Jacobian deformation ellipsoid and Lyapunov stability analysis revisited

TL;DR

This paper revisits Lyapunov stability analysis, addressing its limitations in local instability detection and short-term analysis by introducing the Jacobian deformation ellipsoid and comparing it with existing methods.

Contribution

It introduces the Jacobian deformation ellipsoid and its orthogonal submatrix to improve local stability analysis without long-term integration, enhancing short-term and origin-specific instability detection.

Findings

The Jacobian deformation ellipsoid provides accurate local exponents.

Compared methods show improved detection of instability origins.

Short-term exponents closely match long-term results.

Abstract

The stability analysis introduced by Lyapunov and extended by Oseledec is an excellent tool to describe the character of nonlinear n-dimensional flows by n global exponents if these flows are stable in time. However, there are two main shortcomings: (a) The local exponents fail to indicate the origin of instability where trajectories start to diverge. Instead, their time evolution contains a much stronger chaos than the trajectories, which is only eliminated by integrating over a long time. Therefore, shorter time intervals cannot be characterized correctly, which would be essential to analyse changes of chaotic character as in transients. (b) Moreover, although Oseledec uses an n dimensional sphere around a point x to be transformed into an n dimensional ellipse in first order, this local ellipse has yet not been evaluated. The aim of this contribution is to eliminate these two shortcomings. Problem (a) disappears if the Oseledec method is replaced by a frame with a 'constraint' as performed by Rateitschak and Klages (RK) [Phys. Rev. E 65 036209 (2002)]. The reasons why this method is better will be illustrated by comparing different systems. In order to analyze shorter time intervals, integrals between consecutive Poincare points will be evaluated. The local problems (b) will be solved analytically by introducing the symmetric 'Jacobian deformation ellipsoid' and its orthogonal submatrix, which enable to search in the full phase space for extreme local separation exponents. These are close to the RK exponents but need no time integration of the RK frame. Finally, four sets of local exponents are compared: Oseledec frame, RK frame, Jacobian deformation ellipsoid and its orthogonal submatrix.