Invariance of Lyapunov exponents and Lyapunov dimension for regular and irregular linearizations

TL;DR

This paper clarifies the relationship between different definitions of Lyapunov exponents and proves their invariance under coordinate changes for both regular and irregular linearizations.

Contribution

It demonstrates the invariance of Lyapunov exponents for various linearizations and clarifies the connection between Lyapunov exponents and characteristic exponents.

Findings

Lyapunov exponents are invariant under coordinate transformations.

The relation between Lyapunov exponents and characteristic exponents is clarified.

Invariance holds for both regular and irregular linearizations.

Abstract

Nowadays the Lyapunov exponents and Lyapunov dimension have become so widespread and common that they are often used without references to the rigorous definitions or pioneering works. It may lead to a confusion since there are at least two well-known definitions, which are used in computations: the upper bounds of the exponential growth rate of the norms of linearized system solutions (Lyapunov characteristic exponents, LCEs) and the upper bounds of the exponential growth rate of the singular values of the fundamental matrix of linearized system (Lyapunov exponents, LEs). In this work the relation between Lyapunov exponents and Lyapunov characteristic exponents is discussed. The invariance of Lyapunov exponents for regular and irregular linearizations under the change of coordinates is demonstrated.