What drives transient behavior in complex systems?

Abstract

We study transient behaviour in the dynamics of complex systems described by a set of nonlinear 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 T0. 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 T0 limit where it describes a normal matrix.One of the key problems in studying complex systems is answering the question of stability. The standard linearization approach relies heavily on the large time asymptotics and can be misleading for intermediate times. This is especially pronounced when systems develop transient behaviour -the analysis based on eigenvalues loses its significance and different approach is needed. This shortcoming in physics literature can be traced back to the work of Orr [1] in the hydrodynamical context. Since then, similar ideas were revived in the context of fluid dynamics [2,3], plasma physics [4,5], diffusion in porous media [6] or pattern formation [7,8]. Further motivation for this work is rooted in the ecological literature on biological networks [9][10][11].The physical mechanism of transient behaviour is both relatively simple and quite general. It needs the system's components to interact asymmetrically and be stabilized by an effective dissipation mechanism. Asymmetry is indispensable since only then inter-eigenmodes fluxes of "energy" can be formed. Such an unbalanced flow renders particular eigenmodes overpopulated or amplified. Crucially, this mechanism does not break the overall stability -the dissipation eventually wins over and the amplification effect is only temporary or transient.In the paper we focus on such transient trajectories for a system of non-linear ODE's. On an elementary example we show how eigenvalue-based linearization technique becomes misleading and how simultaneously the transient property is developed. We utilize the May-Wigner model to include this mechanism and inspect its generic features. We find a new transient regime where transient trajectories are present although uncommon. Based on these findings, we identify relevant degrees of freedom driving the transient behaviour and propose a natural extension of the May-Wigner model. Such a modification leads to a generic transient behaviour arising as a robust transient phase. * jacek.grela@lptms.u-psud.fr A. Stability of systemsWe focus on a typical complex dynamical system of a set of N first-order non-linear ordinary differential equa-

Condition Numbers for Real Eigenvalues in the Real Elliptic Gaussian Ensemble

“…Here it is necessary to mention that the interest in statistical properties of the overlap matrix O kl and related objects extends much beyond the issues of eigenvalue stability under perturbation and is driven by numerous applications in theoretical and mathematical physics. In particular, non-orthogonality governs transient dynamics in complex systems [30,32,40] (see also [16,34]), analysis of spectral outliers in non-selfadjoint matrices [36], and, last but not least, the description of the Dyson Brownian motion for non-normal matrices [5,6,31]. Another steady source of interest in the statistics of eigenvector overlaps is due to its role in chaotic wave scattering.…”

On Statistics of Bi-Orthogonal Eigenvectors in Real and Complex Ginibre Ensembles: Combining Partial Schur Decomposition with Supersymmetry

“…Note that the non-orthogonality factors reflect non-normality of the matrix, which in the context of dynamical systems is known to give rise to a long transient behaviour, see a general discussion in [47,26]. In a related setting non-symmetric matrices appear very naturally via linearization around an equilibrium in a complicated nonlinear dynamical system [36,24], and the non-orthogonality factors then control transients in a relaxation towards equilibrium [29]. Nonorthogonality also plays some role in analysis of spectral outliers in non-selfadjoint matrices, see e.g.…”

Condition numbers for real eigenvalues in the real Elliptic Gaussian ensemble

“…Here it is necessary to mention that the interest in statistical properties of the overlap matrix O kl and related objects extends much beyond the issues of eigenvalue stability under perturbation, and is driven by numerous applications in Theoretical and Mathematical Physics. In particular, non-orthogonality governs transient dynamics in complex systems [28,30,38], see also [15,32], analysis of spectral outliers in non-selfadjoint matrices [34], and, last but not least, the description of the Dyson Brownian motion for non-normal matrices [5,6,29]. Another steady source of interest in the statistics of eigenvector overlaps is due to its role in chaotic wave scattering.…”