1 research outputs found
Transverse instability for non-normal parameters
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.Comment: 15 figures, submitted to Nonlinearity, the full paper available at
http://www.maths.qmw.ac.uk/~eo