User:Kuznetsov N.V./Perron effects of Lyapunov exponent sign inversion

Perron effects of Lyapunov exponent sign inversion ...

Time-varying linerization: Lyapunov exponent and stability by the fisrt approximation
To introduce Lyapunov exponent let us consider a fundamental matrix $$ X(t)$$ (e.g., for linearization along stationary solution $$x_0$$ in continuous system the funadamental matrix is $$ \exp\left( \left. \frac{ d f^t(x) }{dx} \right|_{x_0} \right) $$), consisting of the linear-independent solutions of the first approximation system. The singular values $$\{\alpha_j\big(X(t)\big)\}_{1}^{n}$$ of the matrix $$X(t)$$ are the square roots of the eigenvalues of the matrix $$X(t)^*X(t)$$. The largest Lyapunov exponent $$\lambda_{max}$$ is as follows

\lambda_{max}= \max\limits_{j}\limsup _{t \rightarrow \infty}\frac{1}{t}\ln\alpha_j\big(X(t)\big). $$ A.M. Lyapunov proved that if the system of the first appriximaion is regular (e.g., all systems with constant and periodic coefficients are regular) and its largest Lyapunov exponent is negative, then the solution of the original system is asymptotically Lyapunov stable. Later, it was stated by O. Perron that the requirement of regularity of the first approximation is substantial.

Perron effects of largest Lyapunov exponent sign inversion
In 1930 O. Perron constructed an example of the second-order system, the first approximation of which has negative Lyapunov exponents along a zero solution of the original system but, at the same time, this zero solution of original nonlinear system is Lyapunov unstable. Furthermore, in a certain neighborhood of this zero solution almost all solutions of original system have positive Lyapunov exponents. Also it is possible to construct reverse example when first approximation has positive Lyapunov exponents along a zero solution of the original system but, at the same time, this zero solution of original nonlinear system is Lyapunov stable. The effect of sign inversion of Lyapunov 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, counterexample of Perron shows that positive largest Lyapunov exponent doesn't, in general, indicate chaos and negative largest Lyapunov exponent doesn't, in general, indicate stability. Therefore time-varying linearization requires additional justification.