Renormalization Theory for Hamiltonian Systems

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(1)Renormalization Theory for Hamiltonian Systems. Mikhail Pronine. Universit¨ at Bremen 2002.

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(3) Renormalization Theory for Hamiltonian Systems. Vom Fachbereich f¨ ur Physik und Elektrotechnik der Universit¨ at Bremen zur Erlangung des akademischen Grades DOKTOR DER NATURWISSENSCHAFTEN (Dr. rer. nat.) genehmigte Dissertation. von Dipl.-Math. Mikhail Pronine aus Simferopol. Referent: Professor Dr. P. Richter Korreferent: Professor Dr. H. Schwegler Tag des Promotionskolloquiums: 16.12.2002.

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(5) Preface This thesis is devoted to the study of the breakup of invariant tori of irrational winding numbers in Hamiltonian systems with two degrees of freedom. Due to topological reasons, the decay of invariant tori in such systems is closely related to the onset of widespread chaos. To give an estimate of where in parameter space the breakup of an invariant torus occurs, an approximate renormalization scheme is derived. The scheme is applied to a number of systems (the paradigm Hamiltonian of Escande and Doveil, the Walker and Ford model, a model of the ethane molecule, the double pendulum, the Baggott system, lima¸con billiards). The work is organized as follows. Chapter 1 describes the behavior of a generic Hamiltonian system with more than one degree of freedom. The emphasis is put on systems with two degrees of freedom. We introduce the main problem of the work, i.e., the problem of finding the threshold to global chaos in terms of the breakup of the ”last” invariant KAM torus. There exist a number of analytical and numerical methods to deal with the problem. We review these methods in Chapter 2. Our version of the renormalization group approach to the problem is discussed in Chapter 3. The method is applied to various systems in Chapter 4. Chapter 5 summarizes the results of the work. Appendix A is devoted to the normal form for Hamiltonian systems with two degrees of freedom. In Appendix B we discuss classical perturbation theory and its application to the normal form. Appendix C contains some useful formulae for the calculation of derivatives of implicit functions. The realization of the RG approach to the study of the breakup of invariant tori in the Maple computer algebra system is presented in Appendix D. I would like to thank my scientific advisor Prof. Peter H. Richter for supervising the work. I have benefited from useful discussions with former and current members of the group Nichtlineare Dynamik. I particularly thank Dr. Holger Dullin, Jan Nagler, Dr. Hermann Pleteit, Dr. Holger Waalkens.. v.

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(7) Contents 1 Introduction. 1. 2 Criteria for the Breakup of KAM Tori 2.1 Sup map analysis . . . . . . . . . . . . . . 2.2 The method of overlapping resonances . . 2.3 Greene’s method . . . . . . . . . . . . . . 2.4 The renormalization group approach . . . 2.5 Comparison between the different methods. . . . . .. . . . . .. 3 Renormalization Theory 3.1 Normal form . . . . . . . . . . . . . . . . . . . 3.2 The renormalization operator . . . . . . . . . 3.3 The renormalization map for the normal form 3.4 Fixed points . . . . . . . . . . . . . . . . . . . 4 Applications 4.1 Application to the paradigm Hamiltonian . . . 4.2 Application to the Walker and Ford model . . 4.3 Application to a model of the ethane molecule 4.4 Application to the double pendulum problem . 4.4.1 The Lagrange function . . . . . . . . . 4.4.2 Integrable cases . . . . . . . . . . . . . 4.4.3 The Hamilton function . . . . . . . . . 4.4.4 The integrable limit of high energies . 4.5 The Baggott H2 O Hamiltonian . . . . . . . . 4.6 Lima¸con billiards . . . . . . . . . . . . . . . .. . . . . .. . . . .. . . . . . . . . . .. . . . . .. . . . .. . . . . . . . . . .. . . . . .. . . . .. . . . . . . . . . .. . . . . .. . . . .. . . . . . . . . . .. . . . . .. . . . .. . . . . . . . . . .. . . . . .. . . . .. . . . . . . . . . .. . . . . .. . . . .. . . . . . . . . . .. . . . . .. . . . .. . . . . . . . . . .. . . . . .. . . . .. . . . . . . . . . .. . . . . .. . . . .. . . . . . . . . . .. . . . . .. . . . .. . . . . . . . . . .. . . . . .. . . . .. . . . . . . . . . .. . . . . .. . . . .. . . . . . . . . . .. . . . . .. . . . .. . . . . . . . . . .. . . . . .. . . . .. . . . . . . . . . .. . . . . .. 7 7 11 14 16 18. . . . .. 19 19 20 23 26. . . . . . . . . . .. 31 31 36 41 53 53 55 55 57 65 80. 5 Conclusions. 89. A The Normal Form. 91. B Classical Perturbation Theory 95 B.1 Application to the normal form . . . . . . . . . . . . . . . . . . . . . . . . 101 C Derivatives of Implicit Functions. 105 vii.

(8) viii D Maple Program D.1 Renormalization operator . . . . . . . . . . . D.1.1 Operator . . . . . . . . . . . . . . . . . D.1.2 Scaling . . . . . . . . . . . . . . . . . . D.1.3 Numerical approximation . . . . . . . D.1.4 Threshold . . . . . . . . . . . . . . . . D.1.5 Output . . . . . . . . . . . . . . . . . . D.2 The paradigm Hamiltonian . . . . . . . . . . . D.3 Application to a model of the ethane molecule D.4 Application to the Baggott Hamiltonian . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. 107 . 107 . 107 . 108 . 109 . 110 . 112 . 112 . 113 . 114.

(9) Chapter 1 Introduction This thesis deals with the dynamic behavior of classical Hamiltonian systems with two degrees of freedom. Such systems and their stability properties are of interest in diverse fields (celestial mechanics [28, 45], plasma physics [12, 21], chemical physics [28, 24, 25] to name just a few). Hamiltonian systems can be divided into two classes, integrable and non-integrable. Let us recall the notion of integrability which is of utmost importance in the study of Hamiltonian systems. Consider a Hamiltonian system defined by a function H(p, q, t) in the phase space T ∗ Q(p, q). The function H is called the Hamiltonian of the system. The equations of motion are Hamilton’s equations dp ∂H =− , dt ∂q. dq ∂H = . dt ∂p. (1.1). A function F in the phase space is said to be a constant of motion if along any trajectory its value is constant. Constants of motion are also referred to as first integrals. In autonomous systems (i.e., H is explicitly time independent, H = H(p, q)) the Hamiltonian H is a constant of motion. In what follows we restrict ourselves to the case of autonomous systems. The Poisson bracket of two functions F = F (p, q, t) and G = G(p, q, t) is defined to be n X ∂F ∂G ∂F ∂G [F, G] = − . (1.2) ∂pi ∂qi ∂qi ∂pi i=1 The time dependence of a function F = F (p, q, t) is given by dF ∂F = [H, F ] + . dt ∂t. (1.3). Assume that the function F does not depend on time explicitly. It follows from (1.3) that F is a constant of motion if and only if [H, F ] = 0. Two functions F and G are said to be in involution if their Poisson bracket [F, G] vanishes. Two functions F (p, q) and G(p, q) are called (functionally) independent if their gradients (∂F/∂p1 , . . . , ∂F/∂qn ) and (∂G/∂p1 , . . . , ∂G/∂qn ) are linearly independent for almost every point (p, q). We are ready now to introduce the notion of integrability for Hamiltonian systems. A Hamiltonian system of n degrees of freedom is called integrable if there exist n independent 1.

(10) 2. CHAPTER 1. INTRODUCTION. constants of motion F1 , . . . , Fn which are in involution [3]. Integrable systems are also referred to as completely integrable ones. The geometric description of integrable systems is given by the Liouville-Arnold theorem [3]. According to this result the phase space of an integrable Hamiltonian system H0 of n degrees of freedom is foliated by the invariant sets {(p, q) ∈ T ∗ Q : F1 (p, q) = c1 , . . . , Fn (p, q) = cn },. (1.4). with F1 , . . . , Fn being constants of motion. Moreover, in generic situations the motion on these invariant sets is periodic or quasiperiodic. If the energy surface H0 (p, q) = h0 is compact, then connected components of the invariant sets are just n-dimensional tori. Locally there exists a canonical transformation (p, q) → (I, θ) such that in the new coordinates (I, θ) the Hamiltonian H0 does not explicitly depend on the angle variables θ: H0 = H0 (I). (1.5) The coordinates (I, θ) are referred to as action-angle coordinates. The equations of motion are readily solvable in the action-angle coordinates. Indeed, Hamilton’s equations are dI ∂H0 =− = 0, dt ∂θ. dθ ∂H0 = . dt ∂I. (1.6). Thus, the actions I are constants of motion, and the time evolution of the angle variables is periodic or quasiperiodic with constant frequencies ∂H0 /∂I. A generic Hamiltonian system with two or more degrees of freedom is non-integrable. In this case there is no simple geometric description of motion. Moreover, the dynamic behavior of a generic Hamiltonian system is at least partially chaotic. Recall some relevant definitions from the theory of dynamical systems [13]. Let M be an arbitrary set. Consider a one-parametric family of maps f t : M → M from M into itself. The pair (M, f t ) is called a dynamical system. The set M is referred to as the phase space of the dynamical system (M, f t ). The family f t is said to be the dynamics. If the parameter t is continuous, then the dynamical system (M, f t ) is called the flow . In the case of Hamiltonian systems we define the corresponding flow in the following way. Choose the phase space T ∗ Q = {(p, q)} as the phase space of the flow. Given a point (p0 , q 0 ) from T ∗ Q, the map f t assigns to it the solution of Hamilton’s equations with initial conditions (p0 , q 0 ) at time t = 0. A subset M 0 of the phase space M is called an invariant set if f t (M 0 ) lies in M 0 for every t. The notion of invariant set plays an important role in the study of dynamical systems. Indeed, having identified all invariant sets of a given dynamical system, we can investigate the behavior of the dynamics f t on the invariant sets independently from each other. Assume now that the phase space M of some dynamical system (M, f t ) is a metric space. For example, the phase space T ∗ Q of a Hamiltonian system with the Euclidian metric is a metric space. The dynamical system (M, f t ) is said to be topologically transitive if for any two open sets U and V from M there exists t > 0 such that the intersection f t (U ) ∩ V is not empty. Note that a topologically transitive dynamical system has no non-trivial open invariant.

