Critical exponents for crisis-induced intermittency

Abstract

We consider three types of changes that attractors can undergo as a system parameter is varied. The first type leads to the sudden destruction of a chaotic attractor. The second type leads to the sudden widening of a chaotic attractor. In the third type of change, which applies for many systems with symmetries, two (or more) chaotic attractors merge to form a single chaotic attractor and the merged attractor can be larger in phase-space extent than the union of the attractors before the change. All three of these types of changes are termed crises and are accompanied by a characteristic temporal behavior of orbits after the crisis. For the case where the chaotic attractor is destroyed, this characteristic behavior is the existence of chaotic transients. For the case where the chaotic attractor suddenly widens, the characteristic behavior is an intermittent bursting out of the phase-space region within which the attractor was confined before the crisis. For the case where the attractors suddenly merge, the characteristic behavior is an intermittent switching between behaviors characteristic of the attractors before merging.In all cases a time scale \ensuremath{\tau} can be defined which quantifies the observed post-crisis behavior: for attractor destruction, \ensuremath{\tau} is the average chaotic transient lifetime; for intermittent bursting, it is the mean time between bursts; for intermittent switching, it is the mean time between switches. The purpose of this paper is to examine the dependence of \ensuremath{\tau} on a system parameter (call it p) as this parameter passes through its crisis value p=${p}_{c}$. Our main result is that for an important class of systems the dependence of \ensuremath{\tau} on p is \ensuremath{\tau}\ensuremath{\sim}\ensuremath{\Vert}p-${p}_{c}$${\ensuremath{\Vert}}^{\mathrm{\ensuremath{-}}\ensuremath{\gamma}}$ for p close to ${p}_{c}$, and we develop a quantitative theory for the determination of the critical exponent \ensuremath{\gamma}. Illustrative numerical examples are given. In addition, applications to experimental situations, as well as generalizations to higher-dimensional cases, are discussed. Since the case of attractor destruction followed by chaotic transients has previously been illustrated with examples [C. Grebogi, E. Ott, and J. A. Yorke, Phys. Rev. Lett. 57, 1284 (1986)], the numerical experiments reported in this paper will be for crisis-induced intermittency (i.e., intermittent bursting and switching).