Hunting for homoclinic orbits in reversible systems: A shooting technique

1993 ◽  
Vol 1 (1) ◽  
pp. 81-108 ◽  
Author(s):  
Alan R. Champneys ◽  
Alastair Spence
Keyword(s):  
Homoclinic Orbits ◽  
Reversible Systems ◽  
Shooting Technique

scholarly journals Detection of symmetric homoclinic orbits to saddle-centres in reversible systems

2006 ◽  
Vol 214 (2) ◽  
pp. 169-181 ◽  
Author(s):  
Kazuyuki Yagasaki ◽  
Thomas Wagenknecht
Keyword(s):  
Homoclinic Orbits ◽  
Reversible Systems

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 Snaking of Multiple Homoclinic Orbits in Reversible Systems

2008 ◽  
Vol 7 (4) ◽  
pp. 1397-1420 ◽  
Author(s):  
J. Knobloch ◽  
T. Wagenknecht
Keyword(s):  
Homoclinic Orbits ◽  
Reversible Systems

Bifurcation of Homoclinic Orbits to a Saddle-Center in Reversible Systems

2003 ◽  
Vol 13 (09) ◽  
pp. 2603-2622 ◽  
Author(s):  
J. Klaus ◽  
J. Knobloch
Keyword(s):  
Fixed Point ◽  
Homoclinic Orbit ◽  
Vector Fields ◽  
Critical Parameter ◽  
Center Manifold ◽  
Homoclinic Orbits ◽  
Reversible Systems ◽  
Lin's Method ◽  
Two Parameter ◽  
Lin’S Method

We consider two-parameter families of reversible vector fields having (at the critical parameter value) a homoclinic orbit to a nonhyperbolic fixed point. The nonhyperbolicity is due to a pair of purely imaginary eigenvalues. We give a complete description of the bifurcating one-homoclinic orbits to the center manifold. For that purpose we adapt Lin's method.

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