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

1997 ◽  
Vol 17 (2) ◽  
pp. 445-462 ◽  
Author(s):  
HECTOR E. LOMELI
Keyword(s):  
Fixed Point ◽  
Infinite Series ◽  
Variational Approach ◽  
Melnikov Method ◽  
Higher Dimensions ◽  
Heteroclinic Orbits ◽  
Local Description ◽  
Twist Maps ◽  
Hyperbolic Fixed Point ◽  
Stable And Unstable Manifold

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 ${\Bbb{R}}^4$.

HETEROCLINIC ORBITS FOR MONOTONE TWIST MAPS THROUGH A VARIATIONAL PRINCIPLE

2001 ◽  
Vol 11 (09) ◽  
pp. 2451-2461
Author(s):  
TIFEI QIAN
Keyword(s):  
Variational Principle ◽  
Variational Method ◽  
Fixed Points ◽  
Heteroclinic Orbit ◽  
Geometric Method ◽  
Heteroclinic Orbits ◽  
Constructive Approach ◽  
Connecting Orbits ◽  
Relative Ease ◽  
Twist Maps

The variational method has shown many advantages over the geometric method in proving the existence of connecting orbits since it requires much weaker hyperbolicity and less smoothness. Many results known to be difficult to obtain by the geometric method can now be obtained by a variational principle with relative ease. In particular, a variational principle provides a constructive approach to the existence of heteroclinic orbits. In this paper a variational principle is used to construct a heteroclinic orbit between an adjacent minimal pair of fixed points for monotone twist maps on (ℝ/ℤ) × ℝ. Application of our results to a standard map is also given.

The Global Motion of a Rigid Body Subject to Periodic Body-Fixed Torques

1995 ◽  
Author(s):  
X. Tong ◽  
B. Tabarrok
Keyword(s):  
Rigid Body ◽  
Equations Of Motion ◽  
Melnikov Method ◽  
The Body ◽  
Body Motion ◽  
Heteroclinic Orbits ◽  
Coordinate Frame ◽  
Global Motion ◽  
Stable And Unstable Manifolds ◽  
Body Subject

Abstract In this paper the global motion of a rigid body subject to small periodic torques, which has a fixed direction in the body-fixed coordinate frame, is investigated by means of Melnikov’s method. Deprit’s variables are introduced to transform the equations of motion into a form describing a slowly varying oscillator. Then the Melnikov method developed for the slowly varying oscillator is used to predict the transversal intersections of stable and unstable manifolds for the perturbed rigid body motion. It is shown that there exist transversal intersections of heteroclinic orbits for certain ranges of parameter values.

Rigidity of integrable twist maps and a theorem of Moser

1998 ◽  
Vol 18 (3) ◽  
pp. 725-730
Author(s):  
KARL FRIEDRICH SIBURG
Keyword(s):  
Higher Dimensions ◽  
Point Of View ◽  
Compact Set ◽  
Rigidity Result ◽  
Twist Map ◽  
Twist Maps

According to a theorem of Moser, every monotone twist map $\varphi$ on the cylinder ${\Bbb S}^1\times {\Bbb R}$, which is integrable outside a compact set, is the time-1-map $\varphi_H^1$ of a fibrewise convex Hamiltonian $H$. In this paper we prove that if this particular flow $\varphi_H^t$ is also integrable outside a compact set, then $\varphi$ has to be integrable on the whole cylinder (and vice versa, of course). From this dynamical point of view, integrable twist maps appear to be quite rigid.As is shown in the appendix, an analogous rigidity result becomes trivial in higher dimensions.

Construction of invariant measures supported within the gaps of Aubry–Mather sets

1996 ◽  
Vol 16 (1) ◽  
pp. 51-86 ◽  
Author(s):  
Giovanni Forni
Keyword(s):  
Variational Approach ◽  
Invariant Measures ◽  
Positive Entropy ◽  
Almost Periodic ◽  
Invariant Curves ◽  
Mather Theory ◽  
Entropy Measures ◽  
Twist Maps ◽  
Minimization Procedure ◽  
Area Preserving

AbstractThis paper represents a contribution to the variational approach to the understanding of the dynamics of exact area-preserving monotone twist maps of the annulus, currently known as the Aubry–Mather theory. The method introduced by Mather to construct invariant measures of Denjoy type is extended to produce almost-periodic measures, having arbitrary rationally independent frequencies, and positive entropy measures, supported within the gaps of Aubry–Mather sets which do not lie on invariant curves. This extension is based on a generalized version of the Percival's Lagrangian and on a new minimization procedure, which also gives a simplified proof of the basic existence theorem for the Aubry–Mather sets.

