The explosion of singular cycles

Bamón, Rodrigo; Labarca, Rafael; Mañé, Ricardo; Pacífico, Maria José

doi:10.1007/bf02712919

Cited by 32 publications

(36 citation statements)

References 7 publications

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“…In the reverse direction, if a singular cycle as stated exists, the λ-lemma implies that W u (p) accumulates onto W u (q), and small perturbations of the differential equation will create homoclinic orbits to p. Bifurcations from singular cycles form an interesting problem in their own right, and we review relevant material on this subject, which was initiated in [28], in §5.2.4. We note that the singular cycles in Proposition 4.1 are called expanding (and remark that the definitions in [28] consider the time reversed flow).…”

Section: Singular Horseshoesmentioning

confidence: 99%

See 1 more Smart Citation

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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Section: Singular Horseshoesmentioning

confidence: 99%

“…We note that the singular cycles in Proposition 4.1 are called expanding (and remark that the definitions in [28] consider the time reversed flow). Letu = f (u, µ) be a one-parameter family of ODEs on R n with a homoclinic orbit h to an equilibrium p as in the above proposition, so that M s (h) W u (p) along a solution h 1 (t).…”

Section: Singular Horseshoesmentioning

confidence: 99%

Homoclinic and Heteroclinic Bifurcations in Vector Fields

Homburg

Sandstede

2010

Handbook of Dynamical Systems

164

229

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“…There are a number of previous theoretical studies on EP cycles (also called singular cycles) from an ergodic theory point of view, focusing on properties of nonwandering sets (see [1,23,24,26] and references therein). Such properties were studied for EP1t cycles in [25] when the eigenvalues at E are real with a stable leading eigenvalue, and the Floquet multipliers at P are positive.…”

Section: Ep1tmentioning

confidence: 99%

Unfolding a Tangent Equilibrium-to-Periodic Heteroclinic Cycle

Champneys¹,

Kirk²,

Knobloch³

et al. 2009

SIAM J. Appl. Dyn. Syst.

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Abstract. The dynamics occurring near a heteroclinic cycle between a hyperbolic equilibrium and a hyperbolic periodic orbit is analyzed. The case of interest is when the equilibrium has a onedimensional unstable manifold and a two-dimensional stable manifold while the stable and unstable manifolds of the periodic orbit are both two-dimensional. A codimension-two heteroclinic cycle occurs when there are two codimension-one heteroclinic connections, with the connection from the periodic orbit to the equilibrium corresponding to a tangency between the two relevant manifolds. The results are restricted to R 3 , the lowest possible dimension in which such a heteroclinic cycle can occur, but are expected to be applicable to systems of higher dimension as well.A geometric analysis is used to partially unfold the dynamics near such a heteroclinic cycle by constructing a leading-order expression for the Poincaré map in a full neighbourhood of the cycle in both phase and parameter space. Curves of orbits homoclinic to the equilibrium are located in a generic parameter plane, as are curves of homoclinic tangencies to the periodic orbit. Moreover, it is shown how curves of folds of periodic orbits, which have different asymptotics near the homoclinic bifurcation of the equilibrium and the homoclinic bifurcation of the periodic orbit, are glued together near the codimension-two point.A simple global assumption is made about the existence of a pair of codimension-two heteroclinic cycles corresponding to a first and last tangency of the stable manifold of the equilibrium and the unstable manifold of the periodic orbit. Under this assumption, it is shown how the locus of homoclinic orbits to the equilibrium should oscillate in the parameter space, a phenomenon known as homoclinic snaking. Finally, we present several numerical examples of systems that arise in applications, which corroborate and illustrate our theory.Key words: global bifurcation, Shil'nikov analysis, heteroclinic cycle, homoclinic orbit, homoclinic tangency, snaking.AMS subject classifications: 34C23, 34C37, 37C29, 37G20.1. Introduction. We are interested in the dynamics near a heteroclinic cycle connecting a hyperbolic equilibrium solution and a hyperbolic periodic orbit, referred to here as an EP-cycle. Specifically, we consider dynamics in R 3 , and assume that the equilibrium (denoted E) has a one-dimensional unstable manifold W u (E), while the periodic orbit (denoted P ) has a two-dimensional unstable manifold W u (P ). In this case, a heteroclinic connection from E to P generically will be of codimension one, while a connection from P to E generically will be of codimension zero. We denote by EP1-cycle an EP-cycle in R 3 where the connections from E to P and from P to E are both of the appropriate generic type. Such a heteroclinic cycle as a whole is then a codimension-one object, i.e., it occurs at isolated points in a generic one-parameter family of vector fields. In a two-parameter setting, EP1-cycles will occur on onedimensional curves in the two-par...

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“…Indeed, the singular cycle in [1] is in G 1 (M ) but it is not singular Axiom A for, in this case, the critical elements fail to be dense in the nonwandering set. However, such a singular cycle can be approximated by Axiom A flows without cycles.…”

Section: Theorem B a Cmentioning

confidence: 99%

“…Indeed, if we perturb the singular horseshoe, we can get both cases, depending on the way we do such a perturbation (see [1]). By explode we mean that for Y near X, the nonwandering set of Y restricted to t∈R Y t (U ) is a disjoint union of a finite number of transitive sets and this union contains at least two elements.…”

Section: Lemma 2 Let λ Be a Partially Hyperbolic Set Of X Whose Centmentioning

confidence: 99%