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Lorenz attractors through Šil'nikov-type bifurcation. Part I†

Published online by Cambridge University Press:  19 September 2008

Marek Ryszard Rychlik
Marek Ryszard Rychlik
Affiliation:
The Institute for Advanced Study, School of Mathematics, Princeton, NJ 08540, USA
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Abstract

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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 ℝ3 is assumed to have a hyperbolic fixed point, whose eigenvalues satisfy the inequalities λ1 > 0, λ2 > 0, λ3 > 0 and |λ2|>|λ1|>|λ3|. 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.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 10 , Issue 4 , December 1990 , pp. 793 - 821
Copyright
Copyright © Cambridge University Press 1990

References

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