Quasiattractors and homoclinic tangencies

S. V. Gonchenko; L. P. Shil'nikov; D. V. Turaev

doi:10.1016/s0898-1221(97)00124-7

Quasiattractors and homoclinic tangencies

S. V. Gonchenko, L. P. Shil'nikov, D. V. Turaev

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

38 Scopus citations

Abstract

Recent results describing nontrivial dynamical phenomena in systems with homoclinic tangencies are represented. Such systems cover a large variety of dynamical models known from natural applications and it is established that so-called quasiattractors of these systems may exhibit rather nontrivial features which are in a sharp distinction, with that one could expect in analogy with hyperbolic or Lorenz-like attractors. For instance, the impossibility of giving a finite-parameter complete description of dynamics and bifurcations of the quasiattractors is shown. Besides, it is shown that the quasiattractors may simultaneously contain saddle periodic orbits with different numbers of positive Lyapunov exponents. If the dimension of a phase space is not too low (greater than four for flows and greater than three for maps), it is shown that such a quasiattractor may contain infinitely many coexisting strange attractors.

Original language	English
Pages (from-to)	195-227
Number of pages	33
Journal	Computers and Mathematics with Applications
Volume	34
Issue number	2-4
DOIs	https://doi.org/10.1016/s0898-1221(97)00124-7
State	Published - 1 Jan 1997
Externally published	Yes

Keywords

Attractor
Dynamical models
Homoclinic bifurcations

ASJC Scopus subject areas

Modeling and Simulation
Computational Theory and Mathematics
Computational Mathematics

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10.1016/s0898-1221(97)00124-7

Cite this

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title = "Quasiattractors and homoclinic tangencies",

abstract = "Recent results describing nontrivial dynamical phenomena in systems with homoclinic tangencies are represented. Such systems cover a large variety of dynamical models known from natural applications and it is established that so-called quasiattractors of these systems may exhibit rather nontrivial features which are in a sharp distinction, with that one could expect in analogy with hyperbolic or Lorenz-like attractors. For instance, the impossibility of giving a finite-parameter complete description of dynamics and bifurcations of the quasiattractors is shown. Besides, it is shown that the quasiattractors may simultaneously contain saddle periodic orbits with different numbers of positive Lyapunov exponents. If the dimension of a phase space is not too low (greater than four for flows and greater than three for maps), it is shown that such a quasiattractor may contain infinitely many coexisting strange attractors.",

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AU - Gonchenko, S. V.

AU - Shil'nikov, L. P.

AU - Turaev, D. V.

N1 - Funding Information: This work was supported by EC-Russia Collaborative Project No. 93-0570 ext. and by the Russian Foundation of Fundamental

PY - 1997/1/1

Y1 - 1997/1/1

N2 - Recent results describing nontrivial dynamical phenomena in systems with homoclinic tangencies are represented. Such systems cover a large variety of dynamical models known from natural applications and it is established that so-called quasiattractors of these systems may exhibit rather nontrivial features which are in a sharp distinction, with that one could expect in analogy with hyperbolic or Lorenz-like attractors. For instance, the impossibility of giving a finite-parameter complete description of dynamics and bifurcations of the quasiattractors is shown. Besides, it is shown that the quasiattractors may simultaneously contain saddle periodic orbits with different numbers of positive Lyapunov exponents. If the dimension of a phase space is not too low (greater than four for flows and greater than three for maps), it is shown that such a quasiattractor may contain infinitely many coexisting strange attractors.

AB - Recent results describing nontrivial dynamical phenomena in systems with homoclinic tangencies are represented. Such systems cover a large variety of dynamical models known from natural applications and it is established that so-called quasiattractors of these systems may exhibit rather nontrivial features which are in a sharp distinction, with that one could expect in analogy with hyperbolic or Lorenz-like attractors. For instance, the impossibility of giving a finite-parameter complete description of dynamics and bifurcations of the quasiattractors is shown. Besides, it is shown that the quasiattractors may simultaneously contain saddle periodic orbits with different numbers of positive Lyapunov exponents. If the dimension of a phase space is not too low (greater than four for flows and greater than three for maps), it is shown that such a quasiattractor may contain infinitely many coexisting strange attractors.

KW - Attractor

KW - Dynamical models

KW - Homoclinic bifurcations

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JO - Computers and Mathematics with Applications

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ER -

Quasiattractors and homoclinic tangencies

Abstract

Keywords

ASJC Scopus subject areas

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