Title: What drives transient behaviour in complex systems?,
Abstract: We study transient behaviour in the dynamics of complex systems described by
a set of non-linear ODE's. 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 behaviour as connected to the eigenvectors and encoded in a
non-orthogonality 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.