Cycling chaos

Saddle connections between equilibria can occur structurally stable in systems with symmetry, and these saddle connections can cycle so that a given equilibrium is connected to itself by a sequence of connections. These cycles provide a way of generating intermittency, as a trajectory will spend some time near each saddle before quickly moving to the next saddle. Guckenheime and Holmes (1988) showed that cycles of saddle connections can appear via bifurcation. In this paper, we show numerically that the equilibria in the Guckenheimer-Holmes example can be replaced by chaotic sets, such as those that appear in a Chua circuit or a Lorenz attractor. Consequently, there are trajectories that behave chaotically, but where the spatial location of the chaos cycles. We call this phenomenon cycling chaos.>