Cascades of Lorenz attractors in the Shimizu-Morioka model

Abstract

The Lorenz attractor is the first example of a robustly chaotic non-hyperbolic attractor. Each orbit of such an attractor has a positive top Lyapunov exponent, and this property persists under small perturbations despite possible bifurcations of the attractor. In this paper, we study the boundary of the Lorenz attractor existence region in the Shimizu-Morioka model. As in the classical Lorenz system, a part of the boundary is associated with the curve $l_{A=0}$, where the first tangency between some Lyapunov subspaces occurs along orbits of the attractor. However, in the Lorenz system, the curve $l_{A=0}$ forms the exact boundary of the Lorenz attractor existence region. Beyond this curve, the attractor is not robustly chaotic, although it may be indistinguishable from the Lorenz attractor in simple numerical experiments. In the Shimizu-Morioka model, the curve $l_{A=0}$ is divided into two parts. The Lorenz attractor existence region adjoins $l_{A=0}$ along the first part of this curve, as in the Lorenz system. Near the second part, as we show, the region of the existence of the Lorenz attractor is fractal. We describe two infinite cascades of disjoint subregions with the Lorenz attractor. One cascade occurs along the curve $l_{A=0}$, another -- in the transversal direction. We show that along the cascades, the Lorenz attractor undergoes ``doubling bifurcations'', leading to a complication of its topological structure.