Existence of a homoclinic point for the H�non map

1980 ◽  
Vol 75 (3) ◽  
pp. 285-291 ◽  
Author(s):  
Micha? Misiurewicz ◽  
Boles?aw Szewc
Keyword(s):  
Homoclinic Point

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.

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

1999 ◽  
Vol 09 (10) ◽  
pp. 2081-2090 ◽  
Author(s):  
ERIK M. BOLLT
Keyword(s):  
Careful Consideration ◽  
Brute Force ◽  
Homoclinic Point ◽  
Finite Radius ◽  
Uniform Topology ◽  
Linear Map ◽  
Geometric Similarity ◽  
Simple Scaling ◽  
Route To Chaos ◽  
Stability Results

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.

Solitary Waves and Chaotic Behavior in Large-Deflection Beam

2010 ◽  
Vol 29-32 ◽  
pp. 28-34
Author(s):  
Yi Qing Zhou ◽  
Zhi Fang Liu ◽  
Shan Yuan Zhang
Keyword(s):  
Elliptic Function ◽  
Wave Solution ◽  
Large Deflection ◽  
Chaotic Behavior ◽  
Solitary Wave Solution ◽  
Heteroclinic Orbit ◽  
Flexural Wave ◽  
Threshold Condition ◽  
Jacobi Elliptic Function ◽  
Homoclinic Point

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.

The transversal homoclinic point of a saddle-node implies horseshoe

1999 ◽  
Vol 42 (7) ◽  
pp. 699-703
Author(s):  
Weigu Li
Keyword(s):  
Homoclinic Point ◽  
Transversal Homoclinic Point ◽  
Saddle Node

Transport and Diffusion Around a Homoclinic Point

2017 ◽  
Author(s):  
Patrice Meunier ◽  
Peter Huck ◽  
Clément Nobili ◽  
Emmanuel Villermaux
Keyword(s):  
Homoclinic Point ◽  
Transport And Diffusion ◽  
And Diffusion

scholarly journals A generalization of a theorem of marotto

1981 ◽  
Vol 82 ◽  
pp. 83-97 ◽  
Author(s):  
Kenichi Shiraiwa ◽  
Masahiro Kurata
Keyword(s):  
Euclidean Space ◽  
Periodic Point ◽  
Real Line ◽  
Chaotic Behavior ◽  
Dimensional Euclidean Space ◽  
Homoclinic Point ◽  
Minimal Period ◽  
The Real ◽  
Continuous Map ◽  
Transversal Homoclinic Point

In 1975, Li and Yorke [3] found the following fact. Let f: I→ I be a continuous map of the compact interval I of the real line R into itself. If f has a periodic point of minimal period three, then f exhibits chaotic behavior. The above result is generalized by F.R. Marotto [4] in 1978 for the multi-dimensional case as follows. Let f: Rn → Rn be a differentiate map of the n-dimensional Euclidean space Rn (n ≧ 1) into itself. If f has a snap-back repeller, then f exhibits chaotic behavior.In this paper, we give a generalization of the above theorem of Marotto. Our theorem can also be regarded as a generalization of the Smale’s results on the transversal homoclinic point of a diffeomorphism.

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

2021 ◽  
Vol 8 (3) ◽  
pp. 406-416
Author(s):  
Ekaterina V. Vasil’eva ◽  
Keyword(s):  
Finite Order ◽  
Unstable Manifold ◽  
Small Neighborhood ◽  
Periodic Points ◽  
Homoclinic Point ◽  
Hyperbolic Point ◽  
Stable Periodic ◽  
Nontransversal Homoclinic Point ◽  
Stable And Unstable Manifold ◽  
Infinite Set

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.

Smooth diffeomorphisms of the plane with stable periodic points in a neighborhood of a homoclinic point

2012 ◽  
Vol 48 (10) ◽  
pp. 1335-1340
Author(s):  
E. V. Vasil’eva
Keyword(s):  
Periodic Points ◽  
Homoclinic Point ◽  
Stable Periodic

“Syncope Implies Chaos” in Walking-stick Maps

1977 ◽  
Vol 32 (6) ◽  
pp. 607-613 ◽  
Author(s):  
Otto E. Rössler
Keyword(s):  
Cross Sections ◽  
Monotone Sequence ◽  
Homoclinic Point ◽  
One Dimensional ◽  
Variable Chemical ◽  
The One ◽  
Diagnostic Criterion ◽  
Special Case ◽  
Walking Stick ◽  
Modest Degree

Abstract A number of 3-variable chemical and other systems capable of showing 'nonperiodic' oscillations are governed by walking-stick shaped maps as Poincare cross-sections in state space. The 2-dimensional simple walking-stick diffeomorphism contains the one-dimensional 'single-humped' Li-Yorke map (known to be chaos producing) as a 'degenerate' special case. To prove that chaos is possible also in strictly 2-dimensional walking-stick maps, it suffices to show that a homoclinic point (and hence an in­ finite number of periodic solutions) is possible in these maps. Such a point occurs in the second iterate at a certain (modest) 'degree of overlap' of the walking-stick map. At a slightly larger degree, a 'nonlinear horseshoe map' is formed in the second iterate. It implies presence of periodic trajectories of all even periodicities (at least) in the walking-stick map. At the same time, two major, formerly disconnected, chaotic subregimes merge into one. Diagnostic criterion: presence of 'syncopes' in an otherwise non-monotone sequence of amplitudes.

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