Symplectic Twist Maps Research Articles

On the fragility of periodic tori for families of symplectic twist maps

In this article we study the fragility of Lagrangian periodic tori for symplectic twist maps of the 2d-dimensional annulus and prove a rigidity result for completely integrable ones.More specifically, we consider 1-parameter families of symplectic twist maps (fε)ε∈R, obtained by perturbing the generating function of an analytic map f by a family of potentials {εG}ε∈R. Firstly, for an analytic G and for (m,n)∈Zd×N⁎ with m and n coprime, we investigate the topological structure of the set of ε∈R for which fε admits a Lagrangian periodic torus of rotation vector (m,n). In particular we prove that, under a suitable non-degeneracy condition on f, this set consists of at most finitely many points. Then, we exploit this to deduce a rigidity result for integrable symplectic twist maps, in the case of deformations produced by a C2 potential.Our analysis, which holds in any dimension, is based on a thorough investigation of the geometric and dynamical properties of Lagrangian periodic tori, which we believe is of its own interest.

Non-differentiable irrational curves for twist map

AbstractWe construct a $C^1$ symplectic twist map g of the annulus that has an essential invariant curve $\Gamma $ such that $\Gamma $ is not differentiable and g restricted to $\Gamma $ is minimal.

Torsion of instability zones for conservative twist maps on the annulus

For a twist map f of the annulus preserving the Lebesgue measure, we give sufficient conditions to assure the existence of a set of positive measure of points with non-zero asymptotic torsion. In particular, we deduce that every bounded instability region for f contains a set of positive measure of points with non-zero asymptotic torsion. Moreover, for an exact symplectic twist map f, we provide a simple, geometric proof of a result by Cheng and Sun (1996 Sci. China A 39 709) which characterizes -integrability of f by the absence of conjugate points.

Global behaviors of weak KAM solutions for exact symplectic twist maps

We investigated several global behaviors of the weak KAM solutions uc(x,t) parametrized by c∈H1(T,R). For the suspended Hamiltonian H(x,p,t) of the exact symplectic twist map, we could find a family of weak KAM solutions uc(x,t) parametrized by c(σ)∈H1(T,R) with c(σ) continuous and monotonic and∂tuc+H(x,∂xuc+c,t)=α(c),a.e.(x,t)∈T2, such that sequence of weak KAM solutions {uc}c∈H1(T,R) is 1/2-Hölder continuity of parameter σ∈R. Moreover, for each generalized characteristic (no matter regular or singular) solving{x˙(s)∈co[∂pH(x(s),c+D+uc(x(s),s+t),s+t)],x(0)=x0,(x0,t)∈T2, we evaluate it by a uniquely identified rotational number ω(c)∈H1(T,R). This property leads to a certain topological obstruction in the phase space and causes local transitive phenomenon of trajectories. Besides, we discussed this applies to high-dimensional cases.

The integrability of symplectic twist maps without conjugate points

It is proved that a symplectic twist map of the cotangent bundle $T^{\ast }\mathbb{T}^{d}$ of the $d$-dimensional torus that is without conjugate points is $C^{0}$-integrable, that is $T^{\ast }\mathbb{T}^{d}$ is foliated by a family of invariant $C^{0}$ Lagrangian graphs.

Invariant curves for exact symplectic twist maps of the cylinder with Bryuno rotation numbers

Since Moser's seminal work it is well known that the invariant curves of smooth nearly integrable twist maps of the cylinder with Diophantine rotation number are preserved under perturbation. In this paper we show that, in the analytic class, the result extends to Bryuno rotation numbers. First, we will show that the series expansion for the invariant curves in powers of the perturbation parameter can be formally defined, then we shall prove that the series converges absolutely in a neighbourhood of the origin. This will be achieved using multiscale analysis and renormalisation group techniques to express the coefficients of the series as sums of values which are represented graphically as tree diagrams and then exploit cancellations between terms contributing to the same perturbation order. As a byproduct we shall see that, when perturbing linear maps, the series expansion for an analytic invariant curve converges for all perturbations if and only if the corresponding rotation number satisfies the Bryuno condition.

