Existence of heterodimensional cycles near shilnikov loops in systems with A Z2 symmetry

Dongchen Li; Dmitry V. Turaev

doi:10.3934/dcds.2017189

Existence of heterodimensional cycles near shilnikov loops in systems with A Z₂ symmetry

Dongchen Li, Dmitry V. Turaev

Research output: Contribution to journal › Article › peer-review

5 Scopus citations

Abstract

We prove that a pair of heterodimensional cycles can be born at the bifurcations of a pair of Shilnikov loops (homoclinic loops to a saddle-focus equilibrium) having a one-dimensional unstable manifold in a volume-hyperbolic flow with a ℤ₂ symmetry. We also show that these heterodimensional cycles can belong to a chain-transitive attractor of the system along with persistent homoclinic tangency.

Original language	English
Pages (from-to)	4399-4437
Number of pages	39
Journal	Discrete and Continuous Dynamical Systems- Series A
Volume	37
Issue number	8
DOIs	https://doi.org/10.3934/dcds.2017189
State	Published - 1 Aug 2017
Externally published	Yes

Keywords

Chaotic dynamics
Heterodimensional cycle
Homoclinic bifurcation
Homoclinic tangency
Saddle-focus
Strange attractor

ASJC Scopus subject areas

Analysis
Discrete Mathematics and Combinatorics
Applied Mathematics

Access to Document

10.3934/dcds.2017189

Cite this

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title = "Existence of heterodimensional cycles near shilnikov loops in systems with A Z2 symmetry",

abstract = "We prove that a pair of heterodimensional cycles can be born at the bifurcations of a pair of Shilnikov loops (homoclinic loops to a saddle-focus equilibrium) having a one-dimensional unstable manifold in a volume-hyperbolic flow with a ℤ2 symmetry. We also show that these heterodimensional cycles can belong to a chain-transitive attractor of the system along with persistent homoclinic tangency.",

keywords = "Chaotic dynamics, Heterodimensional cycle, Homoclinic bifurcation, Homoclinic tangency, Saddle-focus, Strange attractor",

author = "Dongchen Li and Turaev, {Dmitry V.}",

note = "Funding Information: This work was supported by grant RSF 14-41-00044. The authors also acknowledge support by the Royal Society grant IE141468 and EU Marie-Curie IRSES Brazilian-European partnership in Dynamical Systems (FP7-PEOPLE-2012-IRSES 318999 BREUDS).",

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AU - Turaev, Dmitry V.

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AB - We prove that a pair of heterodimensional cycles can be born at the bifurcations of a pair of Shilnikov loops (homoclinic loops to a saddle-focus equilibrium) having a one-dimensional unstable manifold in a volume-hyperbolic flow with a ℤ2 symmetry. We also show that these heterodimensional cycles can belong to a chain-transitive attractor of the system along with persistent homoclinic tangency.

KW - Chaotic dynamics

KW - Heterodimensional cycle

KW - Homoclinic bifurcation

KW - Homoclinic tangency

KW - Saddle-focus

KW - Strange attractor

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