T from now on, since it turns out that any parameter that occurs in in an analytic way, can be directly carried through the whole proof, ending up analytically in the map I8 In this way the space z is replaced by an open domain of ; 8 A compact-open neighborhood. First of all we note that the compact-open topology on holomorphic extensions corresponds to uniform convergence on compact complex domains. We specify the form of a compact-open neighbor- and X denote hood W of the family F 8 For given NK ¾ YX j ª%[Z ¾ M \] such that M ª 7«] M X for ã ã g l 8 Let be a compact neighborhood of a in ; and ^ a compact neighborhood of : ; Ž_ in Z A@CBED > { ; Ž Z ;. Without loss of generality we choose of the form ^ : ; aò 9Ž U YXm (3.4) with constants `=Ø k and kcb X 8 For sufficiently small, ` and X, the unperturbed family F has a holomorphic extension to ^ 8 So much for the compact part of the compact-open neighborhood. Next we come to the open part. A family belongs to W if it has the form F(U Á a œ a Ãd ( with real analytic œ that can be extended holomorphically to ^ in the supremum norm on ^ and such that M œ Me b Ùgf 8 (3.5) A more technical reformulation of the Main Theorem 3.1 now claims the existence of a constant fc such that for F ^ the conclusions of the theorem hold true. It turns out that f is independent of <qù and X 8 For later use we introduce the set ih j aü /M dist a j? Ü Ù l 8 (3.6) In any case we need that ÙØ k is sufficiently small to let h À Þ contain a Cantor set of positive measure.

22 Ù s å 5 5 å å Þ m F 22 I Survey on dissipative KAM theory Remark As can be seen from (3.5), the Diophantine constant Ù also enters in the smallness condition of the perturbation œ 8 Regarding the measure of the Diophantine Cantor set of invariant tori, we like Ù4Ø k to be as small as possible, which gives a certain conflict of interest with the tolerance for perturbations. In the case where a perturbation parameter is used and the perturbation is denoted œ W 3 we take Ù in dependence of in particular Í as e k 8 Compare with [23] for further discussion Idea behind the proof The nonlinear conjugacy equation. Our goal is to find a map : ; Ž h À Þ e : ; Ž z, preserving the projection to z, which conjugates F to F/ i.e., such that 6 F F 8 (3.7) Taking of the form a lk0 Ê a a m a n the conjugacy equation (3.7) translates to Ê a a m a? œ k0 Ê a 3 a Ê and m a n 8 (3.8) This is a nonlinear equation to be solved in as far as possible, by the Newtonian iteration process mentioned earlier. The corresponding linearizations look like Ê a a, m a E œ a (3.9) which can be solved directly by Fourier series, compare with Section o In this linearization intuitively we think of M œ M as an error that has to be diminuished in the iteration, in the limit yielding the expression (3.7). A Whitney-smooth limit. As said earlier, given a perturbation of F the map solving ½ the conjugacy equation 6 F F, see (3.7), will be obtained as a Whitney- limit of a sequence j l of real analytic diffeomorphisms. Here < is a near-identity map, defined on a complex neighborhood & of : ; Žp Þ h D 8 We ensure that & 5 & for all D & á : ; ` a] â Ž á3ih rq â where j ] l is any geometric sequence with ratio less than q Í ] 5 D 8 The sequence j ] l 5 which directly follows from the following specification. (3.10) and where will be fixed later in such a way Again we speak of a 1-bite small divisor problem.

23 5 5 M e 5 u u 5 e I. 3 Towards a KAM Theory of vector fields 23 that the iteration process converges in the sense of the Inverse Approximation Lemma A.2. In this process the are constructed inductively, starting with ct ƒ 8 For Ø k, whenever < is defined, by q aw we denote the components of the inverse æ and define F á æ â 6 F. Subsequently we introduce a k. Ê a 3 a m a n and F À Uvu Á a Nœ a Ãd ( u 8 Assuming that both F and F have holomorphic extensions to a set ^ (see (3.4)), in the induction process we have to ensure that both < and F have holomorphic extensions to the complex domain & Kw^ 8 Also it follows for the error M œ M that M œ k e as ³ in a rapid way, as this is suitable for a Newtonian iteration process The iteration step In order to explain how the induction works, we assume that and F are known while we want to construct < and therefore 5 F from this. Here 5 we take D 5 where for k we take xt ƒ and hence have F) F 8 Putting < =< 5 ÅyO we so have to construct the map O & & and then have F ázo æ â 6 F 8 The last expression, in another tensorial 6 shorthand, can be rewritten as O F F 6 5 where O O æ 6Ÿ8 Summarizing, for all D we have 5 < m 5 aw ce 5 I{ m ae ce } a 3 which means that OI xå Å~O 8 To simplify things a bit, we introduce the plus-notation. This means that we suppress the index and write a and Ë instead of q aw and 5 a 5 respectively. Also we replace œ by œ and œ 5 by œ 5, & by & and & by 5 & 5 etc. The map O will be taken of the form Ë ceä Ê Ë 3 Ë m Ë $ where the parameter shift Ë ce Ë m Ë is needed to keep track of the frequency vectors that are invariant under near-identity conjugacy. Compare with Section The unknown functions Ê m and are obtained as solutions of a homological equation Ê Ë Ë} m Ë W Nœ Ë (3.11)

