## 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

### Keywords:

geometric Lorenz attractors
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### References:

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