Chaotic Dynamics in an Impact Problem

We consider the model describing the vertical motion of a ball falling with constant acceleration on a wall and elastically reflected. The wall is supposed to move in the vertical direction according to a given periodic function $f$. We show that a modification of a method of Angenent based on sub and super solutions can be applied in order to detect chaotic dynamics. Using the theory of exact symplectic twist maps of the cylinder one can prove the result under "natural" conditions on the function $f$.

Boundaries of instability zones for symplectic twist maps

AbstractVery few things are known about the curves that are at the boundary of the instability zones of symplectic twist maps. It is known that in general they have an irrational rotation number and that they cannot be KAM curves. We address the following questions. Can they be very smooth? Can they be non-${C}^{1} $?Can they have a Diophantine or a Liouville rotation number? We give a partial answer for${C}^{1} $and${C}^{2} $twist maps.In Theorem 1, we construct a${C}^{2} $symplectic twist map$f$of the annulus that has an essential invariant curve$\Gamma $such that$\bullet $ $\Gamma $is not differentiable;$\bullet $the dynamics of${f}_{\vert \Gamma } $is conjugated to the one of a Denjoy counter-example;$\bullet $ $\Gamma $is at the boundary of an instability zone for$f$.Using the Hayashi connecting lemma, we prove in Theroem 2 that any symplectic twist map restricted to an essential invariant curve can be embedded as the dynamics along a boundary of an instability zone for some${C}^{1} $symplectic twist map.

A nondifferentiable essential irrational invariant curve for a $C^1$ symplectic twist map

We construct a $C^1$ symplectic twist map $f$ of the annulus that has an essential invariant curve $\Gamma$ such that: •$\Gamma$ is not differentiable; •$f$ ↾ $\Gamma$ is conjugate to a Denjoy counterexample.

A nondifferentiable essential irrational invariant curve for a $C^1$ symplectic twist map

We construct a $C^1$ symplectic twist map $f$ of the annulus that has an essential invariant curve $\Gamma$ such that: •$\Gamma$ is not differentiable; •$f$ ↾ $\Gamma$ is conjugate to a Denjoy counterexample.

Three results on the regularity of the curves that are invariant by an exact symplectic twist map

A theorem due to G. D. Birkhoff states that every essential curve which is invariant under a symplectic twist map of the annulus is the graph of a Lipschitz map. We prove: if the graph of a Lipschitz map h:T→R is invariant under a symplectic twist map, then h is a little bit more regular than simply Lipschitz (Theorem 1); we deduce that there exists a Lipschitz map h:T→R whose graph is invariant under no symplectic twist map (Corollary 2). Assuming that the dynamic of a twist map restricted to a Lipschitz graph is bi-Lipschitz conjugate to a rotation, we obtain that the graph is even C 1 (Theorem 3). Then we consider the case of the C 0 integrable symplectic twist maps and we prove that for such a map, there exists a dense G δ subset of the set of its invariant curves such that every curve of this G δ subset is C 1 (Theorem 4).

Generating forms for exact volume-preserving maps

We study the group of volume-preserving diffeomorphisms on a manifold. We develop a general theory of implicit generating forms. Our results generalize the classical formulas for generating functions of symplectic twist maps.

Drift by coupling to an anti-integrable limit

A symplectic twist map near an anti-integrable limit has an invariant set that is conjugate to a full shift on a set of symbols. We couple such a system to another twist map in such a way that the resulting system is symplectic. At the anti-integrable limit we construct a set of nonzero measure of orbits of the second map that drifts arbitrarily far, even when the coupling is arbitrarily small. Moreover, these drifting orbits persist near the anti-integrable limit.

