Twisted homoclinic orbits in Lorenz and Chen systems: rigorous proofs from universal normal form

TL;DR

This paper rigorously analyzes the universal normal form of Lorenz and Chen systems, proving the existence of infinite homoclinic orbits with different topologies, and explaining the twisted nature of Chen attractors.

Contribution

It provides a rigorous proof of the existence of diverse homoclinic orbits in the universal normal form, clarifying the topological differences between Lorenz and Chen attractors.

Findings

Existence of infinite homoclinic orbits with varying topological structures.

Inheritance of rotational topological features by Chen attractors.

Explanation of the twisted nature of Chen-type attractors.

Abstract

The properties common to the Lorenz and Chen attractors, as well as their fundamental differences, have been studied for many years in a vast number of works and remain a topic far from a rigorous and complete description. In this paper we take a step towards solving this problem by carrying out a rigorous study of the so-called universal normal form to which we have reduced the systems of both of these families. For this normal form, we prove the existence of infinite set of homoclinic orbits with different topological structure defined by the number of rotations around axis of symmetry. We show that these rotational topological features are inherited by the attractors of Chen-type systems and give rise to their twisted nature - the generic difference from attractors of Lorenz type.