Ergod Th & Dynam Sys (1989), 10, 793-821 Printed in Great Britain Lorenz attractors through Sil'nikov-type bifurcation. Part It MAREK RYSZARD RYCHLIK The Institute for Advanced Study, School of Mathematics, Princeton, NJ 08540, USA (Received 23 March 1988 and revised 10 January 1989) Abstract The main result of this paper is a construction of geometric Lorenz attractors (as axiomatically denned by J Guckenheimer) by means of an fl-explosion The unperturbed vector field on U 3 is assumed to have a hyperbolic fixed point, whose eigenvalues satisfy the inequalities A,>0, A 2 <0, A 3 <0 and |A 2 |> |A,|> |A 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 contain- ing 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 0 Introduction In this paper we attempt to establish rigorous methods for verifying the existence of geometric Lorenz attractors in concrete examples of differential equations By a geometric Lorenz attractor we mean an object satisfying the axioms of J Gucken- heimer [4] Our method is based on a particular type of fl-explosion We start with a vector field in R 3 with a hyperbolic fixed point The eigenvalues of the linearization at this fixed point will be denoted by A,, A 2 and A 3 We assume that A 2 and A 3 are negative and A, > 0 and that they satisfy the following inequality |A 2 |>|A,|>|A 3 | (0 1) Therefore the unstable manifold of our point has dimension 1 and the stable manifold has dimension 2 The unstable manifold is assumed to be a part of the stable manifold, and as a result it forms a double loop (figure 1) (The reader should notice now that our vector fields are assumed to be symmetric with respect to the reflection through the eigendirection corresponding to the eigenvalue A 3 ) In the unperturbed system the fl-set contains the double loop of the unstable manifold It is a consequence of this assumption concerning the eigenvalues at the equilibrium t Supported in part by NSF grants DMS-8601978 and DMS-8701789