Linearization Based on Eigenvalue Estimates

Abstract

For the linearization of a system $\dot x = f\left( x \right)$ of nonlinear ordinary differential equations near a stationary state x_o there exists a method from invariant manifold theory having the property that in any case the linearized system has the qualitatively same phase portrait near x_o as the given nonlinear system. This method is based on the assumption that for each eigenvalue of the Jacobian of f at x_o one is able to decide whether the real part is exactly zero or not. For critical systems with eigenvalues very close to the imaginary axis this generally cannot be done in practice since only estimates for the eigenvalues are available. In this paper we present a modification of the above-mentioned method to the case where the location of eigenvalues is known only approximately.