Homoclinic orbits in a piecewise system and their relation with invariant sets

R O Medrano-T, M S Baptista, I L Caldas

Research output: Contribution to journalArticle

17 Citations (Scopus)

Abstract

Basic phenomena in chaos can be associated with homoclinic and heteroclinic orbits. In this paper, we present a general numerical method to demonstrate the existence of these orbits in piecewise-linear systems. We also show that the tangency of the stable and unstable manifolds, at the onset of the chaotic double-scroll attractor, changes the basin boundaries of two alpha-limit sets. These changes are evidence of homoclinicity in the dynamical system. These basins give complete information about the stable manifolds around the fixed points. We show that trajectories that depart from these boundaries (for backward integration) are bounded sets. Moreover, we also show that the unstable manifolds are geometrically similar to the existing attracting sets. In fact, when no homo- (hetero-)clinic orbits exist, the attractors are omega-limit sets of initial conditions on the unstable manifolds. (C) 2003 Elsevier B.V. All rights reserved.

Original languageEnglish
Pages (from-to)133-147
Number of pages15
JournalPhysica. D, Nonlinear Phenomena
Volume186
Issue number3-4
Early online date27 Oct 2003
DOIs
Publication statusPublished - 15 Dec 2003

Keywords

  • homoclinic orbits
  • bifurcation
  • nonlinear piecewise systems
  • numerical computation
  • Lorentz
  • model

Cite this

Homoclinic orbits in a piecewise system and their relation with invariant sets. / Medrano-T, R O ; Baptista, M S ; Caldas, I L .

In: Physica. D, Nonlinear Phenomena, Vol. 186, No. 3-4, 15.12.2003, p. 133-147.

Research output: Contribution to journalArticle

@article{0e6616e105d74f0da128c2463f492567,
title = "Homoclinic orbits in a piecewise system and their relation with invariant sets",
abstract = "Basic phenomena in chaos can be associated with homoclinic and heteroclinic orbits. In this paper, we present a general numerical method to demonstrate the existence of these orbits in piecewise-linear systems. We also show that the tangency of the stable and unstable manifolds, at the onset of the chaotic double-scroll attractor, changes the basin boundaries of two alpha-limit sets. These changes are evidence of homoclinicity in the dynamical system. These basins give complete information about the stable manifolds around the fixed points. We show that trajectories that depart from these boundaries (for backward integration) are bounded sets. Moreover, we also show that the unstable manifolds are geometrically similar to the existing attracting sets. In fact, when no homo- (hetero-)clinic orbits exist, the attractors are omega-limit sets of initial conditions on the unstable manifolds. (C) 2003 Elsevier B.V. All rights reserved.",
keywords = "homoclinic orbits, bifurcation, nonlinear piecewise systems, numerical computation, Lorentz, model",
author = "Medrano-T, {R O} and Baptista, {M S} and Caldas, {I L}",
year = "2003",
month = "12",
day = "15",
doi = "10.1016/j.physd.2003.08.002",
language = "English",
volume = "186",
pages = "133--147",
journal = "Physica. D, Nonlinear Phenomena",
issn = "0167-2789",
publisher = "Elsevier",
number = "3-4",

}

TY - JOUR

T1 - Homoclinic orbits in a piecewise system and their relation with invariant sets

AU - Medrano-T, R O

AU - Baptista, M S

AU - Caldas, I L

PY - 2003/12/15

Y1 - 2003/12/15

N2 - Basic phenomena in chaos can be associated with homoclinic and heteroclinic orbits. In this paper, we present a general numerical method to demonstrate the existence of these orbits in piecewise-linear systems. We also show that the tangency of the stable and unstable manifolds, at the onset of the chaotic double-scroll attractor, changes the basin boundaries of two alpha-limit sets. These changes are evidence of homoclinicity in the dynamical system. These basins give complete information about the stable manifolds around the fixed points. We show that trajectories that depart from these boundaries (for backward integration) are bounded sets. Moreover, we also show that the unstable manifolds are geometrically similar to the existing attracting sets. In fact, when no homo- (hetero-)clinic orbits exist, the attractors are omega-limit sets of initial conditions on the unstable manifolds. (C) 2003 Elsevier B.V. All rights reserved.

AB - Basic phenomena in chaos can be associated with homoclinic and heteroclinic orbits. In this paper, we present a general numerical method to demonstrate the existence of these orbits in piecewise-linear systems. We also show that the tangency of the stable and unstable manifolds, at the onset of the chaotic double-scroll attractor, changes the basin boundaries of two alpha-limit sets. These changes are evidence of homoclinicity in the dynamical system. These basins give complete information about the stable manifolds around the fixed points. We show that trajectories that depart from these boundaries (for backward integration) are bounded sets. Moreover, we also show that the unstable manifolds are geometrically similar to the existing attracting sets. In fact, when no homo- (hetero-)clinic orbits exist, the attractors are omega-limit sets of initial conditions on the unstable manifolds. (C) 2003 Elsevier B.V. All rights reserved.

KW - homoclinic orbits

KW - bifurcation

KW - nonlinear piecewise systems

KW - numerical computation

KW - Lorentz

KW - model

U2 - 10.1016/j.physd.2003.08.002

DO - 10.1016/j.physd.2003.08.002

M3 - Article

VL - 186

SP - 133

EP - 147

JO - Physica. D, Nonlinear Phenomena

JF - Physica. D, Nonlinear Phenomena

SN - 0167-2789

IS - 3-4

ER -