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Incomplete crossing and semi-topological horseshoes

Junfeng Cheng, Xiao-Song Yang

TL;DR

The work extends topological horseshoe theory by incorporating incomplete crossing and semi-topological horseshoes through a symbolic dynamics lens, enabling both chaos detection and quantitative entropy bounds via crossing matrices and subshifts of finite type. It constructs invariant sets semi-conjugate to subshifts when an f_A-connected framework exists and provides conditions under which chaos and entropy lower bounds follow, notably h(f) ≥ ln ρ(A). The authors validate the framework with explicit applications to a perturbed Duffing system and a Chen polynomial system, obtaining tangible entropy bounds even when complete Smale horseshoes fail. This approach offers a practical, elementary route to detect chaos and estimate entropy in complex dynamical systems, with potential broad applicability across nonlinear dynamics and applied mathematics.

Abstract

This paper enriches the topological horseshoe theory using finite subshift theory in symbolic dynamical systems, and develops an elementary framework addressing incomplete crossing and semi-horseshoes. Two illustrative examples are provided: one from the perturbed Duffing system and another from a polynomial system proposed by Chen, demonstrating the prevalence of semi-horseshoes in chaotic systems. Moreover, the semi-topological horseshoe theory enhances the detection of chaos and improves the accuracy of topological entropy estimation.

Incomplete crossing and semi-topological horseshoes

TL;DR

The work extends topological horseshoe theory by incorporating incomplete crossing and semi-topological horseshoes through a symbolic dynamics lens, enabling both chaos detection and quantitative entropy bounds via crossing matrices and subshifts of finite type. It constructs invariant sets semi-conjugate to subshifts when an f_A-connected framework exists and provides conditions under which chaos and entropy lower bounds follow, notably h(f) ≥ ln ρ(A). The authors validate the framework with explicit applications to a perturbed Duffing system and a Chen polynomial system, obtaining tangible entropy bounds even when complete Smale horseshoes fail. This approach offers a practical, elementary route to detect chaos and estimate entropy in complex dynamical systems, with potential broad applicability across nonlinear dynamics and applied mathematics.

Abstract

This paper enriches the topological horseshoe theory using finite subshift theory in symbolic dynamical systems, and develops an elementary framework addressing incomplete crossing and semi-horseshoes. Two illustrative examples are provided: one from the perturbed Duffing system and another from a polynomial system proposed by Chen, demonstrating the prevalence of semi-horseshoes in chaotic systems. Moreover, the semi-topological horseshoe theory enhances the detection of chaos and improves the accuracy of topological entropy estimation.
Paper Structure (9 sections, 8 theorems, 79 equations, 15 figures, 1 table)

This paper contains 9 sections, 8 theorems, 79 equations, 15 figures, 1 table.

Table of Contents

  1. Introduction
  2. Topological Horseshoe
  3. Subshift of finite type
  4. A generalization of topological horseshoe lemma
  5. Application of semi-topological horseshoe theory in dynamical systems
  6. Horseshoes involving two blocks
  7. Horseshoe and semi-horseshoe in the perturbed Duffing system
  8. Semi-topological horseshoe in a polynomial system
  9. Summary

Key Result

Theorem 1

Yang2004 Suppose that the map $f:D\to \mathbb{R}^n$ satisfies the following assumptions: (1) There exist $m$ mutually path-connected disjoint compact subsets $B_1,B_2,\dots$ and $B_m$ of $D$, the restriction of $f$ to each $B_i$, i.e., $f|_{B_i}$ is continuous. (2) The dimension one crossing relatio

Figures (15)

  • Figure 1: The crossing matrix corresponding to this case is $\left( 1111 \right)$.
  • Figure 2: The crossing matrix $A_1$ corresponding to this case is $\left( 1001 \right)$.
  • Figure 3: The crossing matrix $A_2$ corresponding to this case is $\left( 1011 \right)$.
  • Figure 4: The crossing matrix $A_3$ corresponding to this case is $\left( 0110 \right)$.
  • Figure 5: The crossing matrix corresponding to this case is $\left( 1110 \right)$.
  • ...and 10 more figures

Theorems & Definitions (23)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Theorem 1
  • Remark 1
  • Definition 5
  • Definition 6
  • Definition 7
  • Theorem 2
  • ...and 13 more