Appearance of discrete Lorenz attractors in the transitions from saddle to saddle-focus

TL;DR

This paper investigates the emergence of discrete Lorenz attractors during the transition from saddle to saddle-focus in dynamical systems, identifying parameter domains where such attractors occur, including Belyakov and 3DL bifurcations.

Contribution

It provides the first analysis of the transition from saddle to saddle-focus, proving the existence of Lorenz-like attractors in this scenario, completing the classification of simple bifurcations.

Findings

Existence of parameter domains with Lorenz attractors during saddle to saddle-focus transition

Includes analysis of Belyakov bifurcation and 3DL bifurcation cases

Completes the classification of bifurcations involving homoclinic and heteroclinic tangencies

Abstract

Triply degenerate fixed points appear in global bifurcations -- homoclinic and heteroclinic tangencies. In order to get Lorenz-like attractors, the dynamics of the first return map along the homoclinic or heteroclinic cycle should be effectively at least three-dimensional, i.e. there should not exist lower-dimensional invariant manifolds. This can be achieved by adding some special conditions, global or local. Global degeneracies are related to the existence of non-simple homoclinic tangencies or non-simple heteroclinic orbit in a cycle, these cases were studied before. Local conditions either require the cycle to contain at least one saddle-focus, or add certain relations on the multipliers of the fixed point such that the leading stable direction of the saddle either disappears or alternates. All these cases were already studied before except one, related to the transition from saddle to saddle-focus. In the present paper this case is investigated, and the existence of a cascade of parameter domains containing systems with discrete Lorenz attractors is proved. In particular, it includes the Belyakov bifurcation, when the saddle becomes a saddle-focus through a collision of the eigenvalues, and the 3DL bifurcation, when the dimension of the leading stable subset alternates between 1 and 2. This paper completes the list of the simplest bifurcations of homoclinic and heteroclinic tangencies by studying the last possible case.