Lyapunov Exponent Sign Reversal: Stability and Instability by the First Approximation
This chapter is a concise and updated version of authors’ survey Time-Varying Linearization and the Perron effects [52], devoted to the rigorous mathematical justification of the use of Lyapunov exponents to investigate the stability, instability, and cha
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1 Introduction This chapter is a concise and updated version of authors’ survey Time-Varying Linearization and the Perron effects [52], devoted to the rigorous mathematical justification of the use of Lyapunov exponents to investigate the stability, instability, and chaos. In his thesis A.M. Lyapunov [57] proved that if the first approximation system is regular and its largest Lyapunov exponent is negative, then the solution of the original system is asymptotically stable. Then it was stated by O. Perron [62] that the requirement of regularity is substantial: he constructed an example of second-order system such that a solution of the first approximation system has negative largest Lyapunov exponent while the solution of the original system with the same initial data has positive largest Lyapunov exponent. The effect of Lyapunov exponent sign reversal of solutions of the first approximation system and of the original system under the same initial data, we shall call the Perron effect. Later, [14, 58, 60, 63] there were obtained sufficient conditions of stability by the first approximation for nonregular linearizations generalizing the Lyapunov theorem. At the same time, according to [58]: “: : : The counterexample of Perron shows that the negativeness of Lyapunov exponents is not a sufficient condition of stability by the first approximation. In the general case necessary and sufficient conditions of stability by the first approximation are not obtained.” Recently, it was also shown [47, 52] that, in general, the positiveness of the largest Lyapunov exponent is not a sufficient condition of instability by the first approximation and chaos. In the 1940s N.G. Chetaev [15] published the criterion of instability by the first approximation for regular linearizations. However, in the proof of these criteria a
G.A. Leonov • N.V. Kuznetsov () Saint Petersburg State University, Russia; University of Jyväskylä, Finland e-mail: [email protected]; [email protected] V. Afraimovich et al. (eds.), Nonlinear Dynamics and Complexity, Nonlinear Systems and Complexity 8, DOI 10.1007/978-3-319-02353-3__2, © Springer International Publishing Switzerland 2014
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G.A. Leonov and N.V. Kuznetsov
flaw was discovered [48, 52] and, at present, a complete proof of Chetaev theorems is given for a more weak condition in comparison with that for instability in the sense of Lyapunov, namely, for instability in the sense of Krasovsky. The discovery of strange attractors and chaos in the investigation of complex nonlinear dynamical systems led to the use and study of instability by the first approximation. At present, many specialists in chaotic dynamics use various numerical methods for computation of Lyapunov exponent (see, e.g., [3, 8, 10, 12, 13, 16, 17, 24, 26–28, 30, 55, 56, 64, 65, 68, 70–72, 75–77, 79], and others) and believe that the positiveness of the largest Lyapunov exponent of linear first approximation system implies the instability of solutions of the original system. As a rule, the authors ignore the justification
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