Similarity signature curves for forming periodic orbits in the Lorenz system

TL;DR

This paper introduces a new method using similarity signature curves derived from equivariant moving frame theory to systematically identify short periodic orbits in the Lorenz system, improving detection and analysis.

Contribution

The paper proposes a novel approach combining similarity signature curves with the sliding window method to detect and analyze short periodic orbits in the Lorenz system.

Findings

All periodic orbits with period p ≤ 8 are identified.

Similarity signature curves exhibit more regular behavior than original data.

The method effectively detects quasi-periodic orbits.

Abstract

In this paper, we systematically investigate the short periodic orbits of the Lorenz system by the aid of the similarity signature curve, and a novel method to find the short-period orbits of the Lorenz system is proposed. The similarity invariants are derived by the equivariant moving frame theory and then the similarity signature curve occurs along with them. The similarity signature curve of the Lorenz system presents a more regular behavior than the original one. By combining the sliding window method, the quasi-periodic orbits can be detected numerically, all periodic orbits with period $p \leqslant 8$ in the Lorenz system are found, and their period lengths and symbol sequences are calculated.