Global bifurcation structure of a four-dimensional Lorenz-like system with a wild chaotic attractor
Juan Patiño-Echeverría, Bernd Krauskopf, and Hinke M. Osinga
Abstract
Wild chaos is a higher-dimensional form of chaotic dynamics that can only arise in vector fields of dimension at least four. It is characterized by the persistent presence of tangencies between stable and unstable manifolds of an invariant set. We study a four-dimensional extension of the classic Lorenz system, which was recently shown to exhibit a so-called wild pseudohyperbolic attractor for a specific choice of parameters. Pseudohyperbolicity guarantees that every trajectory in the attractor has a positive maximal Lyapunov exponent, and this property persists under small perturbations of the system.
We investigate how wild chaotic attractors arise geometrically by analysing the overall bifurcation structure in the (ρ, μ)-plane. Here, ρ is the standard (Rayleigh) parameter in the classic Lorenz system, and μ is a new parameter that introduces spiralling dynamics near the origin. We begin by identifying the bifurcation structure inherited from the Lorenz system by continuing its homoclinic bifurcations as curves. Kneading diagrams, in combination with Lin's method, allow us to uncover additional curves of global bifurcation intrinsic to the four-dimensional system. We also compute the Lyapunov spectrum of the unstable manifold of the origin to identify different types of attractors. This approach provides insight into the parameter regions where wild chaos may occur.
PDF copy of the paper (11.5MB)