Invariance of Lyapunov exponents and Lyapunov dimension for 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.

Keywords: chaos, Lyapunov exponent, Lyapunov characteristic exponent, Lyapunov dimension of attractor, time-varying linearization, regular and non regular linearization, time reparametrization, change of coordinates, Perron effect of Lyapunov exponent sign reversal

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• The Lyapunov dimension and its computation for self-excited and hidden attractors in the Glukhovsky-Dolzhansky fluid convection model
• Estimation of Lyapunov dimension for the Chen and Lu systems
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• Time-delay control for stabilization of the Shapovalov mid-size firm model
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• Study of irregular dynamics in an economic model: attractor localization and Lyapunov exponents