PHYSICAL REVIEW A VOLUME 36, NUMBER 11 DECEMBER 1, 1987 Critical exponents for crisis-induced intermittency Celso Grebogi Laboratory for Plasma and Fusion Energy Studies, University of Maryland, College Park, Maryland 20742 Edward Ott Laboratory for Plasma and Fusion Energy Studies and Departments of Electrical Engineering and Physics, University of Maryland, College Park, Maryland 20742 Filipe Romeiras* Laboratory for Plasma and Fusion Energy Studies, University of Maryland, College Park, Maryland 20742 James A. Yorke Institute for Physical Science and Technology and Department of Mathematics, University of Maryland, College Park, Maryland 20742 (Received 1 July 1987) 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 sys- tems 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 be- fore 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 be- tween behaviors characteristic of the attractors before merging. In all cases a time scale ~ can be defined which quantifies the observed post-crisis behavior: for attractor destruction, ~ is the aver- age chaotic transient lifetime; for intermittent bursting, it is the mean time between bursts; for in- termittent switching, it is the mean time between switches. The purpose of this paper is to exam- ine the dependence of ~ on a system parameter (call it p) as this parameter passes through its crisis value p =p, . Our main result is that for an important class of systems the dependence of ~ on p is r- ~ p — p, ~ r for p close to p„and we develop a quantitative theory for the determination of the critical exponent y. Illustrative numerical examples are given. In addition, applications to experi- mental 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 ex- amples [C. Grebogi, E. Ott, and J. A. Yorke, Phys. Rev. Lett. 57, 1284 11986)], the numerical ex- periments reported in this paper will be for crisis-induced intermittency (i.e. , intermittent bursting and switching). I. INTRODUCTION Crises' are a common manifestation of chaotic dy- namics for dissipative systems and have been seen in many experimental and numerical studies. In a crisis, one observes a sudden discontinuous change in a chaotic attractor as a system parameter is varied. The discon- tinuous changes are typically of three types: in the first a chaotic attractor is suddenly destroyed as the parame- ter passes through its critical crisis value; in the second the size of the chaotic attractor in phase space suddenly increases; in the third type (which can occur in systems with symmetries) two or more chaotic attractors merge to form one chaotic attractor. [The inverse of these pro- cesses (i.e. , the sudden creation, shrinking, or splitting of a chaotic attractor) occur as the parameter is varied in the other direction. ] For all three types of crisis there is an associated characteristic temporal dependence of typical orbits for parameter values near the crisis. The characteristic tem- poral dependence can be quantified by a characteristic time which we denote ~. The quantity ~ is here taken to have the following meanings for the three different types of crisis. (I) Attractor destruction Let p denote th.e relevant system parameter, and let p, denote the value of p at the crisis, with the destruction of the chaotic attractor occurring as p increases through p, . Let p be slightly 36 5365 1987 The American Physical Society