(11) 3 sets. Remember that if the dynamics f t of a dynamical system has a dense orbit, then the system is topologically transitive. We say that the dynamics f t of a dynamical system (M, f t ) has sensitive dependence on initial conditions if there exists δ > 0 such that, for any x from M and any neighborhood O(x) of x, there exist y in O(x) and t > 0 such that |f t (x) − f t (y)| > δ. Recall that we assume M to be a metric space. Thus, the distance |f t (x) − f t (y)| is well-defined. Finally, a dynamical system (M, f t ) is said to be chaotic if (i) f t has sensitive dependence on initial conditions, (ii) f t is topologically transitive, and (iii) periodic points are dense in M , see [13]. Integrable systems and chaotic ones are exceptions among Hamiltonian systems with two or more degrees of freedom. A generic system contains both regular and chaotic regions in the phase space. Given an arbitrary Hamiltonian system H, the question arises as to how the complicated interplay of its chaotic and regular motions can be described. The usual approach based on perturbation theory consists of introducing the so-called perturbation parameter and finding an integrable limit H0 for the system H. The integrable limit corresponds to some value, say zero, of the perturbation parameter. As the perturbation parameter varies, we obtain a family of Hamiltonian systems. The family includes the integrable system H0 and the initial system H. Using the knowledge of the dynamics of the former system, we can now try to describe the dynamic behavior of the latter system. For example, one can represent solutions to the initial problem as series with respect to the perturbation parameter. Formally, for every Hamiltonian system H and every integrable Hamiltonian system H0 the former system can be represented as perturbation to the latter one. Indeed, consider the family H0 + ε(H − H0 ) of Hamiltonian systems. The initial system H corresponds to the value one of the perturbation parameter ε. In the limit ε → 0 we obtain the integrable system H0 . If for the given H the integrable limit H0 is chosen in an arbitrary way, then the behavior of H can be very different from that of H0 . In other words, there is almost no hope that the system H can be described on the basis of H0 . Perturbation theory works well only provided that perturbation is small . The choice of an integrable limit for a given non-integrable system is thus by no means trivial. Sometimes, especially working with physical systems, a number of possible integrable limits arise in a natural way. Let us consider the behavior of an integrable system under small perturbation. We refer to such systems as near-integrable ones. In what follows we confine ourselves to the case of Hamiltonian systems with two degrees of freedom. Moreover, assume that the energy surface H0 = h is compact so that according to the Liouville -Arnold theorem it is foliated by two-dimensional invariant tori. Introducing action-angle coordinates (I1 , I2 , θ1 , θ2 ) for the integrable system H0 , we can express the frequencies of a given invariant torus as (ω1 , ω2 ) = (∂H0 /∂I1 , ∂H0 /∂I2 ). The ratio W = ω1 /ω2 is said to be the winding number of the torus. The winding number of a torus is also called the winding ratio. We have to distinguish between tori of rational and of irrational winding numbers. We will refer to invariant tori of rational winding numbers as rational invariant tori, or resonant invariant tori. An invariant torus of irrational winding number is said to be non-resonant, or irrational . The behavior of rational invariant tori under perturbation is described by the Poincaré -Birkhoff theorem applied to the Poincaré map which the continuous flow generates on.

(12) 4. CHAPTER 1. INTRODUCTION. a suitable closed surface of section, see Chapter 2. Note that a rational invariant torus of winding number W = p/q is foliated by an infinite number of periodic orbits. The Poincaré -Birkhoff theorem states that only a finite number of them survive under perturbation. Half of these periodic orbits are elliptic, and the other half are hyperbolic. In a Poincaré surface of section θ2 = 0 mod 2π, the rational torus itself gives rise to a chain of kq islands with k being 1, or 2, or 3, . . . . The islands are separated by chaotic bands which are located in a neighborhood of the stable and unstable manifolds, or separatrices, of the hyperbolic periodic orbits. The behavior of irrational invariant tori can be studied using the Kolmogorov-ArnoldMoser theorem [26, 2, 32]. In order to formulate this result we need a number of preliminary definitions. Consider a near-integrable Hamiltonian system with n degrees of freedom. We assume that the Hamiltonian is written in the form H = H0 + εH1 , where H0 is an integrable system, and ε is a perturbation parameter. Introducing action-angle variables (I, θ) for the unperturbed part H0 , we obtain the following expression for the Hamiltonian: H(I, θ) = H0 (I) + εH1 (I, θ). (1.7) The system H0 is said to be nondegenerate if the condition det. ∂ω ∂ 2 H0 6= 0 2 = det ∂I ∂I. (1.8). is satisfied. We denote in (1.8) the frequencies of the unperturbed system by ω. The system H0 is said to be iso-energetically nondegenerate if one of the frequencies does not vanish and the ratios of the remaining n − 1 frequencies to it are functionally independent on the energy surface H0 = h. Formally, the last condition can be written in the form ∂2H 0 ω 2 ∂I det 6= 0. (1.9) ωt 0 The determinant in (1.9) represents the Jacobian of the mapping of the (n−1)-dimensional surface H0 = h into the (n−1)-dimensional projective space (∂H0 /∂I1 , . . . , ∂H0 /∂In ). Let us discuss this condition in the case of Hamiltonian systems with two degrees of freedom. The winding number W is given by W =. ω1 , ω2. ω1 =. ∂H0 (I1 , I2 ) , ∂I1. ω2 =. ∂H0 (I1 , I2 ) . ∂I2. (1.10). Consider the three-dimensional space (I1 , I2 , h). The Hamiltonian H0 (I1 , I2 ) gives rise to a two-dimensional surface (I1 , I2 , H0 (I1 , I2 )) in the space (I1 , I2 , h). The vector v 1 = (ω1 , ω2 , 1) is orthogonal to this surface. The one-dimensional ”energy surface” H0 = h0 may be viewed as the intersection of the surface (I1 , I2 , H0 (I1 , I2 )) with the plane h = h0 whose normal is given by the vector v 2 = (0, 0, 1). Finally, the gradient of the function W = W (I1 , I2 ) reads 1 ∂ω1 ∂ω2 1 ∂ω1 ∂ω2 ∂W ∂W , ,1 = ω2 − ω1 , 2 ω2 − ω1 , 1 . (1.11) v3 = ∂I1 ∂I2 ω22 ∂I1 ∂I1 ω2 ∂I2 ∂I2.

(13) 5 The linear independence of the vectors v 1 , v 2 , v 3 ensures that the winding number W varies smoothly on every energy surface. It is easy to show that the condition for the vectors v 1 , v 2 , v 3 to be linearly independent coincides with the isoenergetic non-degeneracy condition. We are ready now to formulate the KAM theorem, see, for example, [4]. Theorem. Consider a near-integrable Hamiltonian system which is nondegenerate or iso-energetically nondegenerate. Then for a sufficiently small perturbation irrational invariant tori with Diophantine frequency vectors do not decay but are only deformed. Recall that a vector (ω1 , . . . , ωn ) is called Diophantine if for every nonzero integervalued vector (k1 , . . . , kn ) the following inequality is satisfied: |k1 ω1 + · · · + kn ωn | ≥. K(ω) , kkk. (1.12). with kkk = maxi |ki | being the norm of the vector (k1 , . . . , kn ) and K(ω) being a constant. We refer to the tori from the KAM theorem as KAM tori. Note further that given a KAM torus, angle coordinates for this torus can be introduced, so that the motion on the torus is quasiperiodic with the same frequencies as for the motion on the initial unperturbed torus. If the isoenergetic non-degeneracy condition is fulfilled, then the KAM torus lies on the energy surface {H = h} with the same value h of energy as the initial unperturbed torus. The KAM theorem is of utmost theoretical importance. To name just a few consequences of this result, let us mention that the KAM theorem elucidates the problem of the small divisors in classical perturbation theory, clarifies the questions concerning the convergence of series in perturbation theory, explains the stability of elliptic periodic orbits in nonlinear Hamiltonian systems with two degrees of freedom. From the practical point of view, the KAM theorem has a significant restriction. The theorem is valid only for sufficiently small perturbation. Its original version was restricted to perturbations of order ε ∼ 10−48 . Though the subsequent versions have dramatically improved this estimate, to find a realistic value of perturbation corresponding to the breakup of a given KAM torus remains an open problem. In the case of Hamiltonian systems with two degrees of freedom the KAM theorem has an interesting geometric interpretation. Consider an unperturbed non-resonant torus with Diophantine frequencies on the energy surface {H0 = h}. If the isoenergetic nondegeneracy condition is satisfied, then, in accordance with the KAM theorem, for sufficiently small perturbation there exists a KAM torus with the same winding number as the initial one on the energy surface {H = h}. This energy surface can be viewed as a three-dimensional surface in the four-dimensional phase space. The KAM torus is of dimension two. For topological reasons the energy surface {H = h} is divided by the KAM torus into two invariant three-dimensional regions (at least locally). The regions are invariant sets of the Hamiltonian system in question, so that it is impossible for a trajectory starting in one region to reach the other. Recall that according to the Poincaré -Birkhoff theorem and related results resonant unperturbed tori give rise to chaotic regions in the phase space of the perturbed system. In the case of Hamiltonian systems with two degrees of freedom the chaotic regions corresponding to different resonant tori are separated one from another by KAM tori. Thus,.

