Homoclinic orbits for area preserving diffeomorphisms of surfaces

2021 ◽  
pp. 1-44
Author(s):  
PATRICE LE CALVEZ ◽  
MARTÍN SAMBARINO
Keyword(s):  
Periodic Point ◽  
Homoclinic Orbit ◽  
Homoclinic Orbits ◽  
Compact Surface ◽  
Area Preserving

Abstract We show that $C^r $ generically in the space of $C^r$ conservative diffeomorphisms of a compact surface, every hyperbolic periodic point has a transverse homoclinic orbit.

scholarly journals Homoclinic orbits and Lie rotated vector fields

2000 ◽  
Vol 24 (3) ◽  
pp. 187-192
Author(s):  
Jie Wang ◽  
Chen Chen
Keyword(s):  
Limit Cycles ◽  
Homoclinic Orbit ◽  
Vector Fields ◽  
Homoclinic Orbits ◽  
Singular Points ◽  
Definition Of

Based on the definition of Lie rotated vector fields in the plane, this paper gives the property of homoclinic orbit as parameter is changed and the singular points are fixed on Lie rotated vector fields. It gives the conditions of yielding limit cycles as well.

scholarly journals Interaction of homoclinic solutions and Hopf points in functional differential equations of mixed type

2009 ◽  
Vol 139 (1) ◽  
pp. 123-155 ◽  
Author(s):  
Marc Georgi
Keyword(s):  
Hopf Bifurcation ◽  
Differential Equations ◽  
Homoclinic Orbit ◽  
Mixed Type ◽  
Functional Differential Equations ◽  
Homoclinic Orbits ◽  
Homoclinic Solution ◽  
Equations Of Mixed Type ◽  
Functional Differential ◽  
Equation Of Mixed Type

We study a homoclinic bifurcation in a general functional differential equation of mixed type. More precisely, we investigate the case when the asymptotic steady state of a homoclinic solution undergoes a Hopf bifurcation. Bifurcations of this kind are diffcult to analyse due to the lack of Fredholm properties. In particular, a straightforward application of a Lyapunov–Schmidt reduction is not possible.As one of the main results we prove the existence of centre-stable and centre-unstable manifolds of steady states near homoclinic orbits. With their help, we can analyse the bifurcation scenario similar to the case for ordinary differential equations and can show the existence of solutions which bifurcate near the homoclinic orbit, are decaying in one direction and oscillatory in the other direction. These solutions can be visualized as an interaction of the homoclinic orbit and small periodic solutions that exist on account of the Hopf bifurcation, for exactly one asymptotic direction t→8 or t→−∞.

scholarly journals Josephson's junction, annulus maps, Birkhoff attractors, horseshoes and rotation sets

1986 ◽  
Vol 6 (2) ◽  
pp. 205-239 ◽  
Author(s):  
Kevin Hockett ◽  
Philip Holmes
Keyword(s):  
Symbolic Dynamics ◽  
Homoclinic Orbits ◽  
Irrational Rotation ◽  
Cantor Sets ◽  
Rotation Numbers ◽  
Twist Maps ◽  
Irrational Rotation Number ◽  
Rotation Set ◽  
Area Preserving

AbstractWe investigate the implications of transverse homoclinic orbits to fixed points in dissipative diffeomorphisms of the annulus. We first recover a result due to Aronsonet al.[3]: that certain such ‘rotary’ orbits imply the existence of an interval of rotation numbers in the rotation set of the diffeomorphism. Our proof differs from theirs in that we use embeddings of the Smale [61] horseshoe construction, rather than shadowing and pseudo orbits. The symbolic dynamics associated with the non-wandering Cantor set of the horseshoe is then used to prove the existence of uncountably many invariant Cantor sets (Cantori) of each irrational rotation number in the interval, some of which are shown to be ‘dissipative’ analogues of the order preserving Aubry-Mather Cantor sets found by variational methods in area preserving twist maps. We then apply our results to the Josephson junction equation, checking the necessary hypotheses via Melnikov's method, and give a partial characterization of the attracting set of the Poincaré map for this equation. This provides a concrete example of a ‘Birkhoff attractor’ [10].

Homoclinic Orbits in the Complex Domain

1997 ◽  
Vol 07 (02) ◽  
pp. 253-274 ◽  
Author(s):  
V. F. Lazutkin ◽  
C. Simó
Keyword(s):  
Homoclinic Orbits ◽  
Homoclinic Tangency ◽  
Complex Domain ◽  
Standard Map ◽  
Homoclinic Points ◽  
Promising Tool ◽  
Theoretical Support ◽  
Area Preserving Maps ◽  
Area Preserving

We consider the standard map, as a paradigm of area preserving map, when the variables are taken as complex. We study how to detect the complex homoclinic points, which cannot dissappear under a homoclinic tangency. This seems a promising tool to understand the stochastic zones of area preserving maps. The paper is mainly phenomenological and includes theoretical support to the observed phenomena. Several conjectures are stated.

scholarly journals A NUMERICAL TOOLBOX FOR HOMOCLINIC BIFURCATION ANALYSIS

1996 ◽  
Vol 06 (05) ◽  
pp. 867-887 ◽  
Author(s):  
A.R. CHAMPNEYS ◽  
YU. A. KUZNETSOV ◽  
B. SANDSTEDE
Keyword(s):  
Numerical Analysis ◽  
Chemical Kinetics ◽  
Bifurcation Analysis ◽  
Homoclinic Orbit ◽  
Homoclinic Orbits ◽  
Homoclinic Bifurcation ◽  
Codimension Two ◽  
Codimension One ◽  
Homoclinic Bifurcations ◽  
Theoretical Results

