Time-Varying Linearization and the Perron Effects

In 1892, the general problem of stability by the first approximation was considered by Lyapunov. He proved that if the system of the first approximation is regular and all its Lyapunov exponents (or characteristic exponents) are negative, then the solution of the original system is asymptotically Lyapunov stable. In 1930, it was stated by O. Perron that the requirement of regularity of the first approximation is substantial. He constructed an example of the second-order system of the first approximation, which has negative characteristic exponents along a zero solution of the original system but, at the same time, this zero solution of original system is Lyapunov unstable. Furthermore, in a certain neighborhood of this zero solution almost all solutions of original system have positive characteristic exponents. The effect of a sign reversal of characteristic exponents of solutions of the original system and the system of first approximation with the same initial data was subsequently called the Perron effect. So positive largest Lyapunov exponent doesn’t, in general, indicate chaos and negative largest Lyapunov exponent doesn’t, in general, indicate stability! The counterexample of Perron impressed on the contemporaries and gave an idea of the difficulties arising in the justification of the first approximation theory for nonautonomous and nonperiodic linearizations. Later, Persidskii [1947], Massera [1957], Malkin [1966], and Chetaev [1955], obtained sufficient conditions of stability by the first approximation for nonregular linearizations generalizing the Lyapunov theorem. At the same time, according to Malkin (Theory of Motion Stability, 1966): ”... The counterexample of Perron shows that the negativeness of Lyapunov exponents (or characteristic 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.” For certain situations, the results on stability by the first approximation can be found in the books [Bellman & Cooke, 1963, Davies & James, 1966, Willems, 1970, Coppel, 1965, Wasow, 1965, Bellman, 1953, Bacciotti & Rosier, 2005, Yoshizawa, 1966, Lakshmikantham et al., 1989, Hartman, 1984, Halanay, 1966, Sansone & Conti 1964]. In the 1940s Chetaev [1948] published the criterion of instability by the first approximation for regular linearizations. However, in the proof of these criteria a flaw was discovered and, at present, the complete proof of Chetaev theorems is given for a more weak condition in comparison with that for instability by Lyapunov, namely, for instability by Krasovsky only. The discovery of strange attractors was made obvious with the study of instability by the first approximation. Nowadays the problem of the justification of the nonstationary