Homotopy Classes for Stable Connections between Hamiltonian Saddle-Focus Equilibria

Kalies, William D.; Kwapisz, Jarosław; Vandervorst, Robert

doi:10.1007/s002200050332

Cited by 70 publications

(112 citation statements)

References 26 publications

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“…• This last corollary is of the same flavour of a similar result known for the waterwave equation (1.1) (see [9]) and is a contribution to the understanding of the chaotic dynamics of (EFK) (see [22,23,31]). …”

Section: Restricting M and Sm To An Appropriate Closed Subset Of Y Tsupporting

confidence: 60%

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Shooting methods and topological transversality

Buffoni¹

1999

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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We show that shooting methods for homoclinic or heteroclinic orbits in dynamical systems may automatically guarantee the topological transversality of the stable and unstable manifolds. The interest of such results is twofold. First, these orbits persist under perturbations which destroy the structure allowing the shooting method and, second, topological transversality is often sufficient when some kind of transversality is required to obtain chaotic dynamics. We shall focus on heteroclinic solutions in the extended Fisher-Kolmogorov equation.

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Section: Restricting M and Sm To An Appropriate Closed Subset Of Y Tsupporting

confidence: 60%

“…We shall also need the following lemma. • This gives a new proof that the extended Fisher-Kolmogorov equation is chaotic for all values of 7 > | (see [22,23,31]). …”

Section: A)mentioning

confidence: 91%

Shooting methods and topological transversality

Buffoni¹

1999

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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“…Analogous to (21) and (22), but now for sequences with index starting at 0 rather than −1, we define…”

Section: The Boundsmentioning

confidence: 99%

“…Those equilibria are stable solutions of the PDE (1) for α > 3 2 , and saddle-foci for the ODE (2) in the same parameter range. It is well known that saddle-foci may act as organizing centers for complicated dynamics [15,21], and this inspires us to focus our attention on the dynamics in the energy level E = 0. Our main result is to establish rigorously that the Swift-Hohenberg ODE has chaotic dynamics in the energy level E = 0 for a large continuous range of parameter values.…”

Section: Introductionmentioning

confidence: 99%

Chaotic Braided Solutions via Rigorous Numerics: Chaos in the Swift–Hohenberg Equation

Berg

Lessard

2008

SIAM J. Appl. Dyn. Syst.

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We prove that the stationary Swift-Hohenberg equation has chaotic dynamics on a critical energy level for a large (continuous) range of parameter values. The first step of the method relies on a computer assisted, rigorous, continuation method to prove the existence of a periodic orbit with certain geometric properties. The second step is topological: we use this periodic solution as a skeleton, through which we braid other solutions, thus forcing the existence of infinitely many braided periodic orbits. A semi-conjugacy to a subshift of finite type shows that the dynamics is chaotic.

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“…for all parameter values where ξ 3,4 are saddle-foci. Further existence results might for instance be obtained by variational techniques, as in [15].…”

Section: Discussionmentioning

confidence: 99%

Bifurcation of a reversible Hamiltonian system from a fixed point with fourfold eigenvalue zero

Wagenknecht¹

2002

Dynamical Systems

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We study bifurcations from a fixed point with fourfold eigenvalue zero occurring in a two degrees of freedom Hamiltonian system of second order ODEs which is additionally reversible with respect to two different linear involutions. Using techniques from Catastrophe Theory we are led to a codimension 2 problem and obtain two different unfoldings of the singularity related to the hyperbolic and elliptic umbilic, respectively.The analysis of the unfolded systems is essentially concerned with the existence and properties of homoclinic and heteroclinic orbits. Our studies are motivated by a problem from nonlinear optics concerning the existence of solitons in a χ 2 -medium.

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Homotopy Classes for Stable Connections between Hamiltonian Saddle-Focus Equilibria

Cited by 70 publications

References 26 publications

Shooting methods and topological transversality

Shooting methods and topological transversality

Chaotic Braided Solutions via Rigorous Numerics: Chaos in the Swift–Hohenberg Equation

Bifurcation of a reversible Hamiltonian system from a fixed point with fourfold eigenvalue zero

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