Lyapunov Exponent - Lyapunov Exponent For Time-varying Linearization

Lyapunov Exponent For Time-varying Linearization

To introduce Lyapunov exponent let us consider a fundamental matrix (e.g., for linearization along stationary solution in continuous system the fundamental matrix is ), consisting of the linear-independent solutions of the first approximation system. The singular values of the matrix are the square roots of the eigenvalues of the matrix . The largest Lyapunov exponent is as follows

A.M. Lyapunov proved that if the system of the first approximation 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.