Grebogi, Celso; Ott, Edward; Yorke, James A. Crises, sudden changes in chaotic attractors, and transient chaos. (English) Zbl 0561.58029 Order in chaos, Proc. int. Conf., Los Alamos/N.M. 1982, Physica 7D, 181-200 (1983). [For the entire collection see Zbl 0536.00007.] From the authors’ summary: The occurrence of sudden qualitative changes of chaotic dynamics as a parameter is varied is discussed and illustrated. It is shown that such changes may result from the collision of an unstable periodic orbit and a coexisting chaotic attractor. We call such collisions crises. Phenomena associated with crises include sudden changes in the size of chaotic attractors, sudden appearances of chaotic attractors (a possible route to chaos), and sudden destructions of chaotic attractors and their basins. This paper presents examples illustrating that crisis events are prevalent in many circumstances and systems, and that, just past a crisis, certain characteristic statistical behavior (whose type depends on the type of crisis) occurs. In particular the phenomenon of chaotic transients is investigated. The examples discussed illustrate crises in progressively higher dimension and include the one-dimensional quadratic map, the (two-dimensional) Hénon map, systems of ordinary differential equations in three dimensions and a three-dimensional map. Reviewer: G.Keller Cited in 2 ReviewsCited in 229 Documents MSC: 37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior 37A99 Ergodic theory 37C70 Attractors and repellers of smooth dynamical systems and their topological structure Keywords:chaotic attractor; transient chaos Citations:Zbl 0536.00007 PDF BibTeX XML Full Text: DOI OpenURL References:  Marsden, J.E.; McCracken, M.; Ruelle, D.; Takens, F., The Hopf bifurcation and its applications, Comm. math. phys., 20, 167, (1971), Springer Berlin  Feigenbaum, M.J.; Collet, P.; Eckmann, J.-P.; May, R.M.; Ott, E., Iterated maps on the interval as dynamical systems, J. stat. phys., Nature, Rev. mod. phys., 53, 655, (1981), Birkhauser Boston  Grebogi, C.; Ott, E.; Yorke, J.A., Phys. rev. lett., 48, 1507, (1982)  Chang, S.-J.; Wright, J., Phys. rev. A, 23, 1419, (1981)  Simo’, C., J. stat. phys., 21, 465, (1979)  Kaplan, J.L.; Yorke, J.A., Comm. math. phys., 67, 93, (1979)  Eckmann, J.-P., Rev. mod. phys., 53, 643, (1981)  Pianigiani, G.; Yorke, J.A.; Lasota, A.; Yorke, J.A., Trans. amer. math. soc., Rend. sem. univ. Padova, 64, 141, (1981)  G. Pianigiani, J. Math. Anal. Appl., in press.  Yorke, J.A.; Yorke, E.D., J. stat. phys., 21, 263, (1979)  Guckenheimer, J.; Gumowski, I.; Mira, C., Recurrences and discrete dynamic systems, Comm. math. phys., 70, 133, (1980), Springer-Verlag New York  Pomeau, Y.; Manneville, P., Comm. math. phys., 74, 189, (1980)  Newhouse, S., Publ. math. IHES, 50, 101, (1980)  Smale, S.; Treve, Y., Topics in nonlinear dynamics, (), 73, 147, (1967)  Testa, J.; Perez, J.; Jeffries, C.; Jeffries, C.; Perez, J., Phys. rev. lett., Phys. rev., A27, 601, (1983)  Lorenz, E.N., J. atmos. sci., 20, 130, (1963)  Russell, D.A.; Ott, E., Phys. fluids, 24, 1976, (1981)  Huberman, B.A.; Crutchfield, J.P., Phys. rev. lett., 43, 1743, (1979)  Tresser, C.; Coullet, P.; Arneodo, A., J. physique-lett., 41, L-243, (1980), for a discussion of discontinuous transitions to chaos involving hysteresis as in figs. 27 and 28.  C. Grebogi, E. Ott and J.A. Yorke, to be published.  J.L. Kaplan, J. Mallet-Paret and J.A. Yorke, to be published. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.