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.

9 pages, 5 figures

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• Condition numbers for real eigenvalues in the real Elliptic Gaussian ensemble
• Universal transient behavior in large dynamical systems on networks
• From synaptic interactions to collective dynamics in random neuronal networks models: critical role of eigenvectors and transient behavior
• Full Dysonian dynamics of the complex Ginibre ensemble
• Infinite stable Boltzmann planar maps are subdiffusive
• Complete diagrammatics of the single ring theorem
• Condition numbers for real eigenvalues of real elliptic ensemble: weak non-normality at the edge