What drives transient behavior in complex systems?

Abstract

We study transient behavior in the dynamics of complex systems described by a set of nonlinear ordinary differential equations. Destabilizing nature of transient trajectories is discussed and its connection with the eigenvalue-based linearization procedure. The complexity is realized as a random matrix drawn from a modified May-Wigner model. Based on the initial response of the system, we identify a novel stable-transient regime. We calculate exact abundances of typical and extreme transient trajectories finding both Gaussian and Tracy-Widom distributions known in extreme value statistics. We identify degrees of freedom driving transient behavior as connected to the eigenvectors and encoded in a nonorthogonality matrix T_{0}. We accordingly extend the May-Wigner model to contain a phase with typical transient trajectories present. An exact norm of the trajectory is obtained in the vanishing T_{0} limit where it describes a normal matrix.