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Stability and Lyapunov stability of dynamical systems: a differential approach and a numerical method. (English) Zbl 0656.34045
Author’s abstract: A set of differential equations for the eigenvalues and eigenvectors of the stability matrix of a dynamical system, as well as for the Lyapunov exponents and the corresponding eigenvectors, is derived. The rate of convergence of the Lyapunov eigenvectors is shown to be exponential. The eigenvectors of the stability matrix can be grouped into sets, each spanning a subspace which converges at an exponential rate It is demonstrated that, generically, the real parts of the eigenvalues of the stability matrix equal the corresponding Lyapunov exponents. This statement has been tested numerically. The values of the Lyapunov exponents, \(\mu_ i\), are shown to be related to the corresponding finite time values of the Lyapunov exponents (e.g. those deduced from a finite time numerical simulation), \(\mu_ i(t)\), by: \(\mu_ i(t)=\mu_ i+(b_ i+\xi_ i(t))/t.\) The \(b_ i's\) are constants and \(\xi_ i(t)\) are “noise” terms of zero mean. This observation leads to a method of extrapolation, which has been used to predict Lyapunov exponents from a finite amount of data. It is shown that the use of the standard (numerical) methods to compute Lyapunov exponents introduces an error \(a_ i/t\) in the value of \(\mu_ i(t)\), where the \(a_ i's\) are constants. Thus the standard method has a rate of convergence which is the same as that of the exact \(\mu_ i(t)'s\). Finally, we have shown how one can compute the eigenvectors associated with each of the eigenvalues of the stability matrix as well as the Lyapunov eigenvectors.
Reviewer: J.H.Tian

34D20 Stability of solutions to ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
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