Cascades of homoclinic orbits to a saddle-centre for reversible and perturbed Hamiltonian systems

Champneys, Alan R; Härterich, Jörg

doi:10.1080/026811100418701

Cited by 25 publications

(19 citation statements)

References 29 publications

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“…Homoclinic bifurcations to equilibria with a linear part of saddle-focus or saddle-center type in generic four-dimensional reversible systems, i.e. without SO(2)-symmetry or Hamiltonian structure, were considered recently in [Ha r98,ChH98]. In such systems one typically has sequences of n-homoclinic solutions at the same value of parameter or only on one side of the parameter value where the primary homoclinic orbit exists.…”

Section: Introductionmentioning

confidence: 99%

Bifurcation of Homoclinic Orbits to a Saddle-Focus in Reversible Systems with SO(2)-Symmetry

Afendikov

Mielke

1999

Journal of Differential Equations

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We study reversible, SO(2)-invariant vector fields in R 4 depending on a real parameter = which possess for ==0 a primary family of homoclinic orbits T : H 0 , : # S 1 . Under a transversality condition with respect to = the existence of homoclinic n-pulse solutions is demonstrated for a sequence of parameter values = (n)The existence of cascades of 2 l 3 m -pulse solutions follows by showing their transversality and then using induction. The method relies on the construction of an SO(2)-equivariant Poincare map which, after factorization, is a composition of two involutions: A logarithmic twist map and a smooth global map. Reversible periodic orbits of this map corresponds to reversible periodic or homoclinic solutions of the original problem. As an application we treat the steady complex Ginzburg Landau equation for which a primary homoclinic solution is known explicitly. Academic Press

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Section: Introductionmentioning

confidence: 99%

Bifurcation of Homoclinic Orbits to a Saddle-Focus in Reversible Systems with SO(2)-Symmetry

Afendikov

Mielke

1999

Journal of Differential Equations

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“…It is worthwhile to remark that it is the assumption (5.22) that discriminates between the two cases: indeed, (5.17) shows that the level set H −1 (0) is tangent to W cs (0) in Hamiltonian system; see [72,Lemma 2]. We refer to [72] for a comprehensive discussion and unfolding results in the situation where the Hamiltonian structure is broken while reversibility is retained; see also [219,427] for results on homoclinic orbits to saddle-centers in reversible systems and to [361] for infinite-dimensional conservative systems.…”

Section: Theorem 554 ([156]mentioning

confidence: 99%

Homoclinic and Heteroclinic Bifurcations in Vector Fields

Homburg

Sandstede

2010

Handbook of Dynamical Systems

164

229

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An overview of homoclinic and heteroclinic bifurcation theory for autonomous vector fields is given. Specifically, homoclinic and heteroclinic bifurcations of codimension one and two in generic, equivariant, reversible, and conservative systems are reviewed, and results pertaining to the existence of multi-round homoclinic and periodic orbits and of complicated dynamics such as suspended horseshoes and attractors are stated. Bifurcations of homoclinic orbits from equilibria in local bifurcations are also considered. The main analytic and geometric techniques such as Lin's method, Shil'nikov variables and homoclinic center manifolds for analyzing these bifurcations are discussed. Finally, a few related topics, such as topological moduli, numerical algorithms, variational methods, and extensions to singularly perturbed and infinitedimensional systems, are reviewed briefly.

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“…Here, general results are available which explain the accumulation of such solutions on parameter values where the primary orbit exists, see [7] for the reversible case and [27,23] for the case of systems that are additionally Hamiltonian. An interesting point is that these studies show differences in the behaviour of systems that are purely reversible and of those are also Hamiltonian.…”

Section: Discussionmentioning

confidence: 99%

“…This will certainly be reflected in results concerning bifurcating n-homoclinic orbits. Thus, our geometric approach to the analysis may also give further insight into qualitative differences between reversible and Hamiltonian systems, see [7] for some related remarks.…”

Section: Discussionmentioning

confidence: 99%

When gap solitons become embedded solitons: a generic unfolding

Wagenknecht

Champneys

2003

Physica D: Nonlinear Phenomena

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A two-parameter unfolding is considered of single-pulsed homoclinic orbits to an equilibrium with two real and two zero eigenvalues in fourth-order reversible dynamical systems. One parameter controls the linearisation, with a transition occurring between a saddle-centre and a hyperbolic equilibrium. In the saddle-centre region, the homoclinic orbit is of codimension-one, which is controlled by the second generic parameter, whereas when the equilibrium is hyperbolic the homoclinic orbit is structurally stable. A geometric approach reveals the homoclinic orbits to the saddle to be generically destroyed either by developing an algebraically decaying tail or through a fold, depending on the sign of the perturbation of the second parameter. Special cases of different actions of Z 2 -symmetry are considered, as is the case of the system being Hamiltonian. Application of these results is considered to the transition between embedded solitons (corresponding to the codimension-one homoclinic orbits) and gap solitons (the structurally stable ones) in nonlinear wave systems. The theory is shown to match numerical experiments on two models arising in nonlinear optics and on a form of 5th-order Korteweg de Vries equation.

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Cascades of homoclinic orbits to a saddle-centre for reversible and perturbed Hamiltonian systems

Cited by 25 publications

References 29 publications

Bifurcation of Homoclinic Orbits to a Saddle-Focus in Reversible Systems with SO(2)-Symmetry

Bifurcation of Homoclinic Orbits to a Saddle-Focus in Reversible Systems with SO(2)-Symmetry

Homoclinic and Heteroclinic Bifurcations in Vector Fields

When gap solitons become embedded solitons: a generic unfolding

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