Transverse instability for non-normal parameters

Abstract

We consider the behaviour of attractors near invariant subspaces on varying a parameter that changes the dynamics in the invariant subspace of a dynamical system. We refer to such a parameter as `non-normal'. In the presence of chaos that is fragile, we find blowout bifurcations that are blurred over a range of parameter values. We demonstrate that this can occur on a set of positive measure in the parameter space. Under an assumption that the dynamics is not of skew product form, these blowout bifurcations can create attractors displaying `in-out intermittency', a generalized form of on-off intermittency. We characterize in-out intermittency both in terms of its structure in phase space and statistically by means of a Markov model. We discuss some other dynamical and bifurcation effects associated with non-normal parameters, in particular non-normal bifurcation to riddled basins and transition between on-off and in-out intermittency.