Spatio-temporal intermittency in the spatial unfolding of Andronov’s Bifurcations

Abstract

A simple mechanism for spatiotemporal intermittency is described. It occurs in systems that exhibit bistability between a homogeneous stationary state and an oscillatory one, close to parameter values where the oscillation disappears through an Andronov homoclinic bifurcation. Spatiotemporal chaos denotes a complex dynamical behavior which is observed in spatially extended physical systems. From a mathematical point of view, it is associated with a solution of a partial differential equation characterized by a number of positive Lyapunov exponents that increases with the size of the domain. Phase turbulence [1], defect mediated turbulence [2], self-focusing turbulence [3], and spatiotemporal intermittency [5] provide different forms of spatiotemporal chaos. From the theoretical side, very few is known about the possible mechanisms leading to spatiotemporal chaos. In this paper, a generic mechanism leading to spatiotemporal intermittency is described. Several models that display this kind of behavior are reviewed in the first part. The second part is devoted to the analysis of a simple model that is based on the normal form of a degenerate bifurcation. Spatiotemporal intermittency appears as the consequence of the bistability between an oscillatory behavior and a stationary state. Spatiotemporal intermittency has been described in the context of partial differential equations, coupled maps and cellular automatas [5]. Roughly speaking, it is a complex spatiotemporal behavior that involves two phases: a simple one, that describes a so-called laminar state and a complex one, that has been called “turbulent state”. It is reminiscent of the phenomenon of intermittency observed in low dimensional dynamical systems [6]. In spatially extended systems, the laminar state is nucleated from the turbulent phase. This laminar state is metastable and disappears through the propagation of fronts (see Fig. 1). In low dimensional dynamical systems, the duration of the laminar phase, close to the onset of intermittency also obeys to a simple scaling law [6]. In extended systems, the transition to turbulence is characterized by critical exponents associated with the fraction of turbulent phase into the laminar one.KeywordsSaddle Node BifurcationHomoclinic BifurcationStable Fixed PointSpatiotemporal ChaosLaminar StateThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.