24 Ó Þ 24 I Survey on dissipative KAM theory where denotes the Fourier truncation of œ at order s8 The integer )' C has to be determined in the final book keeping. Compare with the earlier linearization (3.9) of the conjugacy equation (3.7). The equation (3.11) once more can be solved directly by Fourier series, yielding Ê Ë Ê Ë Ð œ ¾ Ë u 2 n ¾ À ˆ m ƒ8 ¾ Ë and Ë `gqp œy Ë 3 (3.12) where Ê Ë is arbitrary. Notice that by truncation of the Fourier series we only need finitely many Diophantine conditions on Ë and we obtain Ê as a trigonometric polynomial in?8 The conjugacy relation O œ 5 Ë Ê 6 F F 5 now translates to Ë Ë œ 5 Ë { m Ë? œ Ê Ë 3 Ë m Ë { 3 which allows for e error -estimates of M œ 5 M on & in terms of 5 M œ M on & etc. As said before, as ³ we want these errors M œ M on & to decay rapidly. The rest of the proof consists of thorough analytic book keeping, where the sequence j ] l ÑiÒŠ the truncation orders i D 5 and the final constant f have to be chosen appropriately. The Paley-Wiener estimate, see the Appendix, is essential for controlling the tails of the ever longer trigonometric polynomials. We refer to [23] (pp ) for further details of this convergence proof. Remarks - In the solution (3.12) we need only finitely many Diophantine conditions of n the form M Ë ogqp.m Ü šÿm gem æ namely only for kb M gem ã s8 A crucial Lemma [23] (p. 147) ensures that this holds for all Ë rq compare the Exercises 5 and 6. - The analytic book keeping mentioned above includes many applications of the Mean Value Theorem and the Cauchy Integral Formula. The latter serves to express the derivatives of a (real) analytic ½ function in terms of this functions, leading to useful estimates of the ½ -norm in terms of the -norm. - As said earlier, the above proof is a simplification of the Lie algebra proof of [24] and thereby its set-up is characteristic for many other contexts, like for KAM Theorems in the Hamiltonian, the reversible context, etc. Compare [73, 22] and many references in [23]. Compare with Pöschel [74] for a simple version of the proof in the Hamiltonian case. For a review of several KAM proofs also see De la Llave [62]. ñ Another 1-bite small divisor problem.

25 M À œ : A e A 6 I. 3 Towards a KAM Theory of vector fields 25 Exercise 5 (Homothetic role of Ù ) By a scaling of the time that Theorem 3.1 only has to be proven for the case Ù the bounded domain as Ù gets small? and of a show 8 What happens to Exercise 6 (Order of truncation) Following Exercise 5 we restrict to the case where Ù taking the set sufficiently large to contain a nontrivial Cantor set of parameter values corresponding to q0 -Diophantine frequencies. Maintaining the plus-notation consider the complexified domain &Ú : ; ` a]y 9Ž h À rqc see (3.10), assuming that q ]0 5 8 For the order of truncation take )Œ }v Õ Ž á ] æ â 8 (i) Show that for all integer vectors g D ; with k b M gem b one has M gem ã A@ 5 qi æ " (ii) Next show that for all Ë h À q and all g with k(b M gem b one has n Ë `gqp0m Ü M gem æ " (iii) As an example take ] Î D 5 and express the order of truncation i as a function of 8 Exercise 7 (A normal form for families of circle maps) Given a -parameter family of circle maps z : of the form z - œ is counted mod and where œ k d1 k 8 One has to show that by successive transformations of the form ce 6? the -dependence of z can be pushed away to higher and higher order in 8 For this appropriate conditions on will be needed. Carry out the corresponding inductive process. What do you think the first step is? Then, concerning the th step, consider z8 ¹ º W Nœ º? NÍ M M 5 3 with œ A and look for a transformation Qt ƒ with 6W 6? º A such that in æ Å7z8 'Å the th order part in is -independent. Formulate sufficient conditions on < ensuring that the corresponding equation can be formally solved, in terms of Fourier series. Finally give conditions 6 on < such that in the real analytic case we obtain real analytic solutions 8 Explain your arguments.

