Abstract:
I will discuss the phenomenon of shear-induced chaos in driven dynamical systems. The unforced system is assumed to have certain simple structures, such as attracting periodic solutions or equilibria undergoing Hopf bifurcations. Specifics of the defining equations are unimportant. A geometric mechanism for producing chaos is proposed. In the case of periodic kicks followed by long relaxations, rigorous results establishing the presence of strange attractors with SRB measures are presented. These attractors are in a class of chaotic systems that can be modeled (roughly) by countable-state Markov chains. From this I deduce information on their statistical properties. In the last part of this talk, I will return to the phenomenon of shear-induced chaos, to explore numerically the range of validity of the geometric ideas. Examples including randomly forced coupled oscillators will be discussed. I am reporting on joint works with a number of co-authors.