Magnetic field lines, Hamiltonian dynamics, and nontwist systems

Magnetic field lines typically do not behave as described in the symmetrical situations treated in conventional physics textbooks. Instead, they behave in a chaotic manner; in fact, magnetic field lines are trajectories of Hamiltonian systems. Consequently the quest for fusion energy has interwoven, for 50 years, the study of magnetic field configurations and Hamiltonian systems theory. The manner in which invariant tori breakup in symplectic twist maps, maps that embody one and a half degree-of-freedom Hamiltonian systems in general and describe magnetic field lines in tokamaks in particular, will be reviewed, including symmetry methods for finding periodic orbits and Greene’s residue criterion. In nontwist maps, which describe, e.g., reverse shear tokamaks and zonal flows in geophysical fluid dynamics, a new theory is required for describing tori breakup. The new theory is discussed and comments about renormalization are made.

Stability of minimal periodic orbits

Symplectic twist maps are obtained from a Lagrangian variational principle. It is well known that nondegenerate minima of the action correspond to hyperbolic orbits of the map when the twist is negative definite and the map is two-dimensional. We show that for more than two dimensions, periodic orbits with minimal action in symplectic twist maps with negative definite twist are not necessarily hyperbolic. In the proof we show that in the neighborhood of a minimal periodic orbit of period n, the nth iterate of the map is again a twist map. This is true even though in general the composition of twist maps is not a twist map.

Applications of the Melnikov method to twist maps in higher dimensions using the variational approach

We work with symplectic diffeomorphisms of the $n$-annulus ${\Bbb{A}}^n=T^*({\Bbb{R}}^n/{\Bbb{Z}}^n)$. Using the variational approach of Aubry and Mather, we are able to give a local description of the stable (and unstable) manifold for a hyperbolic fixed point. We use this in order to get a Melnikov-like formula for exact symplectic twist maps. This formula involves an infinite series that could be computed in some specific cases. We apply our formula to prove the existence of heteroclinic orbits for a family of twist maps in ${\Bbb{R}}^4$.

Optical Hamiltonians and symplectic twist maps

This paper concentrates on optical Hamiltonian systems of T ∗ T n , i.e., those for which H pp is a positive definite matrix, and their relationship with symplectic twist maps. We present theorems of decomposition by symplectic twist maps and existence of periodic orbits for these systems. The novelty of these results resides in the fact that no explicit asymptotic condition is imposed on the system.

Continuity of the phonon gap

The phonon gap G for an invariant set Λ of a symplectic twist map of R d × R d with action functional W is the infimum of ‖D 2 W x( z) ξ‖ 2 over zϵΛ and variations ξ with ‖ ξ‖ 2=1. It is proved here that if Λ k , kϵ N , is a sequence of compact invariant sets converging in Hausdorff topology to a compact invariant set Λ, then G( Λ k ) converges to G( Λ). The result implies that the phonon gap is an excellent quantifier of uniform hyperbolicity. Several generalisations are sketched.

Exit times and transport for symplectic twist maps.

The exit time decomposition of a set yields a description of the transport through the set as well as a visualization of the invariant structures inside it. We construct several sets computationally easier to deal with than the construction of resonances, based on the ordering properties for orbits of twist maps. Furthermore these sets can be constructed for four- and higher-dimensional twist mappings. For the four-dimensional case-using the example of Froeshle-we find "practically" invariant volumes surrounding elliptic fixed points. The boundaries of these regions are remarkably sharp; however, the regions are threaded by "tubes" of escaping orbits.

Equivalence of uniform hyperbolicity for symplectic twist maps and phonon gap for Frenkel-Kontorova models

There is a well-known correspondence between the dynamics of symplectic twist maps which represent an important class of Hamiltonian systems and the equilibrium states of a class of variational problems in solid-state physics, known as Frenkel-Kontorova models. In this paper it is shown that the key concepts of uniform hyperbolicity in the first context and phonon gap in the second context are equivalent. This allows one to transfer many ideas between the two and hence to deduce new results. For example, we prove the uniform hyperbolicity of certain invariant sets for symplectic twist maps constructed from so-called non-degenerate anti-integrable limits by Aubry and Abramovici, using the concept of phonon gap and deduce that they have measure zero; we prove existence of many multi-defect equilibrium states for a Frenkel-Kontorova system if it has a single non-degenerate defect equilibrium state, via standard results in the theory of uniform hyperbolicity.