26 ½ ec z 8 26 I Survey on dissipative KAM theory Exercise 8 (A problem of Sternberg) On :<Ÿ with coordinates a vector ½ field F ½ is given, with the following property. If denotes the circle j kml then the Poincaré return map z ½ e ½ with to F is a rigid rotation 2 everything counted mod 8 From now on we abbreviate  8 Let œ be the return time of the integral curve connecting the points and z ½ in 8 A priori, œ does ½ not have to be constant. The problem now is to construct a(nother) circle that does have a constant return time. Let denote the flow of F and express z in terms of and œ 8 ½ Let us look for a circle of the form j 0 ( é è k 7M ½ l 8 So the search is for a (periodic) function $ and a constant š, such that 4 ½ ½ Rewrite this equation in terms of $ and š 8 Solve this equation formally in terms of Fourier series. What condition on in general will be needed? Give conditions on < such that for a real analytic function œ a real analytic solution $ exists. 4 The normal linear part of quasi-periodic tori Until this moment we focussed on a family of quasi-periodic attractors. The ambient dynamics was of less interest to us, since the attracting tori were normally hyperbolic and therefore persistent as invariant manifolds. For persistence of the dynamics we restricted to the -tori as center manifolds. A central theme in the theory of dynamical systems is formed by bifurcations of attractors. In the last part of this course we address elements of the bifurcation theory regarding quasi-periodic attractors. The bifurcations at hand are all related to the loss of normal hyperbolicity. 4.1 Setting of the problem In this section we consider certain classes of integrable and nearly integrable vector fields with invariant tori and their normal linear part. A more thorough discussion involving the normal bundle of the torus can be found in [24]. For simplicity we assume that a vector field F has : ; Ž as its phase space where the -torus : ; Ž j kml : ; Ž is invariant. As before : ; has angular coordinates mod 8 The coordinates on are As before, and again for simplicity, we assume all

27 e I. 4 The normal linear part of quasi-periodic tori 27 dependences to be real analytic. Expanding F F in powers of we get the following general expression, a W NÍ M M (4.1) NÍ M M 3 as e k which after truncating away the Í -terms, is called the normal linear part of F at : ; Ž j kml 8 Here a : ; e e š ; and : ; P œ are real analytic functions. If the coordinates : ; can be chosen in such a way that a does not depend on the torus is called parallel. See Section 1.2. The Floquet problem asks whether we can adapt the coordinates further in such a way that also does not depend on W8 Such an adaptation is called reduction to Floquet form. In the case where Ÿ i.e., the periodic case, the affirmative answer is provided by Floquet Theory [1, 39, 46]. In the multi- the problem is not so simple, in general there are open frequency case Ü classes of systems for which non-reducibility holds, where the obstructions may be of geometrical (topological) nature [44, œ 26, 27, 80, 82, 37, 54]. The following exercise shows that also for Ÿ i.e., for the case of codimension 1 tori, an affirmative answer can be given, provided that the frequency vector a is Diophantine. Compare with [23] pp Remark Observe that if the vector field F is integrable, which here again amounts to -independence, automatically the normal linear part (4.1) has Floquet form. Exercise 9 (Floquet problem on a codimension 1 torus) Consider a smooth system œ with : ; Ž 8 Assume that œ a -Í M M which implies that U k is a invariant -torus, with on it a constant vector field with frequencyvector al8 Hence the torus k is parallel. Put Í M M y 3 for a map : ; œ gl 8 The present problem is to find a transformation : ; Ž e : ; Ž of the form ce ª 3 for some map : ; œ GL with the following property: The transformed system a NÍ m ª NÍ M ª M M ª M

28 Ð ; e 8 Í 28 I Survey on dissipative KAM theory is on Floquet form, meaning œ m that the matrix is -independent. From now on, we restrict to the case 8 By a computation show that m, ae P <š v From this derive an equation, expressing that m is constant in W8 Formally solve this equation in given 8 Give conditions on a ensuring a formal solution. Also explain how to obtain a real analytic solution, assuming real analyticity of 8 œ m Remark For the expression for becomes more complicated, since then the matrices do not commute. Apart from this, as said earlier, in this case there can be topological obstructions against the existence of a Floquet form. 4.2 The perturbative point of view The discussion in Section 4.1 raises the question of persistence of reducibility under small perturbation of the system. As in the Sections 2 and 3 we need parameters to study this problem systematically. We here include an outline of the main result in [24, 23] for the present dissipative situation. Therefore consider a family F F 3 where : ; Ž and where z for an open subset z of a Euclidean space. Assume that F Á a W W œ W Þ ( 'Á W W Þ@Ÿ* (4.2) for a e : ; Ž z ; and : ; Ž œ and with œ M M and Í M M as e k 8 We again assume real analytic dependence on all variables and parameters. Assume that all the eigenvalues of W are simple and different from k " by continuity this holds for an open, bounded subdomain Küz 8 Moreover we assume nondegeneracy in the sense that the map ceä a W 3 Õ W { (4.3) is a submersion (if necessary please take smaller).` If the eigenvalues of are given by f f i 8 with Ø k for ã ã then we define Õ 7 fc 2{? 8 By the Inverse Function Theorem it is sufficient that the derivative at a certain point is surjective.