Transverse instability for non-normal parameters

Suppose a smooth dynamical system has an invariant subspace and a parameter that leaves the dynamics in the invariant subspace invariant while changing the normal dynamics. Then we say the parameter is a normal parameter, and much is understood of how attractors can change with normal parameters. Unfortunately, normal parameters do not arise very often in practise. We consider the behaviour of attractors near invariant subspaces on varying a parameter that does not preserve the dynamics in the invariant subspace but is otherwise generic, in a smooth dynamical system. We refer to such a parameter as “non-normal”. If there is chaos in the invariant subspace that is not structurally stable, this has the effect of “blurring out” blowout bifurcations over a range of parameter values that we show can have positive measure in parameter space. Associated with such blowout bifurcations are bifurcations to attractors displaying a new type of intermittency that is phenomenologically similar to on-off intermittency, but where the intersection of the attractor by the invariant subspace is larger than a minimal attractor. The presence of distinct repelling and attracting invariant sets leads us to refer to this as “in-out” intermittency. Such behaviour cannot appear in systems where the transverse dynamics is a skew product over the system on the invariant subspace. We characterise in-out intermittency in terms of its structure in phase space and in terms of invariants of the dynamics obtained from a Markov model of the attractor. This model predicts a scaling of the length of laminar phases that is similar to that for on-off intermittency but which has some differences. Finally, we discuss some other bifurcation effects associated with non-normal parameters, in particular a bifurcation to riddled basins.