(14) 6. CHAPTER 1. INTRODUCTION. KAM tori may be considered as barriers to widespread chaos. The following scenario for the transition from regular to chaotic motion can often be observed. Consider a near-integrable Hamiltonian system with two degrees of freedom. Let us confine ourselves to a fixed value h of energy. For small perturbation the larger part of the energy surface consists of KAM tori. Resonant tori of the unperturbed system give rise to chaotic regions. These regions are relatively small for small perturbation. Different chaotic regions are separated by KAM tori. As perturbation increases, the chaotic regions become larger and larger. Simultaneously, more and more KAM tori are destroyed. The chaotic regions corresponding to different resonant tori can merge provided that all the KAM tori between them broke up. In some systems for sufficiently large perturbation the chaotic regions form two invariant sets divided by a KAM torus. As perturbation increases further, this KAM torus, which is called the last KAM torus, is also destroyed. The two chaotic regions merge into one, so that a transition to global chaos can be observed. The scenario just described holds, e. g., in the case of a double pendulum, see [37]. We are ready now to formulate the main problem of this work. Consider a nearintegrable Hamiltonian system H0 + εH1 with two degrees of freedom. Here H0 is an integrable Hamiltonian system, ε is the perturbation parameter. Consider two resonant tori, or resonances, of the system H0 on the energy surface H0 = h. The tori give rise to two chaotic regions on the energy surface H = h. For small values of the perturbation parameter ε these two chaotic regions are separated by KAM tori. As perturbation increases, more and more of these KAM tori decay. At some value εcrit of the perturbation parameter ε the last KAM torus between the chaotic regions in question is destroyed. We refer to this value of perturbation as the critical value. The problem is as follows. Given two resonances on the energy surface H0 = h, find the critical value εcrit of the perturbation parameter. The following empirical rule has been observed studying various systems. The winding number of the last KAM torus is a noble number . Recall the definition of a noble number. An arbitrary real number x can be represented in a unique way by a continued fraction x = a0 +. 1 1 a1 + a2 +.... (1.13). with a0 being an integer, and ai , i > 0, positive integers, see [23, 33]. We will write the continued fraction expansion (1.13) for x in the form x = [a0 , a1 , . . . ]. Note that a rational number has a finite continued fraction. The number x is said to be noble if its continued fraction x = [a0 , a1 , . . . ] satisfies the property that ai = 1 if i > i0 for some i0 . In the next chapter we review methods for studying the breakup of KAM tori and the transition to widespread chaos..

(15) Chapter 2 Criteria for the Breakup of KAM Tori In this chapter we discuss a number of methods dealing with the study of the breakup of KAM tori in Hamiltonian systems with two degrees of freedom. The majority of the methods described below are concerned with two-dimensional areapreserving mappings. In order to make use of these approaches in the case of Hamiltonian systems with two degrees of freedom we employ the technique of Poincaré sections. This technique reduces, at least locally, the study of these systems to that of two-dimensional area-preserving mappings. Consider a Hamiltonian system H with two degrees of freedom. Take the energy surface H = h corresponding to some value h of energy. Consider some two-dimensional surface S contained in the three-dimensional energy surface. Take some point p on the surface S. Let γ be the trajectory with initial condition p and p0 the next intersection of γ with the surface S. The mapping P : S → S of the surface S into itself defined in this way is said to be the return map, or the Poincaré map. The Poincaré map is a two-dimensional map. It is always possible to introduce some regular measure on S so that the return map becomes area-preserving with respect to this measure. Thus, we reduce the problem of the breakup of KAM tori in Hamiltonian systems with two degrees of freedom to the problem of the breakup of KAM circles in two-dimensional area-preserving mappings.. 2.1. Sup map analysis. This section is devoted to the so-called sup map analysis, see [20] and references therein. We describe the sup map analysis for the case of two-dimensional mappings. Consider the famous standard map In+1 = In + K sin θn , θn+1 = θn + In+1. (2.1) (2.2). with K being the perturbation parameter. The phase space is assumed to be the twodimensional torus {(I, θ)|0 ≤ I < 2π, 0 ≤ θ < 2π, }. The standard map is also referred 7.

(16) 8. CHAPTER 2. CRITERIA FOR THE BREAKUP OF KAM TORI. K=0.0000. I. K=0.5000. 6.28. 6.28. 5.23. 5.23. 4.18. 4.18 I. 3.14. 3.14. 2.09. 2.09. 1.04. 1.04. 0.00 0.00. 1.04. 2.09. 3.14 θ. 4.18. 5.23. 0.00 0.00. 6.28. K=1.0000. I. 6.28. 5.23. 5.23. 4.18. 4.18 I. 3.14. 2.09. 1.04. 1.04. 2.09. 3.14 θ. 4.18. 5.23. 6.28. 3.14 θ. 4.18. 5.23. 6.28. 3.14. 2.09. 1.04. 2.09. K=1.5000. 6.28. 0.00 0.00. 1.04. 3.14 θ. 4.18. 5.23. 6.28. 0.00 0.00. 1.04. 2.09. Figure 2.1: The phase portrait for the standard map with K = 0 (top left), K = 0.5 (top right), K = 1 (bottom left), K = 1.5 (bottom right)..

(17) 2.1. SUP MAP ANALYSIS. 9. to as the Chirikov-Taylor map. We present the phase portrait for the standard map in Figure 2.1. Note that the value K = 0 corresponds to the integrable case. Indeed, if K = 0, then the standard map becomes In+1 = In , θn+1 = θn + In+1 ,. (2.3) (2.4). so that I is a constant of motion and I/2π the winding number. An analog of the KAM theorem for Hamiltonian maps, Moser’s twist theorem [32], guarantees the existence of one-dimensional KAM tori (circles) for small values of perturbation K. With the help of Figure 2.1 one can easily identify two large chaotic regions in the vicinity of the resonances which correspond to the resonant circles I = 0 and I = π in the integrable limit. As perturbation becomes larger, these two chaotic regions grow, and more and more KAM √ tori are destroyed. The last KAM torus to be destroyed has the winding number g = ( 5 − 1)/2, where g is the golden mean. Note that the last KAM torus does not divide the phase space into two invariant sets. Indeed, because of the periodicity of the action variable I it is possible for a trajectory starting in the bottom part of the phase space to reach the upper one. Nevertheless, as long as the last KAM torus exists, this kind of transition is the only possible one. The transition through the last KAM torus is not possible. If we omit the periodicity condition on the action variable I, then the phase space becomes a cylinder R × S 1 . In this case, the last KAM torus divides the phase space into two different invariant sets. Actually, there are then an infinite number of last KAM tori. All of them are just copies of the last KAM torus in question. The torus becomes a barrier to widespread chaos. In the following we consider the standard map as lifted to the cylinder. As we have just mentioned, as long as the last KAM torus exists, a trajectory with initial condition in the invariant set below the last KAM torus remains in the same invariant set. More precisely, let us fix some value of the perturbation parameter K and choose a point (I0 , θ0 ) as initial condition of some trajectory. Assume that the point (I0KAM , θ0 ) belongs to the last KAM torus, and that the inequality I0 < I0KAM holds. Then the trajectory with initial condition (I0 , θ0 ) lies in the bottom invariant set, and the inequality

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(21) max{In

(22) n = 0, 1, . . . } < max{InKAM

(23) n = 0, 1, . . . } (2.5) is satisfied. Here In denotes the I-values of the images of (I0 , θ0 ) under the standard map. Let Imax (I, θ, n) be the maximal value of I among the first n iterates of the orbit with initial conditions (I, θ). Figure 2.2 shows Imax (I, θ, n) as a function of I for θ = 0 and n = 1000 in the case of the standard map. The following three structures are clearly visible: (i) monotonically increasing variations, (ii) noisy variations, and (iii) V-shaped structures, see [20]. The monotonic variations correspond to the regions where many KAM tori exist. The crossing into a chaotic region gives rise to a jump in Imax followed by a noisy variation. V-shaped structures are related to island chains..

