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Similarity signature curves for forming periodic orbits in the Lorenz system

Jindi Li, Yun Yang

TL;DR

By combining the sliding window method, the quasi-periodic orbits can be detected numerically and all periodic orbits with period $p \leqslant 8$ in the Lorenz system are found, and their period lengths and symbol sequences are calculated.

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.

Similarity signature curves for forming periodic orbits in the Lorenz system

TL;DR

By combining the sliding window method, the quasi-periodic orbits can be detected numerically and all periodic orbits with period in the Lorenz system are found, and their period lengths and symbol sequences are calculated.

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 in the Lorenz system are found, and their period lengths and symbol sequences are calculated.
Paper Structure (12 sections, 1 theorem, 31 equations, 11 figures, 2 tables, 1 algorithm)

This paper contains 12 sections, 1 theorem, 31 equations, 11 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Lorenz system and its similarity signature curve
  3. Lorenz system dynamics
  4. Similarity signature curve
  5. Analysis of the similarity signature curve for the Lorenz attractor
  6. Method of finding periodic orbits
  7. Interval methods
  8. Periodic orbits by the symbolic dynamics based approach
  9. Sliding window method based on similarity signature curve
  10. Method summary
  11. Periodic orbits for the Lorenz system
  12. Conclusion

Key Result

Theorem 3.2

For two different periodic orbits $P_1$ and $P_2$, their periodic symbol sequences are ${S_1} = \left( {{s_{11}},{s_{12}}, \ldots ,{s_{1{n_1}}}} \right)$, ${S_2} = \left( {{s_{21}},{s_{22}}, \ldots ,{s_{2{n_2}}}} \right)$, $1 < {n_2} < {n_1}$, and $S_2$ is a ordered subset of $S_1$. The similarity s

Figures (11)

  • Figure 1: A trajectory of the Lorenz system.
  • Figure 2: Consecutive points for signature approximations.
  • Figure 3: Projection of signature curve of Lorenz attractor on coordinate plane.
  • Figure 4: Projection of similarity signature curve of Lorenz attractor on coordinate plane.
  • Figure 5: The Lorenz attractor with self similarity and its similarity signature curve
  • ...and 6 more figures

Theorems & Definitions (5)

  • Remark 2.1
  • Definition 2.2
  • Definition 3.1: bol
  • Theorem 3.2
  • proof