Snaking of Multiple Homoclinic Orbits in Reversible Systems

Knobloch, Jürgen; Wagenknecht, Thomas

doi:10.1137/070695800

Cited by 15 publications

(9 citation statements)

References 19 publications

(31 reference statements)

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“…The above arguments apply to case ͑b͒ as well ͑Fig. 23͒: one finds S-shaped tertiary branches connecting L 19 1+ to itself as well as S-shaped tertiary branches connecting L 19 1− to itself. In addition, one now finds tertiary branches that connect L 19 1− to L 20 2+ .…”

Section: Theoretical Interpretationmentioning

confidence: 84%

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Spatially localized binary fluid convection in a porous medium

2010

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Section: Theoretical Interpretationmentioning

confidence: 84%

“…Such multipulse states also snake. [17][18][19][20] The origin and properties of the behavior just described are now quite well understood at least for single pulse states in variational systems such as SH23 or SH35. [21][22][23][24] In these systems the rung states necessarily correspond to steady solutions.…”

Section: Introductionmentioning

confidence: 99%

Spatially localized binary fluid convection in a porous medium

2010

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“…In the same setting, the existence of multi-pulses was recently considered in [224] under the assumption that u = 0 is also a bi-focus.…”

Section: Heteroclinic Cycles and Snakingmentioning

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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“…However, if the saddle index is greater than unity, then the bifurcation diagram is monotonic. Knobloch and Wagenknecht [23,27] analyse symmetric heteroclinic cycles connecting saddlefocus equilibria in reversible four-dimensional dynamical systems that arise in a number of applications, e.g., in models for water waves in horizontal water channels [28] and in the study of cellular buckling in structural mechanics [29]. In these systems the symmetric heteroclinic cycle organises the dynamics in an equivalent way to the homoclinic solution in Shilnikov's case.…”

Section: Authorsmentioning

confidence: 99%

Collapsed heteroclinic snaking near a heteroclinic chain in dragged meniscus problems

2014

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Abstract. A liquid film is studied that is deposited onto a flat plate that is inclined at a constant angle to the horizontal and is extracted from a liquid bath at a constant speed. We analyse steady-state solutions of a long-wave evolution equation for the film thickness. Using centre manifold theory, we first obtain an asymptotic expansion of solutions in the bath region. The presence of an additional temperature gradient along the plate that induces a Marangoni shear stress significantly changes these expansions and leads to the presence of logarithmic terms that are absent otherwise. Next, we numerically obtain steady solutions and analyse their behaviour as the plate velocity is changed. We observe that the bifurcation curve exhibits collapsed (or exponential) heteroclinic snaking when the plate inclination angle is above a certain critical value. Otherwise, the bifurcation curve is monotonic. The steady profiles along these curves are characterised by a foot-like structure that is formed close to the meniscus and is preceded by a thin precursor film further up the plate. The length of the foot increases along the bifurcation curve. Finally, we prove with a Shilnikov-type method that the snaking behaviour of the bifurcation curves is caused by the existence of an infinite number of heteroclinic orbits close to a heteroclinic chain that connects in an appropriate three-dimensional phase space the fixed point corresponding to the precursor film with the fixed point corresponding to the foot and then with the fixed point corresponding to the bath.

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Snaking of Multiple Homoclinic Orbits in Reversible Systems

Cited by 15 publications

References 19 publications

Spatially localized binary fluid convection in a porous medium

Spatially localized binary fluid convection in a porous medium

Homoclinic and Heteroclinic Bifurcations in Vector Fields

Collapsed heteroclinic snaking near a heteroclinic chain in dragged meniscus problems

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