(24) 10. CHAPTER 2. CRITERIA FOR THE BREAKUP OF KAM TORI. K=0. K=0.5. 10.0. 10.0. 8.00. 8.00. 6.00. 6.00. sup. sup 4.00. 4.00. 2.00. 2.00. 0.00 0.00. 1.05. 2.10. 3.15 I. 4.20. 5.25. 0.00 0.00. 6.30. 1.05. 2.10. K=1. 3.15 I. 4.20. 5.25. 6.30. 4.20. 5.25. 6.30. K=1.5. 10.0. 10.0. 8.00. 8.00. 6.00. 6.00. sup. sup 4.00. 4.00. 2.00. 2.00. 0.00 0.00. 1.05. 2.10. 3.15 I. 4.20. 5.25. 6.30. 0.00 0.00. 1.05. 2.10. 3.15 I. Figure 2.2: The sup map for K = 0 (top left), K = 0.5 (top right), K = 1 (bottom left), K = 1.5 (bottom right)..

(25) 2.2. THE METHOD OF OVERLAPPING RESONANCES. 11. The sup map analysis can be applied to the standard map as follows. Consider the region 0 < I < Imax = π. A number of resonances in this region can easily be identified with the help of Figure 2.1. We study the onset of widespread chaos between the two resonances corresponding to the island chains consisting of one and two islands respectively. First, we choose bracketing bounds for the critical value Kcrit . For example, we can confine ourselves to the interval [0, 5]. For a fixed value of the perturbation parameter K we study the trajectories with initial conditions (Ij , θ0 ) where Ij = I0 + j∆. In our actual computation we take θ0 to be π, I0 = 4π/5, the grid spacing ∆ = π/500, and j = 0, 1, . . . , 100. The initial value I0 is chosen in such a way that the point (I0 , θ0 ) lies below the last KAM torus. For a given value of the perturbation parameter K, a trajectory with initial condition (Ij , θ0 ) is iterated n times. Denote the maximal and minimal (j) (j) (0,1,...,j) values of I among the first n iterates of (Ij , θ0 ) by Imax and Imin respectively. Let Imax (0) (1) (j) be the maximum of {Imax , Imax , . . . , Imax }. Consider the following conditions: (0,...,j−1) (j) , < Imax Imax (0,...,j−2) (j−1) Imax < Imax , (j−1). (j). Imin < Imin , (j) Imax < Imax .. (2.6) (2.7) (2.8) (2.9). If all these conditions are fulfilled for some j > 1, we conclude that for the given K there exists a KAM torus between the resonances under study and the corresponding chaotic regions are localized. Otherwise, the onset of widespread chaos is expected for this value of the perturbation parameter. Let us comment the conditions (2.6-2.9). Inequalities (j) (2.6-2.7) are fulfilled in regions where Imax varies monotonically with respect to j. Thus, these conditions are a strong evidence that the trajectory lies either in a region dominated by KAM tori or in a region corresponding to nested islands. If inequalities (2.6-2.7) are satisfied, then inequality (2.8) guarantees that the trajectory does not lie in a region dominated by nested islands. The last condition (2.9) implies that the trajectory lies outside the top chaotic region. In order to find the critical value of K on the given bracketing interval with prescribed accuracy, we can now use for example the bisection method. The results of the application of the sup map analysis to the study of the onset of widespread chaos for the standard map are presented in Figure 2.3 and in Table 2.1. They are to be compared with the estimate Kcrit = 0.9716 . . . delivered by Greene’s method, see below.. 2.2. The method of overlapping resonances. The method of overlapping resonances is due to Chirikov [12]. We discuss the method in the case of non-autonomous Hamiltonian systems with one degree of freedom. Consider the Hamiltonian H(I, θ, t) = H0 (I) + H1 (I, θ, t). (2.10).

(26) 12. CHAPTER 2. CRITERIA FOR THE BREAKUP OF KAM TORI. 1.1000. 1.0600. 1.0200 K 0.9800. 0.9400. 0.9000 10000. 25000. 40000. 55000. 70000. 85000. 100000 n. Figure 2.3: The critical value of K for the onset of widespread chaos (numerical calculation using sup map analysis). The accuracy of the bisection algorithm is 0.001.. n 10000 20000 30000 40000 50000 60000 70000 80000 90000 100000. Kcrit 1.008 1.008 1.001 0.986 0.983 0.983 0.983 0.979 0.979 0.979. Table 2.1: The critical value of K for the onset of widespread chaos against the number of iterations per orbit (numerical calculation using sup map analysis). The accuracy of the bisection algorithm is 0.001..

(27) 2.2. THE METHOD OF OVERLAPPING RESONANCES. 13. with H1 (I, θ, t) being the perturbation term. Expand the perturbation part in a Fourier series: X H1 (I, θ, t) = Vmn (I) cos(mθ + nt + γmn ). (2.11) m,n∈Z. As an example we consider the standard map. It can be represented as a non-autonomous Hamiltonian system in the following way: H=. X I2 − K cos θ δ(t − n). 2 n∈Z. (2.12). Figure 2.1 may now be viewed as the stroboscopic map of the system given by Hamiltonian (2.12). The corresponding perturbation part is H1 (I, θ, t) = −K cos θ(1 + 2. ∞ X. cos(2πmt)).. (2.13). m=1. Our aim now is to study the onset of widespread chaos between the resonances (m1 , m2 ) = (1, 0) and (p1 , p2 ) = (1, 1). They correspond to the terms of the form V10 cos(θ) and V11 cos(θ +2πt) in the Fourier series expansion for the perturbation H1 . Using the identity 2 cos θ cos(2πt) = cos(θ + 2πt) + cos(θ − 2πt),. (2.14). we obtain H1 = −K. cos θ + cos(θ + 2πt) + cos(θ − 2πt) + 2 cos θ. ∞ X. !. cos(2πmt) .. (2.15). m=2. If we consider the standard map as a mapping of R1 × S 1 into itself, the resonances in question correspond to the two islands centered at (I, θ) = (0, π) and (I, θ) = (2π, π). A numerical study of the system shows that the last KAM torus between the resonances is √ of winding number g = ( 5 − 1)/2 with g being the golden mean. The idea of the method of overlapping resonances consists of the following. To study the onset of widespread chaos between two given resonances we neglect all the other terms in the Fourier expansion of the perturbation part. We replace the initial system with a system whose perturbation part is the sum of two resonance terms. We discuss now these two remaining resonances independently. A Hamiltonian system of the form H(I) = H0 (I) + V cos(mθ + 2nπt). (2.16). is integrable. Here m and n are assumed to be integers. However, compared with Figure 2.1, K = 0, the stroboscopic plot becomes more involved. Actually, it looks qualitatively just like the phase portrait of a pendulum. The initial system can now be thought of as perturbation to the integrable Hamiltonian given by Eq. (2.16). According to the Poincaré-Birkhoff theorem the neighborhood of the separatrix gives rise to a chaotic band in the vicinity of the resonance. Let Ir be the position of the resonance in question in the case of the integrable system H0 (I). The size of the resonance can be described by.

(28) 14. CHAPTER 2. CRITERIA FOR THE BREAKUP OF KAM TORI. the resonance half-width. For the Hamiltonian (2.16) it is defined to be the distance between Ir and the maximal value of I attained by points on the separatrix. Denote the resonance half-width by ∆I. Thus, if we consider the projection of the chaotic region to the coordinate line I, we obtain the interval [Ir − ∆I, Ir + ∆I]. We now repeat our considerations for the second resonance. As a result, we obtain an interval [Ir0 − ∆I 0 , Ir0 + ∆I 0 ] related to the second resonance. Chirikov’s criterion in its basic form states that widespread chaos sets in if and only if the intervals [Ir −∆I, Ir +∆I] and [Ir0 −∆I 0 , Ir0 +∆I 0 ] have a non-empty intersection. Let us return to the standard map. Consider the first resonance independently of the others. We have the Hamiltonian H(I, θ, t) =. I2 − K cos θ. 2. (2.17). The frequency is given by ω = ∂H/∂I = I. The resonance corresponds to the frequency √ zero. Thus, the resonance is centered at the point I = 0. Its half-width is ∆I = 2 K. For the second resonance we obtain H(I, θ, t) =. I2 − K cos(θ + 2πt). 2. (2.18). The resonance corresponds to the value ω = 2π of frequency. Thus, the center of the √ resonance lies at the point I = 2π. The half-width is again given by ∆I = 2 K. The distance between the resonances is equal to 2π. The method of overlapping resonances states that the critical value Kcrit of the perturbation parameter is given by p 2∆I = 4 Kcrit = 2π. (2.19) Thus, π2 ≈ 2.5 (2.20) 4 The estimate obtained is rather far from the value 0.97 which is thought to be the critical value for the standard map. Nevertheless, Chirikov’s method provides us with an estimate of the right magnitude. Note also that there exist modifications of the method of overlapping resonances which deliver considerably better results. Kcrit =. 2.3. Greene’s method. The method is introduced and applied to the standard map in [21]. The idea of this method is to relate the existence of a KAM torus to the stability properties of periodic orbits in the vicinity of the KAM torus. First, let us introduce the notion of the residue of a periodic orbit. Let P be an area-preserving mapping of a two-dimensional region into itself. Assume that a point (x0 , y0 ) is a fixed point of the mapping P . Consider the linearization M of the mapping near the fixed point (x0 , y0 ). It is given by the matrix ∂P (x, y)

(29)

(30) a b M= = . (2.21)

(31) c d ∂(x, y) (x,y)=(x0 ,y0 ).

