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arXiv · 2008.01911

Invariant manifolds of homoclinic orbits and the dynamical consequences of a super-homoclinic: A case study in (mathbb{R}^4) with (mathbb{Z}_2)-symmetry and integral of motion

Sajjad Bakrani·Jeroen S. W. Lamb·Dmitry Turaev
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Abstract

We consider a (mathbb{Z}_2)-equivariant flow in (mathbb{R}^{4}) with an integral of motion and a hyperbolic equilibrium with a transverse homoclinic orbit (Gamma). We provide criteria for the existence of stable and unstable invariant manifolds of (Gamma). We prove that if these manifolds intersect transversely, creating a so-called super-homoclinic, then in any neighborhood of this super-homoclinic there exist infinitely many multi-pulse homoclinic loops. An application to a system of coupled nonlinear Schr\"odinger equations is considered.

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Sajjad Bakrani, Jeroen S. W. Lamb, Dmitry Turaev. 2020-08-05. Invariant manifolds of homoclinic orbits and the dynamical consequences of a super-homoclinic: A case study in (mathbb{R}^4) with (mathbb{Z}_2)-symmetry and integral of motion. https://doi.org/10.1016/j.jde.2022.04.002

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