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A classification of explosions in dimension one

Published online by Cambridge University Press:  01 April 2009

E. SANDER  and
J. A. YORKE
E. SANDER
Affiliation:
Department of Mathematical Sciences, George Mason University, 4400 University Drive, Fairfax, VA 22030, USA (email: [email protected])
J. A. YORKE
Affiliation:
IPST, University of Maryland, College Park, MD 20742, USA (email: [email protected])
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Abstract

A discontinuous change in the size of an attractor is the most easily observed type of global bifurcation. More generally, an explosion is a discontinuous change in the set of recurrent points. An explosion often results from heteroclinic and homoclinic tangency bifurcations. We prove that, for one-dimensional maps, explosions are generically the result of either tangency or saddle-node bifurcations. Furthermore, we give necessary and sufficient conditions for generic tangency bifurcations to lead to explosions.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 2 , April 2009 , pp. 715 - 731
Copyright
Copyright © Cambridge University Press 2008

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