(32) 2.3. GREENE’S METHOD. 15. The absolute value of the determinant of the matrix M equals one because the mapping P preserves area. Assume further that the mapping P is orientation-preserving. Then the determinant of M is equal to one. Consider the eigenvalues λ1 , λ2 of the matrix M . They are given by solutions of the equation λ2 − (a + d)λ + ad − bc = λ2 − tr(M )λ + det(M ) = λ2 − tr(M )λ + 1 = 0. Explicitly, λ1,2 =. trM ±. (2.22). √. tr2 M − 4 . 2. (2.23). The quantity 1 R = (2 − trM ) (2.24) 4 is called the residue of the fixed point (x0 , y0 ). In terms of the residue the eigenvalues become p λ1,2 = 1 − 2R ± 2 R(R − 1). (2.25) A fixed point (x0 , y0 ) of a two-dimensional area-preserving mapping is called elliptic if all the eigenvalues of its linearization operator are complex of magnitude unity. It is easy to see that in the case of orientation-preserving mappings the residue of an elliptic fixed point satisfies the inequality 0 < R < 1. (2.26) A fixed point (x0 , y0 ) of a two-dimensional area-preserving mapping is said to be hyperbolic if the eigenvalues of its linearization operator are real and different from ±1. For a hyperbolic fixed point we have in the case of an orientation-preserving mapping R < 0 or R > 1.. (2.27). Consider a KAM torus with winding number W = [a0 , a1 , . . . ]. It is known that the best rational approximants Pn /Qn to W are given by the truncated continued fractions Pn /Qn = [a0 , a1 , . . . , an ], see [23, 33]. According to the Poincaré-Birkhoff theorem the resonant torus with winding number Pn /Qn gives rise to two periodic orbits. One of them is hyperbolic, the other elliptic. Consider these two orbits. Let RP+n /Qn denote the residue of the elliptic periodic orbit, and RP−n /Qn the residue of the hyperbolic periodic orbit. We are interested in the behavior of the quantities RP+n /Qn and RP−n /Qn as n increases. Based on numerical investigations of the standard map, Greene formulated the following criterion. Given a two-dimensional area-preserving map, there is a smooth invariant KAM curve of winding number W = [a0 , a1 , . . . ] if and only if the sequence RP±n /Qn with Pn /Qn being [a0 , a1 , . . . , an ] converges to zero. Greene’s method seems to be very effective for the study of the breakup of KAM tori or KAM curves. In the case of the standard map, the method predicts the critical value Kcrit to be 0.971635406 . . . . MacKay proposed another method for studying the breakup of noble KAM tori using the notion of residue, see [29, 30] for details. Paul and Richter have applied Greene’s method and the Mackay residue criterion to the double pendulum, see [35]..

(33) 16. 2.4. CHAPTER 2. CRITERIA FOR THE BREAKUP OF KAM TORI. The renormalization group approach. The idea of the renormalization group (RG) approach to the study of the breakup of KAM tori in Hamiltonian systems with two degrees of freedom was developed in different versions by MacKay [29, 30] and Escande [16]. In our presentation we follow the approach of Escande. Consider the following functional space H. Let H be a Hamiltonian defining a Hamiltonian system with two degrees of freedom. A point in H is given by a set {H, m1 , m2 , p1 , p2 , W, h} with m1 , m2 , p1 , p2 being integers, and W and h being real numbers. We interprete the point {H, m1 , m2 , p1 , p2 , W, h} as a Hamiltonian system with two fixed resonances defined by the pairs (m1 , m2 ) and (p1 , p2 ). The value of energy is given by h. The number W corresponds to the winding number of a KAM torus on the energy surface H = h. Assume further that we have found a mapping R : H → H of the space H into itself with the following property. For every point {H, m1 , m2 , p1 , p2 , W, h} in H there exists a KAM torus of winding number W located between the resonances (m1 , m2 ) and (p1 , p2 ) on the energy surface H = h if and only if there exists a KAM torus of winding number W 0 located between the resonances (m01 , m02 ) and (p01 , p02 ) on the energy surface H 0 = h0 where the point {H 0 , m01 , m02 , p01 , p02 , W 0 , h0 } is the image of the point {H, m1 , m2 , p1 , p2 , W, h} under the mapping R. Suppose that the mapping R has two invariant stable surfaces in the functional space H corresponding to the cases of integrable and strongly chaotic systems respectively. Assume that almost every orbit of the mapping R converges to one of these surfaces. Having found a mapping R with the property just described, we could study the breakup of KAM tori as follows. Given a system H, two resonances (m1 , m2 ) and (p1 , p2 ), an irrational number W , and a value of energy h, consider the orbit of the point {H, m1 , m2 , p1 , p2 , W, h} ∈ H under the mapping R. If the orbit converges to the invariant surface corresponding to integrable systems, then the KAM torus of winding number W exists on the energy surface H = h. Indeed, the property of the system H to contain the KAM torus of winding number W on the energy surface H = h remains invariant under the mapping R. After a sufficient number of iterations, the orbit of the point {H, m1 , m2 , p1 , p2 , W, h} is located in the neighborhood of an integrable system where the KAM theorem is valid. Thus, for all the points along the orbit there exists a KAM torus of corresponding winding number. In particular, the system H itself contains the KAM torus of winding number W on the energy surface H = h. If the orbit of a point {H, m1 , m2 , p1 , p2 , W, h} ∈ H is attracted by the invariant stable surface corresponding to chaotic systems, then it is reasonable to assume that the KAM torus is destroyed. Unfortunately, till now a mapping R on H with the properties introduced above has not been discovered. It is also not clear whether such a mapping can exist. The usual way to avoid this difficulty is to replace the infinite dimensional space H by some finite dimensional subspace N of H. Given a point {H, m1 , m2 , p1 , p2 , W, h} from H, we consider the projection of the point into the subspace N . We refer to the resulting Hamiltonian system as the normal form of {H, m1 , m2 , p1 , p2 , W, h}. In some cases the initial system is already written as a normal form so that the process of projection into N can be omitted..

(34) 2.4. THE RENORMALIZATION GROUP APPROACH. 17. A number of various normal forms have been suggested by different authors. Escande [16] discusses the so-called paradigm Hamiltonian H(v, x, t) =. v2 − M cos x − P cos k(x − t). 2. (2.28). The system is a non-autonomous Hamiltonian system with one degree of freedom. It can easily be rewritten as an autonomous Hamiltonian system with two degrees of freedom. The parameters of the Hamiltonian are the magnitudes M , P , and the wave number k. Thus, normal forms make up a finite dimensional space N . Paul [34] chooses the function 1 H(I1 , I2 , θ1 , θ2 ) = W I2 + (I12 + 2βI1 I2 + γI22 ) − M cos(θ1 + θ2 ) − P cos(k1 θ1 + k2 θ2 ) (2.29) 2 as the normal form for a Hamiltonian system with two degrees of freedom. Chandre et al. [11] consider the family 1 H = ωI1 − I2 + (I1 + kI2 )2 − M cos θ1 − P cos θ2 . 2. (2.30). We choose as the normal form the Hamiltonian H = ωI1 + I2 + aI12 + 2bI1 I2 + cI22 − M cos θ1 − P cos θ2. (2.31). with a2 + b2 + c2 = 1. The next step of the RG analysis is to introduce the renormalization map on the space N . Two approaches have been used for this purpose. Chandre et al. [11, 8] make use of Lie transformations. Escande et al. [16, 19, 18, 17], and Paul [34] employ the technique of generating functions in the framework of classical perturbation theory. We also use the latter approach in our work. Qualitatively the renormalization map can be described as follows. Given a point from N , transform its Hamiltonian in such a way that the transformed Hamilton function contains no term linear in perturbation parameters. The resulting Hamiltonian does not belong to N anymore. Generally, it has an infinite number of resonances. We define the projection of the transformed Hamiltonian into N in the following way. The two initial resonances may be viewed as the first rational approximants to the irrational winding number in question. Out of an infinite number of resonances presented in the new Hamiltonian, we take into account only two of them. These two resonances correspond to the next rational approximants of the winding number. In this way we obtain a near-integrable system whose Hamiltonian contains only two resonances. Rescaling the angle variables we come to resonances of the same form as initial ones. Using Taylor expansion about the KAM torus in question, the integrable part of this new Hamiltonian can be approximated by the integrable part of the corresponding normal form. The magnitudes of the resonances have also to be approximated. These simplifications lead to a new Hamilton function in normal form. The renormalization map assigns this resulting system to the initial point from N . The qualitative behavior of the renormalization map is thought to be rather simple. If we restrict ourselves to the two-dimensional plane given by the magnitudes of the.