This paper presents extensions and improvements of recently developed algorithms for the numerical analysis of orbits homoclinic to equilibria in ODEs and describes the implementation of these algorithms within the standard continuation package AUTO86. This leads to a kind of toolbox, called HOMCONT, for analysing homoclinic bifurcations either as an aid to producing new theoretical results, or to understand dynamics arising from applications. This toolbox allows the continuation of codimension-one homoclinic orbits to hyperbolic or non-hyperbolic equilibria as well as detection and continuation of higher-order homoclinic singularities in more parameters. All known codimension-two cases that involve a unique homoclinic orbit are supported. Two specific example systems from ecology and chemical kinetics are analysed in some detail, allowing the reader to understand how to use the the toolbox for themselves. In the process, new results are also derived on these two particular models.

scholarly journals Homoclinic orbits to invariant tori near a homoclinic orbit to center–center–saddle equilibrium

2005 ◽  
Vol 201 (3-4) ◽  
pp. 268-290 ◽  
Author(s):  
Oksana Koltsova ◽  
Lev Lerman ◽  
Amadeu Delshams ◽  
Pere Gutiérrez
Keyword(s):  
Homoclinic Orbit ◽  
Invariant Tori ◽  
Homoclinic Orbits

Stability and Perturbations of Homoclinic Loops in a Class of Piecewise Smooth Systems

2015 ◽  
Vol 25 (09) ◽  
pp. 1550114 ◽  
Author(s):  
Shuang Chen ◽  
Zhengdong Du
Keyword(s):  
Limit Cycles ◽  
Homoclinic Orbit ◽  
Piecewise Linear ◽  
Small Neighborhood ◽  
Homoclinic Orbits ◽  
Piecewise Smooth ◽  
Piecewise Smooth Systems ◽  
Planar System ◽  
The Stability ◽  
First Time

Like for smooth systems, a typical method to produce multiple limit cycles for a given piecewise smooth planar system is via homoclinic bifurcation. Previous works only focused on limit cycles that bifurcate from homoclinic orbits of piecewise-linear systems. In this paper, we consider for the first time the same problem for a class of general nonlinear piecewise smooth systems. By introducing the Dulac map in a small neighborhood of the hyperbolic saddle, we obtain the approximation of the Poincaré map for the nonsmooth homoclinic orbit. Then, we give conditions for the stability of the homoclinic orbit and conditions under which one or two limit cycles bifurcate from it. As an example, we construct a nonlinear piecewise smooth system with two limit cycles that bifurcate from a homoclinic orbit.

SUBSIDIARY HOMOCLINIC ORBITS TO A SADDLE-FOCUS FOR REVERSIBLE SYSTEMS

1994 ◽  
Vol 04 (06) ◽  
pp. 1447-1482 ◽  
Author(s):  
A.R. CHAMPNEYS
Keyword(s):  
Homoclinic Orbit ◽  
Water Waves ◽  
Numerical Experiments ◽  
Homoclinic Orbits ◽  
Reversible Systems ◽  
Reversible System ◽  
Type Analysis ◽  
Reversible Transformation ◽  
Positive Integers ◽  
Good Agreement

A dynamical system is said to be reversible if there is an involution of phase space that reverses the direction of the flow. Examples are classical Hamiltonian systems with quadratic kinetic energy. For reversible systems, homoclinic orbits that are invariant under the reversible transformation typically persist as parameters are varied. This paper concerns reversible systems for which a primary homoclinic orbit to a saddle-focus is assumed to exist. The problem under investigation is a characterisation of the subsidiary homoclinic orbits which then exist in a neighbourhood of the primary one. Such orbits have applications as solitary water waves and as buckling solutions of nonlinear struts. A Shil’nikov-type analysis is performed for four-dimensional linearly reversible systems. It is shown that each subsidiary homoclinic orbit can be labelled by a symmetric string of positive integers. All possible strings of length one, two or three correspond to the existence of a homoclinic orbit, whereas only certain of those of length four or greater do. This situation contrasts with known results if the reversible system is also Hamiltonian. The analysis is supported by performing careful numerical experiments on the equation [Formula: see text] where P and α are parameters; a good agreement with the theory is found.

scholarly journals On the generic existence of homoclinic points

1987 ◽  
Vol 7 (4) ◽  
pp. 567-595 ◽  
Author(s):  
Fernando Oliveira
Keyword(s):  
Periodic Point ◽  
Invariant Manifolds ◽  
Compact Manifolds ◽  
Homoclinic Points ◽  
Symplectic Diffeomorphisms ◽  
Generic Existence ◽  
Area Preserving ◽  
Orientable Surfaces

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.

Differential Equations with Bifocal Homoclinic Orbits

1997 ◽  
Vol 07 (01) ◽  
pp. 27-37 ◽  
Author(s):  
Paul Glendinning
Keyword(s):  
Differential Equations ◽  
Homoclinic Orbit ◽  
Stationary Point ◽  
Piecewise Linear ◽  
Global Bifurcation ◽  
Homoclinic Orbits ◽  
Bifurcation Phenomena ◽  
Global Bifurcation Theory ◽  
Linear Differential ◽  
Theoretical Results

Global bifurcation theory can be used to understand complicated bifurcation phenomena in families of differential equations. There are many theoretical results relating to systems having a homoclinic orbit biasymptotic to a stationary point at some value of the parameters, and these results depend upon the eigenvalues of the Jacobian matrix of the flow evaluated at the stationary point. Three important cases arise in the theoretical analysis, and there are many examples of systems which illustrate two of these three cases. We describe a construction which can be used to produce examples of the third case (bifocal homoclinic orbits), and use this construction to prove the existence of a bifocal homoclinic orbit in a simple piecewise linear differential equation.

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