Multi-pass stable periodic points of diffeomorphism of a plane with a homoclinic orbit

A diffeomorphism of a plane into itself with a fixed hyperbolic point and a nontransversal point homoclinic to it is studied. There are various ways of touching a stable and unstable manifold at a homoclinic point. Periodic points whose trajectories do not leave the vicinity of the trajectory of a homoclinic point are divided into a countable set of types. Periodic points of the same type are called n-pass periodic points if their trajectories have n turns that lie outside a sufficiently small neighborhood of the hyperbolic point. Earlier in the articles of Sh. Newhouse, L. P. Shil’nikov, B. F. Ivanov and other authors, diffeomorphisms of the plane with a nontransversal homoclinic point were studied, it was assumed that this point is a tangency point of finite order. In these papers, it was shown that in a neighborhood of a homoclinic point there can be infinite sets of stable two-pass and three-pass periodic points. The presence of such sets depends on the properties of the hyperbolic point. In this paper, it is assumed that a homoclinic point is not a point with a finite order of tangency of a stable and unstable manifold. It is shown in the paper that for any fixed natural number n, a neighborhood of a nontransversal homolinic point can contain an infinite set of stable n-pass periodic points with characteristic exponents separated from zero.

Different types of stable periodic points of diffeomorphism of a plane with a homoclinic orbit

A diffeomorphism of the plane into itself with a fixed hyperbolic point is considered; the presence of a nontransverse homoclinic point is assumed. Stable and unstable manifolds touch each other at a homoclinic point; there are various ways of touching a stable and unstable manifold. In the works of Sh. Newhouse, L. P. Shilnikov and other authors, studied diffeomorphisms of the plane with a nontranverse homoclinic point, under the assumption that this point is a tangency point of finite order. It follows from the works of these authors that an infinite set of stable periodic points can lie in a neighborhood of a homoclinic point; the presence of such a set depends on the properties of the hyperbolic point. In this paper, it is assumed that a homoclinic point is not a point at which the tangency of a stable and unstable manifold is a tangency of finite order. Allocate a countable number of types of periodic points lying in the vicinity of a homoclinic point; points belonging to the same type are called n-pass (multi-pass), where n is a natural number. In the present paper, it is shown that if the tangency is not a tangency of finite order, the neighborhood of a nontransverse homolinic point can contain an infinite set of stable single-pass, double-pass, or three-pass periodic points with characteristic exponents separated from zero.

Solitary Waves and Chaotic Behavior in Large-Deflection Beam

The motion equation of nonlinear flexural wave in large-deflection beam is derived from Hamilton's variational principle using the coupling of flexural deformation and midplane stretching as key source of nonlinearity and taking into account transverse, axial and rotary inertia effects. The system has homoclinic or heteroclinic orbit under certain conditions, the exact periodic solutions of nonlinear wave equation are obtained by means of Jacobi elliptic function expansion. The solitary wave solution and shock wave solution is given when the modulus of Jacobi elliptic function in the degenerate case. It is easily thought that the introduction of damping and external load can result in break of homoclinic (or heteroclinic) orbit and appearance of transverse homoclinic point. The threshold condition of the existence of transverse homoclinic point is given by help of Melnikov function. It shows that the system has chaos property under Smale horseshoe meaning.

A SUSPENSION OF AN ORIENTATION PRESERVING DIFFEOMORPHISM OF D2 WITH A HYPERBOLIC FIXED POINT AND UNIVERSAL TEMPLATE

For an orientation preserving homeomorphism φ of the disk into itself, a suspension of a finite union of periodic orbits P of φ represents a link type in the 3-sphere S3. Let φ be a C1 diffeomorphism, and p a hyperbolic fixed point of φ with a homoclinic point. If all the homoclinic points for p are transeverse, then for infinitely many n>0, φn induces all link types, that is, for each link type L in S3, there exists a finite union of periodic orbits [Formula: see text] of φn such that a suspension of [Formula: see text] of φn represents L.

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.