Globally optimal stretching foliations of dynamical systems reveal the organizing skeleton of intensive instabilities

TL;DR

This paper develops a method to identify globally optimal stretching foliations in dynamical systems, revealing the underlying structure of instabilities and providing new insights into chaos and flow analysis.

Contribution

It extends the concept of local stretching to a global optimization framework for foliations, applicable to finite-time and finite-sequence dynamical systems.

Findings

Identified foliations that maximize and minimize global stretching.

Classified nondegenerate singularities of the optimal foliations.

Validated the approach on Hénon map, double-gyre flow, and standard map.

Abstract

Understanding instabilities in dynamical systems drives to the heart of modern chaos theory, whether forecasting or attempting to control future outcomes. Instabilities in the sense of locally maximal stretching in maps is well understood, and is connected to the concepts of Lyapunov exponents/vectors, Oseledec spaces and the Cauchy--Green tensor. In this paper, we extend the concept to global optimization of stretching, as this forms a skeleton organizing the general instabilities. The `map' is general but incorporates the inevitability of finite-time as in any realistic application: it can be defined via a finite sequence of discrete maps, or a finite-time flow associated with a continuous dynamical system. Limiting attention to two-dimensions, we formulate the global optimization problem as one over a restricted class of foliations, and establish the foliations which both maximize and minimize global stretching. A classification of nondegenerate singularities of the foliations is obtained. Numerical issues in computing optimal foliations are examined, in particular insights into special curves along which foliations appear to veer and/or do not cross, and foliation behavior near singularities. Illustrations and validations of the results to the H\'{e}non map, the double-gyre flow and the standard map are provided.