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.