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

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.