An example derived from Lorenz attractor

TL;DR

This paper constructs a novel 4-dimensional dynamical system derived from the Lorenz attractor, exhibiting unique stability and heterodimensional cycles, advancing understanding of complex hyperbolic behaviors.

Contribution

It provides the first example of a singular chain recurrence class that is Lyapunov stable, robustly heterodimensional, and contains a sectionally expanding subbundle within the finest dominated splitting.

Findings

First example of a Lyapunov stable singular chain recurrence class away from homoclinic tangencies.

Existence of robust heterodimensional cycles in the constructed system.

Presence of a robustly sectionally expanding subbundle contained in the finest dominated splitting.

Abstract

We consider a DA-type surgery of the famous Lorenz attractor in dimension 4. This kind of surgeries have been firstly used by Smale [S] and Ma\~n\'e [M1] to give important examples in the study of partially hyperbolic systems. Our construction gives the first example of a singular chain recurrence class which is Lyapunov stable, away from homoclinic tangencies and exhibits robustly heterodimensional cycles. Moreover, the chain recurrence class has the following interesting property: there exists robustly a 2-dimensional sectionally expanding subbundle (containing the flow direction) of the tangent bundle such that it is properly included in a subbundle of the finest dominated splitting for the tangent flow.