Transversal Homoclinic Points Research Articles

STABILITY OF ORDER: AN EXAMPLE OF HORSESHOES "NEAR" A LINEAR MAP

We have been motivated by a question of "anticontrol" of chaos [Schiff et al., 1994], in which recent examples [Chen & Lai, 1997] of controlling nonchaotic maps to chaos have required large perturbations. Can this be done without such brute force? In this paper, we present an example in which a family of maps, Gε, numerically displays a transverse homoclinic point, and hence a horseshoe and chaos, for a fixed value of the parameter ε. We show that these maps converge pointwise to a linear map. Furthermore, a simple scaling conjugacy is shown for a family of maps which even shows geometric similarity of all relevant structures. This is in seeming contradiction to well-known structural stability results concerning horseshoes, but careful consideration reveals that these theorems require convergence in a uniform topology in function space. We show that no such convergence is possible for our family of maps, since it is impossible to find a finite radius disk which contains all of the horseshoes Λε for every ε. Thus, there is no contradiction. Our example may be considered to be a new kind of bifurcation route to chaos by horseshoes, in which rather than creating/destroying a horseshoe by creating/destroying transverse homoclinic points, the horseshoe is sent/brought to/from infinity.

The transversal homoclinic points are dense in the codimension-1 Hénon-like strange attractors

We consider the codimension-1 Hénon-like strange attractors Λ \Lambda . We prove that the transversal homoclinic points are dense in Λ \Lambda , and that hyperbolic periodic points are dense in Λ \Lambda . Moreover the hyperbolic periodic points that are heteroclinically related to the primary fixed point ( transversal intersection of stable and unstable manifolds) are dense in Λ \Lambda .

Chaotic Motions of a Dual-Spin Body

The dynamics of a dual-spinner subject to the action of an internal oscillatory torque and Coulomb friction between the two linked bodies is investigated. The conditions for existence of transverse homoclinic points and homoclinic tangency of chaotic motions are obtained using Melnikov’s method. Through long-term numerical integration of the equations of motion, steady-state chaotic attractors are also found and studied numerically by varying the forcing amplitude.

Chaos in singular impulsive O.D.E.

It is well-known [l] that the existence of a transversal homoclinic orbit of a diffeomorphism implies chaos, i.e. the diffeomorphism possesses a compact domain which is invariant for some of its iterates and such that the dynamics of such an iterate on it is homeomorphic to the Bernoulli shift on a finite number of symbols. However, the location of transversal homoclinic points is generally difficult. One of the most common approaches is a perturbation method based on the derivation of the so-called Melnikov functions [2], i.e. we look for a transversal homoclinic point of a perturbed diffeomorphism when the unperturbed diffeomorphism has a known dynamics. Analytical methods, based on the Lyapunov-Schmidt procedure, are presented in [3-51 for periodically, regularly perturbed autonomous ordinary differential equations (o.d.e.), and in [6-81 for diffeomorphisms. Regularly perturbed impulsive o.d.e. are studied in [8]. Singular perturbation problems are studied analytically in [9, lo] for o.d.e., and in [l 11 for impulsive o.d.e. In all these papers it is assumed that the so-called reduced o.d.e. has a homoclinic orbit. In particular, the following impulsive o.d.e. is studied in [ll]

Discretizations of dynamical systems with a saddle-node homoclinic orbit

We consider a parametrized dynamical system with a homoclinic orbit that connects the center manifold of a saddle node to its strongly stable manifold. This is a codimension 2 homoclinic bifurcation with a well known unfolding. We show that the map obtained by discretizing such a system with a one-step method (the centered Euler scheme), inherits a discrete saddle-node homoclinic orbit. This orbit occurs on the line of saddle nodes and, as the numerical results suggest, there is actually a closed curve of such orbits and almost all of them consist of transversal homoclinic points. Our results complement those of [1], [5] on homoclinic discretization effects in the hyperbolic case.

Bifurcation and chaotic dynamics of homoclinic systems in ℝ3

We consider perturbations which may or may not depend explicitly on time for the three-dimensional homoclinic systems. We obtain the existence and bifurcation theorems for transversal homoclinic points and homoclinic orbits, and illustrate our results with two examples.

Stability and bifurcations of a stationary state for an impact oscillator.

The motion of a vibroimpacting one-degree-of-freedom model is analyzed. This model is motivated by the behavior of a shearing granular material, in which a transitional phenomenon is observed as the concentration of the grains decreases. This transition changes the motion of a granular assembly from an orderly shearing between two blocks sandwiching a single layer of grains to a chaotic shear flow of the whole granular mass. The model consists of a mass-spring-dashpot assembly that bounces between two rigid walls. The walls are prescribed to move harmonically in opposite phases. For low wall frequencies or small amplitudes, the motion of the mass is damped out, and it approaches a stationary state with zero velocity and displacement. In this paper, the stability of such a state and the transition into chaos are analyzed. It is shown that the state is always changed into a saddle point after a bifurcation. For some parameter combinations, horseshoe-like structures can be observed in the Poincare sections. Analyzing the stable and unstable manifolds of the saddle point, transversal homoclinic points are found to exist for some of these parameter combinations. (c) 1994 American Institute of Physics.

Rigorous verification of chaos in a molecular model.

The Thiele-Wilson system, a simple model of a linear, triatomic molecule, has been studied extensively in the past. The system exhibits complex molecular dynamics including dissociation, periodic trajectories, and bifurcations. In addition, it has for a long time been suspected to be chaotic, but this has never been proved with mathematical rigor. In this paper, we present numerical results that, using interval methods, rigorously verify the existence of transversal homoclinic points in a Poincar\'e map of this system. By a theorem of Smale, the existence of transversal homoclinic points in a map rigorously proves its mixing property, i.e., the chaoticity of the system.

