Global aspects of homoclinic bifurcations of vector fields

Homburg, Ale Jan

doi:10.1090/memo/0578

Cited by 68 publications

(101 citation statements)

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“…In order to avoid a geometric configuration of manifolds similar to that arising at an inclination flip in systems without symmetry [16], we require that …”

Section: Setting and Main Resultsmentioning

confidence: 99%

“…Initially, we require only a small perturbation of the vector field restricted to Fix G γ , the existence of which follows from the general theory without symmetry [16]. Those perturbations can be extended to a small equivariant perturbation of f in the phase space in the neighborhood of Fix G γ .…”

Section: Lemma 22 Suppose Hypotheses (H 1) -(H 6) Are Met Then By mentioning

confidence: 99%

“…A direct computation shows that (p, E ss ) is a hyperbolic equilibrium and the unstable directions include the tangent space of the fiber {p} × T E ss G ss (R n ); see e.g. [16].…”

Section: Lemma 22 Suppose Hypotheses (H 1) -(H 6) Are Met Then By mentioning

confidence: 99%

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Bifurcation from codimension one relative homoclinic cycles

Homburg

Jukes

Knobloch

et al. 2011

Trans. Amer. Math. Soc.

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Abstract. We study bifurcations of relative homoclinic cycles in flows that are equivariant under the action of a finite group. The relative homoclinic cycles we consider are not robust, but have codimension one. We assume real leading eigenvalues and connecting trajectories that approach the equilibria along leading directions. We show how suspensions of subshifts of finite type generically appear in the unfolding. Descriptions of the suspended subshifts in terms of the geometry and symmetry of the connecting trajectories are provided.

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“…In order to avoid a geometric configuration of manifolds similar to that arising at an inclination flip in systems without symmetry [16], we require that …”

Section: Setting and Main Resultsmentioning

confidence: 99%

Section: Lemma 22 Suppose Hypotheses (H 1) -(H 6) Are Met Then By mentioning

confidence: 99%

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Bifurcation from codimension one relative homoclinic cycles

Homburg

Jukes

Knobloch

et al. 2011

Trans. Amer. Math. Soc.

Self Cite

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“…Existence and smoothness of the center manifold with the first three properties listed are a consequence of the center-manifold theorem in [25]. Existence of smooth foliations can be proved as in [13], as a consequence of normal hyperbolicity and the fact that the manifold is overflowing after a suitable modification of the flow. We need to verify the assumptions of the results in [25].…”

Section: Proposition 52 There Is ι > 0 So That the System (51) Possmentioning

confidence: 99%

Robustness of Liesegang patterns

Scheel

2009

Nonlinearity

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We propose a conceptual analysis of stationary reaction-diffusion patterns with geometric spatial scaling laws as observed in Liesegang patterns. We give necessary and sufficient conditions for such patterns to occur in a robust fashion. Main ingredients are a skewproduct structure in the kinetics, caused by irreversible chemical reactions, the existence of localized spikes, and slowly decaying boundary layers. The proofs invoke the analysis of homoclinic orbits in orbit-flip position for the spatial dynamics. In particular, we show that there exists a manifold of initial conditions that do not converge to the equilibrium but to the homoclinic orbit as a set.

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“…Bifurcation diagrams for two-parameter unfoldings of a homoclinic bellows in general systems, without symmetry assumptions, are provided in [TurShi87,Tur88]. In [Hom93] the nonwandering set near homoclinic bellows in systems with Z 2 -symmetry has been studied.…”

Section: Introductionmentioning

confidence: 99%

Multiple homoclinic orbits in conservative and reversible systems

Homburg

Knobloch

2005

Trans. Amer. Math. Soc.

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Abstract. We study dynamics near multiple homoclinic orbits to saddles in conservative and reversible flows. We consider the existence of two homoclinic orbits in the bellows configuration, where the homoclinic orbits approach the equilibrium along the same direction for positive and negative times.In conservative systems one finds one parameter families of suspended horseshoes, parameterized by the level of the first integral. A somewhat similar picture occurs in reversible systems, with two homoclinic orbits that are both symmetric. The lack of a first integral implies that complete horseshoes do not exist. We provide a description of orbits that necessarily do exist.A second possible configuration in reversible systems occurs if a non-symmetric homoclinic orbit exists and forms a bellows together with its symmetric image. We describe the nonwandering set in an unfolding. The nonwandering set is shown to simultaneously contain one-parameter families of periodic orbits, hyperbolic periodic orbits of different index, and heteroclinic cycles between these periodic orbits.

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Global aspects of homoclinic bifurcations of vector fields

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Cited by 68 publications

References 0 publications

Bifurcation from codimension one relative homoclinic cycles

Bifurcation from codimension one relative homoclinic cycles

Robustness of Liesegang patterns

Multiple homoclinic orbits in conservative and reversible systems

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