Lorenz attractors through Šill'nikov-type bifurcation. Part I

Research output: Contribution to journalArticle

70 Citations (Scopus)

Abstract

The main result of this paper is a construction of geometric Lorenz attractors (as axiomatically defined by J. Guckenheimer) by means of an Ω-explosion. The unperturbed vector field on 3 is assumed to have a hyperbolic fixed point, whose eigenvalues satisfy the inequalities 1 > 0, 2 > 0, 3 > 0 and |2|>|1|>|3|. Moreover, the unstable manifold of the fixed point is supposed to form a double loop. Under some other natural assumptions a generic two-parameter family containing the unperturbed vector field contains geometric Lorenz attractors. A possible application of this result is a method of proving the existence of geometric Lorenz attractors in concrete families of differential equations. A detailed discussion of the method is in preparation and will be published as Part II.

Original languageEnglish (US)
Pages (from-to)793-821
Number of pages29
JournalErgodic Theory and Dynamical Systems
Volume10
Issue number4
DOIs
StatePublished - 1990
Externally publishedYes

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Lorenz attractor
Bifurcation
Vector Field
Fixed point
Explosions
Unstable Manifold
Differential equations
Concretes
Explosion
Two Parameters
Preparation
Differential equation
Eigenvalue
Family

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Lorenz attractors through Šill'nikov-type bifurcation. Part I. / Rychlik, Marek R.

In: Ergodic Theory and Dynamical Systems, Vol. 10, No. 4, 1990, p. 793-821.

Research output: Contribution to journalArticle

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