(35) 18. CHAPTER 2. CRITERIA FOR THE BREAKUP OF KAM TORI. resonances, then there are two stable and one unstable fixed points. The stable fixed points correspond to the stable surfaces discussed above. Again, the existence of the KAM torus is related to the behavior of iterations of the normal norm under the renormalization map just described.. 2.5. Comparison between the different methods. We end this chapter with some remarks concerning the methods discussed above. We described four methods for the study of the onset of widespread chaos in Hamiltonian systems with two degrees of freedom. Two of them, namely, the sup map analysis and Greene’s methods, are entirely numerical. The other two, the method of overlapping resonances and the RG approach, are of analytical nature. The sup map analysis is a numerical method, easy to implement and intuitively appealing. However, it has a number of disadvantages. First, the method is rather timeconsuming. Even in the case of the standard map one has to carry out many iterations in order to obtain a realistic estimate for the critical value of the perturbation parameter. Second, one needs a good localization of the two main resonances for the method to work properly. Greene’s method is also a numerical one. Usually, the use of the method or its modifications leads to very reliable estimates. The implementation of the method becomes essentially easier if the system in question possesses symmetries as it is the case for the standard map. Chirikov’s method is an analytical, clear, and easy to apply approach. However, the estimates which can be obtained in this way are usually not very precise. The method does not work well if resonances of very different magnitudes are studied. It does not make use of the KAM theory. Another analytical method is based on the RG approach. Like Greene’s method it employs the qualitative picture of the onset of widespread chaos provided by the KAM theory. Like in Chirikov’s method one must carry out a number of essential approximations to the initial Hamilton function. It is by no means obvious whether the study of the transition to global chaos in the simplified system leads qualitatively to the same results as that of the initial system. A number of difficult technical problems must be overcome in order to apply the RG approach to a given system. One of them is to introduce actionangle variables for the unperturbed system and express the perturbation part in terms of these variables. Nevertheless, from a theorist’s point of view, the RG approach seems to be the most justified method for the study of the onset of widespread chaos. We describe our version of the renormalization theory in Chapter 3..

(36) Chapter 3 Renormalization Theory In this section we introduce a new version of the renormalization operator for Hamiltonian systems with two degrees of freedom. This version is essentially based on the renormalization operator of Escande [16], [31], [34]. To rescale the Hamiltonian we also use the classical perturbation theory approach. We choose, however, a different normal form for the Hamilton function. Our choice is like in [11]. We start with a compact integrable Hamiltonian system with two degrees of freedom. According to the Liouville-Arnold theorem [3], one can introduce local action-angles variables (I, θ) = (I1 , I2 , θ1 , θ2 ). In these variables the Hamiltonian H0 is independent of angles. So we have H0 (I, θ) = H0 (I). (3.1) Let us now consider some perturbation of the Hamiltonian system H0 which depends on a small parameter ε. The perturbed Hamiltonian is then H(I, θ) = H0 (I) + εH1 (I, θ). We expand the non-integrable part H1 in a Fourier series, X H(I, θ) = H0 (I) + ε Vn (I) cos(n · θ + γ n ).. (3.2). (3.3). n=(n1 ,n2 ). 3.1. Normal form. Let us consider a KAM torus Tε with frequencies ω = (ω1 , ω2 ), ω1 = ∂H0 /∂I1 , ω2 = ∂H0 /∂I2 on the energy surface H = h. Assume that only two resonances are relevant for the torus Tε . Let m = (m1 , m2 ) and p = (p1 , p2 ) be these resonances in the sense that the frequencies (ω1m , ω2m ) and (ω1p , ω2p ) of the corresponding resonant tori of the unperturbed system satisfy the relations m1 ω1m + m2 ω2m = 0, p1 ω1p + p2 ω2p = 0.. (3.4) (3.5). As shown in Appendix A, it is possible to approximate the given Hamilton function H with the following normal form H(I, θ) = ωI1 + I2 + aI12 + 2bI1 I2 + cI22 + M cos θ1 + P cos θ2 19. (3.6).

(37) 20. CHAPTER 3. RENORMALIZATION THEORY. with 0 < ω < 1, 0 < a, a2 + b2 + c2 = 1. The coefficients ω, a, b, c, M , P depend on the initial Hamilton function, the value of energy h, the torus, and the resonances m, p, see Appendix A for details. Note that ω=. m1 ω1 + m2 ω2 , p1 ω1 + p2 ω2. (3.7). and the torus after normalization lies on the energy surface H = 0. In the integrable case, the torus is located at the origin (I1 , I2 ) = (0, 0).. 3.2. The renormalization operator. Assume that the Hamiltonian system in question has the normal form given by (3.6). Let us discuss changes in qualitative behavior of the system as the parameters M and P vary. If the values of the amplitudes M and P are zero, then one has an integrable case. Figure 3.1 shows a Poincaré section θ1 = θ2 , θ˙1 < θ˙2 , h = 0, for the √ normal form with ω = g, a = (g + 1)/2, b = −1/2, c = g/2, M = 0, P = 0, where g = ( 5 − 1)/2 is the golden mean. In a general case, for non-zero values of M and P , the system becomes non-integrable. Figure 3.2 shows a Poincaré section θ1 = θ2 , θ˙1 < θ˙2 , h = 0, for the normal form with ω = g, a = (g + 1)/2, b = −1/2, c = g/2, M = 0.06, P = 0.06. Obviously, the two resonances M cos θ1 , P cos θ2 give rise to chaotic regions. h =0.00000. M=0.0000. 0.10. I2. -0.25. -0.60 0.00. 1.57. θ2. 3.14. 4.71. 6.28. Figure 3.1: A Poincaré section θ1 − θ2 = 0, θ˙1 − θ˙2 < 0, h = 0, M = 0, P = 0 for the normal form. Let us discuss Figure 3.2 in more detail. A number of nested islands can be seen in the Poincaré section. Consider the large region containing one island structure in the bottom part of Figure 3.2. The numerical investigation of the trajectories from this region shows that it corresponds to the resonances ω1 : ω2 = 0 : 1. In other words, the corresponding torus in the unperturbed Hamilton system is of winding number W = ω1 /ω2 = 0. Another.

(38) 3.2. THE RENORMALIZATION OPERATOR. h =0.00000. 21. M=0.0600. 0.06. I2. -0.60 0.00. 1.57. θ2. 3.14. 4.71. 6.28. Figure 3.2: A Poincaré section θ1 − θ2 = 0, θ˙1 − θ˙2 < 0, h = 0, M = 0.06, P = 0.06 for the normal form in projection to the (I2 ,θ2 ) plane.. h =0.00000. M=0.0600. 0.06. I2. -0.10 0.00. 1.57. θ2. 3.14. 4.71. 6.28. Figure 3.3: Blowup of the top part of Figure 3.2..

(39) 22. CHAPTER 3. RENORMALIZATION THEORY h =0.00000. M=0.0600. 0.04. I2. -0.05 0.00. 1.57. θ2. 3.14. 4.71. 6.28. Figure 3.4: Further blowup of the top part of Figure 3.2.. elliptic region containing one island structure is located in the top part of Figure 3.2. It corresponds to the resonance ω1 : ω2 = 2 : 3 = 2/3. The third one island structure is located between the resonance 2/3 and the large chaotic region corresponding to the resonance 0/1. This structure is related to the resonance ω1 : ω2 = 1 : 2 = 1/2. Consider the blowup of Figure 3.2, see Figure 3.3. The resonances 2/3 and 1/2 are seen in more detail. Two further resonant regions can be identified. Both lie between the resonances 2/3 and 1/2. They correspond to the resonances ω1 : ω2 = 3 : 5 = 3/5 and ω1 : ω2 = 5 : 8 = 5/8. The former consists of two islands, the latter of three. An invariant line corresponding to a KAM torus can clearly be seen in Figure 3.3. The line lies between the resonances 3/5 and 5/8. A further blowup of Figure 3.2 is given by Figure 3.4. The last KAM torus is thought to be of winding ratio ω = g. The rationals w2 = 1/2, w3 = 2/3, w4 = 3/5, w5 = 5/8 corresponding to the resonances discussed are rational approximants of the noble number g given by its continued fraction representation. Figures 3.2-3.4 show that the chaotic regions originating from the resonances given by rationals wn , n = 2, 3, 4, 5, become smaller as n is increased. One of the main assumptions of the RG approach in the study of KAM tori is that the existence of a KAM torus is connected to the perturbation properties of the resonances in the vicinity of the torus. As we see from Figures 3.2-3.4, the resonances defined by the best rational approximants to the winding number of the KAM torus can and usually do lead to the strongest chaotic regions. Note that the best rational approximants for a given irrational number can be obtained using its continued fraction representation, see [23, 33] for details. In what follows only these resonances will be taken into account. The first two rational approximants of the winding number are included in the normal form. The resonances defined by the other rational approximants will be used in the next steps of the renormalization map. Using the classical methods of perturbation theory, one can introduce a canonical transformation such that the Hamilton function in the new variables contains no perturbation terms linear in M or P . In other words, we get rid of the two resonances presented.

