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Global Convergence of the Return Dynamics in the Class $\mathcal{O}_C

Mohammed Barkatou, Mohamed El Morsalani

Abstract

In previous work we introduced a return map generated by a geometric round-trip between the boundary of a convex core \(C\) and the boundary of an admissible domain \(Ω\) belonging to the class \(\mathcal{O}_C\) (see Definition 1.2). The associated transformation \[ F:\partial C\to \partial C \] defines a discrete dynamical system whose first-order expansion reveals a variable-step gradient descent structure for the thickness function \[ d:\partial C\to \mathbb{R}_{+}. \] The purpose of the present paper is to analyze the global dynamical properties of this transformation. Under natural regularity and nondegeneracy assumptions on the thickness function, we prove that every trajectory of the return map converges to a critical point of \(d\). The proof combines the first-order expansion of the return map, a Lyapunov dissipation estimate for the energy \[ V(c)=\frac{1}{2}d(c)^2, \] and a compactness argument based on the isolation of the critical points. These results show that the geometry of the domain \(Ω\) induces a gradient-like dynamical system on \(\partial C\) whose long-term behavior is completely organized by the critical points of the thickness landscape. In particular, the phase space decomposes into basins of attraction associated with these critical points, and nontrivial periodic orbits cannot occur.

Global Convergence of the Return Dynamics in the Class $\mathcal{O}_C

Abstract

In previous work we introduced a return map generated by a geometric round-trip between the boundary of a convex core and the boundary of an admissible domain belonging to the class (see Definition 1.2). The associated transformation defines a discrete dynamical system whose first-order expansion reveals a variable-step gradient descent structure for the thickness function The purpose of the present paper is to analyze the global dynamical properties of this transformation. Under natural regularity and nondegeneracy assumptions on the thickness function, we prove that every trajectory of the return map converges to a critical point of . The proof combines the first-order expansion of the return map, a Lyapunov dissipation estimate for the energy and a compactness argument based on the isolation of the critical points. These results show that the geometry of the domain induces a gradient-like dynamical system on whose long-term behavior is completely organized by the critical points of the thickness landscape. In particular, the phase space decomposes into basins of attraction associated with these critical points, and nontrivial periodic orbits cannot occur.

Paper Structure

This paper contains 38 sections, 7 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Geometric Framework
  3. Convex Core and Admissible Domains
  4. Thickness Function
  5. Radial Map
  6. Reciprocal Map
  7. Return Map
  8. First-Order Structure of the Return Map
  9. First-Order Expansion
  10. Gradient Descent Interpretation
  11. Dynamical Consequences
  12. Lyapunov Structure of the Return Dynamics
  13. Energy Function
  14. Energy Variation Along the Dynamics
  15. Energy Dissipation
  16. ...and 23 more sections

Key Result

Proposition 4.1

There exist constants $a,b>0$ depending only on $d_{\min}$, $d_{\max}$ and the Lipschitz bound of $\nabla_{\partial C}d$ such that for all $c\in\partial C$,

Theorems & Definitions (15)

  • Definition 1.1: C-geometric normal property
  • Definition 1.2: Class $\mathcal{O}_C$
  • Remark 1.3
  • Proposition 4.1
  • Remark 5.1
  • Theorem 5.2: Global convergence
  • proof
  • Proposition 6.1
  • Theorem 6.2
  • Proposition 6.3
  • ...and 5 more