Bifurcations of global reinjection orbits near a saddle-node Hopf bifurcation

Krauskopf, Bernd; Oldeman, Bart E.

doi:10.1088/0951-7715/19/9/010

Cited by 23 publications

(51 citation statements)

References 33 publications

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“…In the parameter plane of the FitzHugh-Nagumo model many different branches of homoclinic orbits appear to have single folds near an EP1t-point, while in the calcium model a single branch of homoclinic orbits folds many times (in a "homoclinic snake") near a pair of EP1t-points. EP1t-points have also been seen in a three-dimensional model motivated by studies of semiconductor lasers with optical reinjection [20]. In this case, folds of a single curve of homoclinic orbits again accumulate on a pair of EP1t-points, in a manner similar to that seen in the nine-dimensional calcium model in [4].…”

Section: Ep1tmentioning

confidence: 64%

“…In §8.1, we describe a system originally designed to model a saddle-node Hopf bifurcation with global reinjection [20]. The bifurcation set for this system contains a snaking E-homoclinic curve that converges in parameter space to a segment connecting two EP1t-points.…”

Section: Case IIImentioning

confidence: 99%

“…Motivated by studies of semiconductor lasers with optical reinjection, the following system of equations was proposed in [20] as a model of a saddle-node Hopf bifurcation with global reinjection:ẋ…”

Section: 1mentioning

confidence: 99%

“…However, the variable ϕ is periodic, which causes reinjection and the existence of new orbits, which wind around a cylinder, that would not be present if ϕ were not periodic. It was shown in [20] that there exists a curve of E-homoclinic orbits to an equilibrium, b, that emanates from a non-central saddle-node homoclinic bifurcation point 3 at the left-hand EP1t-point and 1, 0.4980, and 0.636 × 10 3 at the right-hand point. Thus both EP1t-points conform to Case III of our analysis.…”

Section: 1mentioning

confidence: 99%

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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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Section: Ep1tmentioning

confidence: 64%

Section: Case IIImentioning

confidence: 99%

Section: 1mentioning

confidence: 99%

Section: 1mentioning

confidence: 99%

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Unfolding a Tangent Equilibrium-to-Periodic Heteroclinic Cycle

Champneys¹,

Kirk²,

Knobloch³

et al. 2009

SIAM J. Appl. Dyn. Syst.

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“…A concrete example is a heteroclinic cycle between a saddle equilibrium and a saddle periodic orbit, as can be found, for example, near a saddle-node Hopf bifurcation with global reinjection. Near this global bifurcation one can find large attracting periodic orbits that visit a neighborhood of the equilibrium and also have an arbitrary number of smaller loops around the saddle-periodic orbit; see [130,135].…”

Section: Other Mmo Mechanisms In Odesmentioning

confidence: 99%

Mixed-Mode Oscillations with Multiple Time Scales

Desroches¹,

Guckenheimer

Krauskopf³

et al. 2012

SIAM Rev.

Self Cite

507

635

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Abstract. Mixed-mode oscillations (MMOs) are trajectories of a dynamical system in which there is an alternation between oscillations of distinct large and small amplitudes. MMOs have been observed and studied for over thirty years in chemical, physical, and biological systems. Few attempts have been made thus far to classify different patterns of MMOs, in contrast to the classification of the related phenomena of bursting oscillations. This paper gives a survey of different types of MMOs, concentrating its analysis on MMOs whose small-amplitude oscillations are produced by a local, multiple-time-scale "mechanism." Recent work gives substantially improved insight into the mathematical properties of these mechanisms. In this survey, we unify diverse observations about MMOs and establish a systematic framework for studying their properties. Numerical methods for computing different types of invariant manifolds and their intersections are an important aspect of the analysis described in this paper.

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Structure of saddle-node and cusp bifurcations of periodic orbits near a non-transversal T-point

et al. 2010

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Non-transversal T-points have been recently found in problems from many different fields: electronic circuits, pendula and laser problems. In this work we study a model, based on the construction of a Poincaré map, that describes the behaviour of curves of saddlenode and cusp bifurcations in the vicinity of such a nontransversal T-point. This model is also able to predict, reproduce and explain the numerical results previously obtained in Chua's equation.

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Bifurcations of global reinjection orbits near a saddle-node Hopf bifurcation

Cited by 23 publications

References 33 publications

Unfolding a Tangent Equilibrium-to-Periodic Heteroclinic Cycle

Unfolding a Tangent Equilibrium-to-Periodic Heteroclinic Cycle

Mixed-Mode Oscillations with Multiple Time Scales

Structure of saddle-node and cusp bifurcations of periodic orbits near a non-transversal T-point

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