(40) 3.3. THE RENORMALIZATION MAP FOR THE NORMAL FORM. 23. in the normal form. The next two rational approximants of the winding number are chosen to define the two-resonance approximation of the new Hamiltonian, see Section 3.3 and Appendix B.1 for details. One can then approximate the new Hamiltonian with the normal form H 0 (I1 , I2 , θ1 , θ2 ) = ω 0 I1 + I2 + a0 I12 + 2b0 I1 I2 + c0 I22 + M 0 cos θ1 + P 0 cos θ2 .. (3.8). In this way we define a map R : N → N from the parameter space to itself. The map R is called the renormalization map. Note that normal forms with M = P = 0 correspond to integrable cases. We use the following criterion for the existence of the KAM torus. The KAM torus exists for the value ε if and only if lim R(n) (ω, a, b, c, M, P ) = (ω∞ , a∞ , b∞ , c∞ , 0, 0),. n→∞. (3.9). for some parameters ω∞ , a∞ , b∞ , c∞ . Here ω, a, b, c, M , P are the parameters given by the normal form of the Hamiltonian H = H0 + εH1 . In general, the implementation of the approach just described is not easy. First, for a given Hamiltonian system one has to find a suitable presentation in terms of the perturbation of some integrable case. Second, action-angle coordinates have to be determined which may be a formidable task in practice. The Hamiltonian must then be rewritten in these coordinates. Third, we have to identify KAM tori that are particularly stable against the perturbation. Having fixed one of these tori, one has to determine its most important resonances. At this point, we are ready to write down the normal form. The normal form depends on the energy level, the KAM torus, and the value of the perturbation parameter. One of the possible strategies is now as follows: keep the energy level and the winding number ω of the KAM torus fixed; change the perturbation parameter; obtain a family of normal forms. Using the criterion just described, one can determine the critical value for the perturbation parameter. Having calculated the critical values for different KAM tori, we choose their largest value as the relevant value for the transition to global chaos at the given energy level. In the following, we consider the renormalization operator in more detail. The methods of classical perturbation theory enable us to get rid of the two given resonances in linear order. But the corresponding canonical transformation gives rise to infinitely many resonance terms that are of the second or larger power with respect to the perturbation. This necessitates a procedure for choosing the two relevant resonances out of these infinitely many ones. One of the possible solutions to this problem is to use the continued fraction representation of the torus’ winding number.. 3.3. The renormalization map for the normal form. We discuss now the renormalization map in more detail. Consider the normal form in the slightly more general version H(I1 , I2 , θ1 , θ2 ) = ω1 I1 + ω2 I2 + aI12 + 2bI1 I2 + cI22 + M cos θ1 + P cos θ2 .. (3.10).

(41) 24. CHAPTER 3. RENORMALIZATION THEORY. Apply the canonical transformation (I1 , I2 , θ1 , θ2 ) → (J1 , J2 , φ1 , φ2 ) defined by the generating function F (J1 , J2 , θ1 , θ2 ) = θ1 J1 + θ2 J2 − where w1 (J1 , J2 ) =. M sin θ1 P sin θ2 − , w1 (J1 , J2 ) w2 (J1 , J2 ). ∂H0 (J1 , J2 ) = ω1 + 2aJ1 + 2bJ2 , ∂J1. ∂H0 (J1 , J2 ) = ω2 + 2bJ1 + 2cJ2 ∂J2 are the frequencies of the unperturbed part. (3.11). (3.12). w2 (J1 , J2 ) =. (3.13). H0 (I1 , I2 ) = ω1 I1 + ω2 I2 + aI12 + 2bI1 I2 + cI22 .. (3.14). The generating function F (J1 , J2 , θ1 , θ2 ) is a function of the old angles θ1 , θ2 and new momenta J1 , J2 . The new angles φ1 , φ2 and old momenta I1 , I2 are represented in the following manner. The new angle φ1 has the form (−1) ∂w1 (−1) ∂w2 ∂F = θ1 + (−M sin θ1 ) 2 ∂J1 + (−P sin θ2 ) 2 ∂J1 , ∂J1 w1 (J1 , J2 ) w2 (J1 , J2 ) 2aM sin θ1 2bP sin θ2 + 2 . = θ1 + 2 w1 (J1 , J2 ) w2 (J1 , J2 ). φ1 =. (3.15) (3.16). Similarly, the expression for φ2 can be obtained. Thus, we have 2bP sin θ2 2aM sin θ1 + 2 , 2 w1 (J1 , J2 ) w2 (J1 , J2 ) 2bM sin θ1 2cP sin θ2 = θ2 + 2 + 2 . w1 (J1 , J2 ) w2 (J1 , J2 ). φ1 = θ1 +. (3.17). φ2. (3.18). The old momenta are given by ∂F = J1 − ∂θ1 ∂F = = J2 − ∂θ2. I1 = I2. M cos θ1 , w1 (J1 , J2 ) P cos θ2 . w2 (J1 , J2 ). (3.19) (3.20). Let us express the Hamilton function H in the new coordinates. Assuming M and P to be small, we obtain for the Taylor expansion near the point I = J the following result: H(I1 , I2 , θ1 , θ2 ) = H(I1 (J , θ), I2 (J , θ), θ1 , θ2 ) = H0 (J1 , J2 ) + M cos θ1 P cos θ2 + w2 (J1 , J2 ) − + +w1 (J1 , J2 ) − w1 (J1 , J2 ) w2 (J1 , J2 ) +M cos θ1 + P cos θ2 + O(M 2 + P 2 ) = H0 (J1 , J2 ) + O(M 2 + P 2 ), (3.21).

(42) 3.3. THE RENORMALIZATION MAP FOR THE NORMAL FORM. 25. where O(M 2 + P 2 ) denotes terms of the second and higher order in M and P . Thus, the Hamilton function H expressed in the new coordinates has no terms linear in M and P . Let us consider the Taylor expansion up to the quadratic terms in M and P . H(J1 , J2 , θ1 , θ2 ) = H0 (J1 , J2 ) + 2 1 −M cos θ1 −P cos θ2 ∂ H0 + , 2 w1 w2 ∂J1 ∂J2. −M cos θ1 w1 −P cos θ2 w2. . +. +O((M 2 + P 2 )3/2 ),. (3.22). where O((M 2 + P 2 )3/2 ) denotes terms of the third and higher order in M and P . The Hessian ∂ 2 H0 /∂J1 ∂J2 is given by ∂ 2 H0 a b =2 . (3.23) b c ∂J1 ∂J2 Thus, H(J1 , J2 , θ1 , θ2 ) = H0 (J1 , J2 ) +. . −M cos θ1 −P cos θ2 , w1 w2. . a b b c. . −M cos θ1 w1 −P cos θ2 w2. +O((M 2 + P 2 )3/2 ).. . +. (3.24). From now on, we neglect terms of magnitude (M 2 + P 2 )3/2 and higher: H(J1 , J2 , θ1 , θ2 ) = H0 (J1 , J2 ) +. aM 2 cos2 θ1 2bM P cos θ1 cos θ2 cP 2 cos2 θ2 + + . (3.25) w12 w1 w2 w22. Using the trigonometric formulae 1 + cos 2θi , 2 1 = (cos(θ1 + θ2 ) + cos(θ1 − θ2 )), 2. cos2 θi = cos θ1 cos θ2. (3.26) (3.27). we obtain H(J , θ) = H0 (J ) + +. aM 2 cP 2 aM 2 cP 2 + + cos 2θ + cos 2θ2 + 1 2w12 2w22 2w12 2w22. bM P (cos(θ1 + θ2 ) + cos(θ1 − θ2 )) . w1 w2. (3.28). Note that in order to express the Hamiltonian in the new variables J1 , J2 , φ1 , φ2 , we need to represent the old angles θ1 , θ2 as functions of J1 , J2 , φ1 , φ2 and replace θ1 , θ2 in Eq. (3.28) by these expressions. We will not need the explicit expression for H as a function of J1 , J2 , φ1 , φ2 . From the point of view of the renormalization map, it is enough to represent the integrable part as a function of J1 , J2 and to find the amplitudes of the main resonances. See Appendix B for details. As the new integrable part, we choose 2 2 ˜ 0 (J1 , J2 ) = H0 (J1 , J2 ) + aM + cP . H 2w12 2w22. (3.29). Appendix D contains the realization of the renormalization map for the normal form in the Maple computer algebra system..

(43) 26. 3.4. CHAPTER 3. RENORMALIZATION THEORY. Fixed points. This part is devoted to the study of the stable fixed point (M, P ) = (0, 0) of the renormalization map. As mentioned in Chapter 2, it is expected that the renormalization map has the following qualitative behavior in projection to the (M, P ) plane, see Figure 3.5. There exist two stable fixed points. One of them is the point (M, P ) = (0, 0). Note that in some sufficiently small neighborhood of this point the KAM theorem is valid. Thus, for any point x with (M, P ) sufficiently small the existence of the last KAM torus can be obtained as a consequence of the KAM theorem. Another stable fixed point in the (M, P ) plane is the point (M, P ) = (∞, ∞). This point is thought to correspond to the strong perturbation of the initial integrable part. Thus, the onset of widespread chaos for points in a sufficiently small neighborhood of the fixed point (M, P ) = (∞, ∞) is expected. The two basins of attraction corresponding to the two fixed points are believed to be separated by a one-dimensional critical surface.. P. KAM M Figure 3.5: The renormalization map in the (M, P ) plane.. Note that the fixed point (M, P ) = (0, 0) is super-stable. It means that the corresponding eigenvalues are zero. Indeed, let x = (ω1 , ω2 , a, b, c, M, P ) be a point in the parameter space. Assume that one of its last components, say M , equals zero. Let us consider the image of the Hamilton function H defined by x under the renormalization.

