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Lorenz attractors through Šil’nikov-type bifurcation. I. (English) Zbl 0715.58027

Summary: The main result of this paper is a construction of geometric Lorenz attractors (as axiomatically defined by J. Guckenheimer) by means of an \(\Omega\)-explosion. The unperturbed vector field on \({\mathbb{R}}^ 3\) is assumed to have a hyperbolic fixed point, whose eigenvalues satisfy the inequalities \(\lambda_ 1>0\), \(\lambda_ 2<0\), \(\lambda_ 3<0\) and \(| \lambda_ 2| >| \lambda_ 1| >| \lambda_ 3|.\) 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.

MSC:

37G99 Local and nonlocal bifurcation theory for dynamical systems
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
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References:

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