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Robust upper estimates for topological entropy via nonlinear constrained optimization over adapted metrics

Mikhail Anikushin, Andrey Romanov

Abstract

We present an analytical-numerical method providing robust upper estimates for the topological entropy or, more generally, uniform volume growth exponents of differentiable mappings. By introducing varying metrics, we simplify the analysis at the cost of generally rougher bounds, but keeping the prospect of choosing more relatable metrics to refine the estimates. With any covering of an invariant set by a finite number of cubes, we associate a graph describing overlaps (edges) of the cubes (vertices) under iterates of the mapping. Weighing vertices according to a given metric, we reduce the problem to finding simple cycles with maximal relative weights. Then we develop an algorithm concerned with iterative resolving nonlinear programming problems for optimization of maximal relative weights in general smooth families of metrics which may involve interpolation or neural networks models. We describe applications of the algorithm to compute the largest uniform Lyapunov exponent and uniform Lyapunov dimension for the Hénon and Rabinovich systems justifying the Eden conjecture at stationary and periodic points respectively.

Robust upper estimates for topological entropy via nonlinear constrained optimization over adapted metrics

Abstract

We present an analytical-numerical method providing robust upper estimates for the topological entropy or, more generally, uniform volume growth exponents of differentiable mappings. By introducing varying metrics, we simplify the analysis at the cost of generally rougher bounds, but keeping the prospect of choosing more relatable metrics to refine the estimates. With any covering of an invariant set by a finite number of cubes, we associate a graph describing overlaps (edges) of the cubes (vertices) under iterates of the mapping. Weighing vertices according to a given metric, we reduce the problem to finding simple cycles with maximal relative weights. Then we develop an algorithm concerned with iterative resolving nonlinear programming problems for optimization of maximal relative weights in general smooth families of metrics which may involve interpolation or neural networks models. We describe applications of the algorithm to compute the largest uniform Lyapunov exponent and uniform Lyapunov dimension for the Hénon and Rabinovich systems justifying the Eden conjecture at stationary and periodic points respectively.

Paper Structure

This paper contains 12 sections, 5 theorems, 106 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminary definitions and historical background
  3. Topological entropy and volume expansion
  4. Uniform Lyapunov exponents and their computation
  5. Tracking volume expansion via adapted metrics
  6. Relative weights optimization for estimation of uniform growth exponents of subadditive families
  7. Optimization in smooth families of metrics via Iterative Nonlinear Programming
  8. Computation tips and regularizing optimization strategies
  9. Hénon mapping
  10. Rabinovich system
  11. Lyapunov-Floquet metrics for periodic cocycles
  12. Operators on exterior powers and singular values

Key Result

Proposition 4.1

For any $q \in \mathcal{K}$ choose a sequenceClearly, at least one such a sequence always exists due to the invariance of $\mathcal{K}$.$i_{0}=i_{0}(q), i_{1}=i_{1}(q),i_{2}=i_{2}(q),\ldots$ of vertices in $G$ such that Then

Figures (4)

  • Figure 1: Examples of refined coverings for $\varepsilon = 0.1$ (left) and $\varepsilon = 0.05$ (right).
  • Figure 2: Graphs of the maximal cycle weight (extracted from the maximal path of length $1000$; blue) and the optimized maximal weight over reference cycles w.r.t. reference points (green) versus iteration of the Iterative Nonlinear Programming optimization described below \ref{['EQ: HenonMetricGeneralForm']}.
  • Figure 3: Graphs of the data growing (number of reference cycles and reference points) versus iteration of the Iterative Nonlinear Programming optimization described below \ref{['EQ: HenonMetricGeneralForm']}.
  • Figure 4: Numerical simulation of the attractor of \ref{['EQ: LorenzLikeSytem']} with parameters \ref{['EQ: RabinovichSystemFedorovParameters']} under the change of variables \ref{['EQ: RabLorenzLikeChange']}. It is localized by a trajectory (red) coming from outside (transient time is removed) and trajectories (black and orange) starting respectively in small neighborhoods of $q^{0}$ (black bold dot), $q^{+}$ (orange bold dot) and $q^{-}$ (green bold dot). In agreement with rigorous results, all trajectories are contained in the ellipsoid $\mathcal{E}_{\delta}$ from \ref{['EQ: RabinovichEllipsoidsLocalization']} with $\delta = 125$ (black transparent mesh).

Theorems & Definitions (27)

  • Remark 1.1
  • Remark 2.1
  • Remark 3.1
  • Remark 4.1
  • Remark 4.2
  • Remark 4.3
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • proof
  • ...and 17 more