Symmetric homoclinic tangles in reversible dynamical systems have positive topological entropy

A. J. Homburg; J. S.W. Lamb; D. V. Turaev

doi:10.1016/j.aim.2025.110131

Symmetric homoclinic tangles in reversible dynamical systems have positive topological entropy

A. J. Homburg, J. S.W. Lamb, D. V. Turaev

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

Abstract

We consider reversible vector fields in R²ⁿ such that the set of fixed points of the involutory reversing symmetry is n-dimensional. Let such system have a smooth one-parameter family of symmetric periodic orbits which is of saddle type in normal directions. We establish that the topological entropy is positive when the stable and unstable manifolds of this family of periodic orbits have a strongly-transverse intersection.

Original language	English
Article number	110131
Journal	Advances in Mathematics
Volume	464
DOIs	https://doi.org/10.1016/j.aim.2025.110131
State	Published - 1 Mar 2025
Externally published	Yes

Keywords

Homoclinic tangles
Reversible dynamical systems
Topological entropy

ASJC Scopus subject areas

General Mathematics

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10.1016/j.aim.2025.110131

Cite this

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abstract = "We consider reversible vector fields in R2n such that the set of fixed points of the involutory reversing symmetry is n-dimensional. Let such system have a smooth one-parameter family of symmetric periodic orbits which is of saddle type in normal directions. We establish that the topological entropy is positive when the stable and unstable manifolds of this family of periodic orbits have a strongly-transverse intersection.",

keywords = "Homoclinic tangles, Reversible dynamical systems, Topological entropy",

author = "Homburg, {A. J.} and Lamb, {J. S.W.} and Turaev, {D. V.}",

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AU - Homburg, A. J.

AU - Lamb, J. S.W.

AU - Turaev, D. V.

PY - 2025/3/1

Y1 - 2025/3/1

N2 - We consider reversible vector fields in R2n such that the set of fixed points of the involutory reversing symmetry is n-dimensional. Let such system have a smooth one-parameter family of symmetric periodic orbits which is of saddle type in normal directions. We establish that the topological entropy is positive when the stable and unstable manifolds of this family of periodic orbits have a strongly-transverse intersection.

AB - We consider reversible vector fields in R2n such that the set of fixed points of the involutory reversing symmetry is n-dimensional. Let such system have a smooth one-parameter family of symmetric periodic orbits which is of saddle type in normal directions. We establish that the topological entropy is positive when the stable and unstable manifolds of this family of periodic orbits have a strongly-transverse intersection.

KW - Homoclinic tangles

KW - Reversible dynamical systems

KW - Topological entropy

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U2 - 10.1016/j.aim.2025.110131

DO - 10.1016/j.aim.2025.110131

M3 - Article

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JO - Advances in Mathematics

JF - Advances in Mathematics

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ER -

Symmetric homoclinic tangles in reversible dynamical systems have positive topological entropy

Abstract

Keywords

ASJC Scopus subject areas

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