(44) 3.4. FIXED POINTS. 27. map. We refer to Section 3.3 and Appendix B for notation. In the coordinates J, θ the function H reads cP 2 cP 2 H(J, θ) = H0 (J) + + cos 2θ2 . (3.30) 2w22 2w22 The normal form for this Hamiltonian defines the image x0 = (ω10 , ω20 , a0 , b0 , c0 , M 0 , P 0 ) of the point x under the renormalization map. Consider the linearization R of the renormalization map near the fixed point (M, P ) = (0, 0). Then the new amplitudes are 0 M M 0 2 2 =R + O(M + P ) = R + O(M 2 + P 2 ). (3.31) P0 P P In other words, M 0 and P 0 are proportional to P unless R = 0. However, from Eq. (3.30) it can easily be seen that M 0 and P 0 must be proportional to P 2 . Thus, the linearized part R of the renormalization map is identically zero. In particular, the eigenvalues of the fixed point (M, P ) = (0, 0) are zero, i.e., the fixed point is super-stable. Let (ω1 , ω2 ) be as usual (ω, 1). Consider the representation of ω as a continued fraction: ω = [a0 , a1 , a2 , a3 , . . . ].. (3.32). Assume without restriction of generality that a0 is equal to zero. Note that the first rational approximants to ω are given by a0 = 0,. a0 +. 1 1 = , a1 a1. a0 +. 1 a1 +. 1 a2. =. a2 , a1 a2 + 1. a0 +. 1 a1 +. 1 a2 + a1. =. a2 a3 + 1 . a1 a2 a3 + a1 + a3. 3. (3.33) We are interested in the last two of these approximants. Note that they correspond to the resonances (a1 a2 + 1, −a2 ) and (a1 a2 a3 + a1 + a3 , −a2 a3 − 1). These two resonances represent a reasonable choice for the main resonances corresponding to the KAM torus with the winding ratio ω. Thus, we use them for the two-resonance approximation of the perturbation part in our version of the renormalization theory. Let us now discuss the fixed point (M, P ) = (0, 0) in more detail. First of all, note that the other parameters in the normal form (the other coordinates of the point (ω1 , ω2 , a, b, c, 0, 0)) can change under the iterations of the renormalization map. √ We restrict ourselves to the case ω = ω1 = g, where g = ( 5 − 1)/2 is the golden mean. The normal form defined by x reads H = gI1 + I2 + aI12 + 2bI1 I2 + cI22 .. (3.34). We study the orbit of the point x = (g, 1, a, b, c, 0, 0) under the renormalization map. First, we apply perturbation theory to the normal form H as described in Appendix B. In the new coordinates J1 , J2 , w1 , w2 the Hamiltonian H preserves its form H = gJ1 + J2 + a0 J12 + 2b0 J1 J2 + c0 J22 ,. (3.35). but the main resonances are given by (m1 , m2 ) = (a1 a2 + 1, −a2 ) = (2, −1), (p1 , p2 ) = (a1 a2 a3 + a1 + a3 , −a2 a3 − 1) = (3, −2).. (3.36) (3.37).

(45) 28. CHAPTER 3. RENORMALIZATION THEORY. Second, we bring the new Hamiltonian to the normal form, see Appendix B for details. We transform the resonances (m1 , m2 ) = (2, −1) and (p1 , p2 ) = (3, −2) to (1, 0) and (0, 1). To do so, we use the linear canonical transformation defined by the matrix a1 a2 + 1, −a2 −a2 2 −1 R= = . (3.38) a1 a2 a3 + a1 + a3 −a2 a3 − 1 3 −2 As shown in Appendix A, the coefficients (ω, 1) = (g, 1) of the linear part of the normal form H are transformed in the following way: 0 ω1 ω1 g 2g − 1 =R =R = (3.39) ω20 ω2 1 3g − 2 with (ω10 , ω10 ) being the linear coefficients in the new coordinates. Using the fact that the golden mean g satisfies the equation g 2 + g − 1, one obtains for the winding ratio in the new coordinates ω0 2g − 1 W 0 = 10 = = −g − 1. (3.40) ω2 3g − 2 To bring the linear part of the normal form to ω ˜ J1 + J2 with 0 < ω ˜ < 1, we use the following two linear canonical transformations. First, we interchange the action coordinates. Second, we change the sign of the first action coordinate. Finally, we multiply the resulting Hamiltonian by 1/(2g − 1). It is easy to see that the frequency ω ˜ becomes ω ˜=−. ω20 1 = = g. 0 ω1 g+1. (3.41). As shown in Appendix A, up to multiplication by a constant we obtain the following expression for the quadratic part of the normal form 0 0 a b a b t t =QP R Rt P Q, (3.42) b0 c 0 b c where the matrices P =. . 0 1 1 0. . ,. Q=. . −1 0 0 1. . (3.43). corresponds to the two linear canonical transformations in question. It is easy to check that the transformation can be written as  0       a a 9 −12 4 a  b0  = S  b  =  −6   7 −2 b . (3.44) 0 c c 4 −4 1 c √ √ √ The eigenvalues of the matrix S are −1, 9 + 4 5, 9 − 4 5. The largest eigenvalue 9 + 4 5 corresponds to the stable fixed point in the subspace (a, b, c) of the parameter space. The corresponding eigenvector of length one is given by ((g + 1)/2, −1/2, g/2). We obtain the following result. The point g+1 1 g g, 1, , − , , 0, 0 (3.45) 2 2 2.

(46) 3.4. FIXED POINTS. 29. 1 0 –1 I2. –2 –3 –4 –5 –5. –4. –3. –2. –1 I1. 0. 1. 2. Figure 3.6: The energy surface (h = 0) for the trivial fixed point of the renormalization operator.. is a stable fixed point with respect to the coordinates a, b, c, M , P of the renormalization map. Figure 3.6 shows the energy surface h = 0 of the corresponding normal form. Figure 3.7 shows the results of the numerical investigations of the renormalization map in projection to the (M, P ) plane. Consider a point g+1 1 g , − , , M, P (3.46) x = g, 1, 2 2 2 from the parameter space N . Choose some small neighborhood M 2 + P 2 < δ 2 of the origin. We take δ = 0.01 in our actual computation. We consider three iterates x1 , x2 , x3 of the point x = x0 . Let n be the minimal integer such that the projection of xn to the (M, P ) plane lies in the neighborhood M 2 + P 2 < δ. If the projections of all three iterates are outside this region take n to be 4. The regions in the (M, P ) plane corresponding to different values of n are visualized in Figure 3.7. Three regions can easily be identified. The largest region corresponds to n = 4, that is, to the points in N where the onset of widespread chaos is expected. The region in the neighborhood of the origin contains the points with n = 1 or n = 0. The third region is located between the first two and corresponds to the initial conditions leading to n = 2. The region n = 3 can hardly be seen. The chaotic region and the region n = 2 are separated by a critical surface..

(47) 30. CHAPTER 3. RENORMALIZATION THEORY. 0.5. 0.4. 0.3 P 0.2. 0.1. 0.0 0.0. 0.1. 0.2. 0.3. 0.4. M. Figure 3.7: The renormalization map in the (M, P ) plane.. 0.5.

(48) Chapter 4 Applications In this chapter the RG approach to the breakup of KAM tori is applied to a number of Hamiltonian systems. Section 4.1 deals with the paradigm Hamiltonian of Escande and Doveil. In Section 4.2 we discuss the Walker and Ford model. Section 4.3 is devoted to the study of a simple model of the ethane molecule. Section 4.4 is concerned with the double pendulum problem. The Baggott H2 O Hamiltonian is studied in Section 4.5. Our last example is the family of lima¸con billiards, see Section 4.6. Given a Hamiltonian system with two degrees of freedom, the following steps must be executed in order to make use of the renormalization theory. First, using the Poincaré section technique we visualize the onset of chaos in the system. Usually, many different resonances can be identified. We fix two of them and discuss the onset of widespread chaos between these two resonances. Then we express the Hamilton function as a near-integrable Hamiltonian system. Note that to choose an integrable limiting case which is suitable for the study of the breakup of KAM tori between the resonances is not always an easy task. Having determined such a limiting case, introduce action-angle variables, and express the Hamiltonian in these new coordinates. Using the two-resonance approximation, the centered-resonance approximation, the approximation of the integrable part by its Taylor expansion to the second order, we find the parameters of the normal form corresponding to the pair of resonances, the value of energy, and the winding ratio of the KAM torus which is thought to be the last KAM torus between the two resonances in question. In particular, we need the expression of the Hamilton function in the action-angle coordinates, and the first and second derivatives of the Hamilton function with respect to the action variables. Usually, the corresponding formulae are easily written in implicit form. Appendix C describes how one finds the quantities in question in the case when the energy is given by an implicit function of the action variables. To calculate the magnitudes of the resonances we need formulae for the angle variables.. 4.1. Application to the paradigm Hamiltonian. This section is devoted to the study of the so-called paradigm Hamiltonian of Escande and Doveil , see [16, 28, 36]. 31.