ESTIMATION OF CHAOTIC THRESHOLDS FOR THE RECENTLY PROPOSED ROTATING PENDULUM

2013 ◽  
Vol 23 (04) ◽  
pp. 1350074 ◽  
Author(s):  
N. HAN ◽  
Q. J. CAO ◽  
M. WIERCIGROCH
Keyword(s):  
Nonlinear System ◽  
Nonlinear Behavior ◽  
Melnikov Method ◽  
Original System ◽  
Homoclinic Orbits ◽  
Heteroclinic Orbits ◽  
Chaotic Attractors ◽  
Viscous Damping ◽  
Approximate System ◽  
Smoothness Parameter

In this paper, we investigate the nonlinear behavior of the recently proposed rotating pendulum which is a cylindrically nonlinear system with irrational type having smooth and discontinuous characteristics depending on the value of a smoothness parameter. We introduce a cylindrical approximate system whose analytical solutions can be obtained successfully to reflect the nature of the original system without the barrier of irrationalities. Furthermore, Melnikov method is employed to detect the chaotic thresholds for the homoclinic orbits of the second-type, a pair of homoclinic orbits of the first and second-type and the double heteroclinic orbits under the perturbation of viscous damping and external harmonic forcing within the smooth regime. Numerical simulations show the efficiency of the proposed method and the results presented herein this paper demonstrate the predicated chaotic attractors of pendulum-type, SD-type and their mixture depending on the coupling of the nonlinearities.

NESTED INVARIANT 3-TORI EMBEDDED IN A SEA OF CHAOS IN A QUASIPERIODIC FLUID FLOW

2009 ◽  
Vol 19 (07) ◽  
pp. 2181-2191 ◽  
Author(s):  
HOPE L. WEISS ◽  
ANDREW J. SZERI
Keyword(s):  
Fixed Point ◽  
Fluid Flow ◽  
Cross Sections ◽  
Coherent Structures ◽  
Three Dimensional ◽  
Elliptic Type ◽  
Two Phase ◽  
Dimensional Phase Space ◽  
Hyperbolic Fixed Point ◽  
Stream Functions

Nested invariant 3-tori surrounding a torus braid of elliptic type are found to exist in a model of a fluid flow with quasiperiodic forcing. The Hamiltonian describing the system is given by the superposition of two steady stream functions, one with an elliptic fixed point and the other with a coincident hyperbolic fixed point. The superposition, modulated by two incommensurate frequencies, yields an elliptic torus braid at the location of the fixed point. The system is suspended in a four-dimensional phase space (two space and two phase directions). To analyze this system we define two three-dimensional, global, Poincaré sections of the flow. The coherent structures (cross-sections of nested 2 tori) are found each to have a fractal dimensional of two, in each Poincaré cross-section. This framework has applications to tidal and other mixing problems of geophysical interest.

scholarly journals Hartman’s theorem for hyperbolic sets

1977 ◽  
Vol 67 ◽  
pp. 41-52 ◽  
Author(s):  
Masahiro Kurata
Keyword(s):  
Fixed Point ◽  
Finite Type ◽  
Conjugacy Problem ◽  
Topological Conjugacy ◽  
Shift Of Finite Type ◽  
Linear Map ◽  
Hyperbolic Set ◽  
Slight Extension ◽  
Hyperbolic Fixed Point ◽  
Hyperbolic Sets

Hartman proved that a diffeomorphism is topologically conjugate to a linear map on a neighbourhood of a hyperbolic fixed point ([3]). In this paper we study the topological conjugacy problem of a diffeomorphism on a neighbourhood of a hyperbolic set, and prove that for any hyperbolic set there is an arbitrarily slight extension to which a sub-shift of finite type is semi-conjugate.

Horseshoes in the measure-preserving Hénon map

1995 ◽  
Vol 15 (6) ◽  
pp. 1045-1059 ◽  
Author(s):  
Ray Brown
Keyword(s):  
Fixed Point ◽  
Unstable Manifold ◽  
Recursive Formula ◽  
Homoclinic Point ◽  
Hénon Map ◽  
Henon Map ◽  
Broad Array ◽  
Hyperbolic Fixed Point ◽  
Elementary Methods

AbstractWe show, using elementary methods, that for 0 < a the measure-preserving, orientation-preserving Hénon map, H, has a horseshoe. This improves on the result of Devaney and Nitecki who have shown that a horseshoe exists in this map for a ≥ 8. For a > 0, we also prove the conjecture of Devaney that the first symmetric homoclinic point is transversal.To obtain our results, we show that for a branch, Cu, of the unstable manifold of a hyperbolic fixed point of H, Cu crosses the line y = − x and that this crossing is a homoclinic point, χc. This has been shown by Devaney, but we obtain the crossing using simpler methods. Next we show that if the crossing of Wu(p) and Ws(p) at χc is degenerate then the slope of Cu at this crossing is one. Following this we show that if χc is a degenerate homoclinic its x-coordinate must be greater than l/(2a). We then derive a contradiction from this by showing that the slope of Cu at H-1(χc) must be both positive and negative, thus we conclude that χc is transversal.Our approach uses a lemma that gives a recursive formula for the sign of curvature of the unstable manifold. This lemma, referred to as ‘the curvature lemma’, is the key to reducing the proof to elementary methods. A curvature lemma can be derived for a very broad array of maps making the applicability of these methods very general. Further, since curvature is the strongest differentiability feature needed in our proof, the methods work for maps of the plane which are only C2.

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