Weak disks of Denjoy minimal sets

AbstractWe investigate the existence of Denjoy minimal sets and, more generally, strictly ergodic sets in the dynamics of iterated homeomorphisms. It is shown that for the full two-shift, the collection of such invariant sets with the weak topology contains topological balls of all finite dimensions. One implication is an analogous result that holds for diffeomorphisms with transverse homoclinic points. It is also shown that the union of Denjoy minimal sets is dense in the two-shift and that the set of unique probability measures supported on these sets is weakly dense in the set of all shift-invariant, Borel probability measures.

Rigorous chaos verification in discrete dynamical systems

In this paper we show how interval analysis can be used to calculate rigorously valid enclosures of transversal homoclinic points in discrete dynamical systems. The existence of such points guarantees that a Smale horseshoe is embedded in the mapping which therefore is chaotic. We take special interest in the 2-dimensional case and describe a method to obtain a computer-assisted proof of chaos for these systems. Numerical results are presented for the standard map.

On the existence of chaotic behaviour of diffeomorphisms

For several specific mappings we show their chaotic behaviour by detecting the existence of their transversal homoclinic points. Our approach has an analytical feature based on the method of Lyapunov-Schmidt.

Melnikov method and transversal homoclinic points in the restricted three-body problem

In this paper we show, by Melnikov method, the existence of the transversal homoclinic orbits in the circular restricted three-body problem for all but some finite number of values of the mass ratio of the two primaries. This implies the existence of a family of oscillatory and capture motion. This also shows the non-existence of any real analytic integral in the circular restricted three-body problem besides the well-known Jacobi integral for all but possibly finite number of values of the mass ratio of the two primaries. This extends a classical theorem of Poincaré [10]. Because the resulting singularities in our equation are degenerate, a stable manifold theorem of McGehee [7] is used.

Transversal homoclinic points of a class of conservative diffeomorphisms

We develop a global graph transformation to obtain estimates for certain invariant manifolds of a class of area preserving diffeomorphisms with symmetries. The existence of homoclinic or heteroclinic points and the analyticity of the angle between the invariant manifolds at them is proved for the Hénon and the standard maps. A lower bound of this angle is given for a large set of values of the parameters.

Chaotic Behaviour in Simple Dynamical Systems

In this paper a description is given of the chaotic behaviour generated by a transversal homoclinic point of a plane map. A proof of Smale’s theorem via the shadowing property of hyperbolic sets is provided. The result is related to certain plane periodic systems of ODE’S like the periodically perturbed pendulum equation. To this end the so-called method of Melnikov is derived.

Effects of symmetry-breaking perturbations on the onset of the “Smale-horseshoe” chaos

We shall consider in detail, by means of Melnikov method, the appearance of transverse homoclinic points in 1-dimensional problems whose potential term is given by an even function perturbed by some parity-breaking term. After having described the two possible and essentially different situations one can find, we shall introduce two methods in order to carefully analyze in each case the effects of the symmetry perturbing term on the appearance of homoclinic points (and therefore on the onset of the typical “Smale horseshoe” chaotic motion). As examples for the various cases, we shall consider a perturbed Duffing equation the equation for the Josephson junction, and finally an equation for which we can provide the explicit solution, and which becomes in this way a good test for the results.

Hyperbolic periodic solutions, heteroclinic connections and transversal homoclinic points in autonomous differential delay equations

Hyperbolic periodic solutions, heteroclinic connections and transversal homoclinic points in autonomous differential delay equations

Topological Entropy of Homoclinic Closures

In this paper we study the topological entropy of certain invariant sets of diffeomorphisms, namely the closure of the set of transverse homoclinic points associated with a hyperbolic periodic point, in terms of the growth rate of homoclinic orbits. First we study homoclinic closures which are hyperbolic in $n$-dimensional compact manifolds. Using the pseudo-orbit shadowing property of basic sets we prove a formula similar to Bowen’s one on the growth of periodic points. For the nonuniformly hyperbolic case we restrict our attention to compact surfaces.

Topological entropy of homoclinic closures

In this paper we study the topological entropy of certain invariant sets of diffeomorphisms, namely the closure of the set of transverse homoclinic points associated with a hyperbolic periodic point, in terms of the growth rate of homoclinic orbits. First we study homoclinic closures which are hyperbolic in n n -dimensional compact manifolds. Using the pseudo-orbit shadowing property of basic sets we prove a formula similar to Bowen’s one on the growth of periodic points. For the nonuniformly hyperbolic case we restrict our attention to compact surfaces.

The double-morse well and topological chaos

For a triatomic system with masses m, M and m moving in a symmetric collinear double-Morse well, it is shown that topological chaos exists and determines the flow of scattering trajectories. Three types of hyperbolic periodic orbits are found. Their stable and unstable asymptotic manifolds are numerically demonstrated to cross in transverse homoclinic and heteroclinic points, respectively,leading to chaos. This situation seems to prevail for a wide range of energies and mass ratios, but structural details of the manifolds, and the amount of chaos induced in the scattering, depend in an oscillatory manner on the mass ratio m/M.

On the generic existence of homoclinic points

AbstractThis work is concerned with the generic existence of homoclinic points for area preserving diffeomorphisms of compact orientable surfaces. We give a shorter proof of Pixton's theorem that shows that, Cr-generically, an area preserving diffeomorphism of the two sphere has the property that every hyperbolic periodic point has transverse homoclinic points. Then, we extend Pixton's result to the torus and investigate certain generic aspects of the accumulation of the invariant manifolds all over themselves in the case of symplectic diffeomorphisms of compact manifolds.