Transverse instability for non-normal parameters

Abstract

Abstract. 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.

Dynamics of counterpropagating waves in parametrically forced systems

“…The figure suggests that the solution trajectory spirals within this subspace before being ejected in a direction normal to it. These ejections out of the invariant subspace appear to oceur intermittently, much as might be expected of dynamics associated with a blow-out bifurcation and the so-called in-out intermitteney [27,28]; similar behavior occurs in other partial differential equations with reflection-invariant subspaces such as those studied by Proctor and Lega [29] or Covas et al [30]. Also included in Fig.…”

“…Note, however, that despite the chaotic motion about the A = B subspace both A(x,x) and B(x,x) spend sometimes quite long periods of time very cióse to reflection-invariant subspaces before escaping again. As already mentioned, behavior of this type may be associated with the onset of in-out intermitteney [27,28]. This behavior is not revealed unambiguously by time series shown in Fig.…”

“…9c). Such sporadic excursions from the reflection-invariant subspaces are again likely to be the consequence of a blow-out bifurcation [27,28]. Fig.…”

Crisis-Induced Intermittency Due to Attractor-Widening in a Buoyancy-Driven Solar Dynamo

“…Despite the presence of simplifications in these models, this is of potential importance since it shows the occurrence of another type of intermittency (in addition to Type I Pomeau-Manneville, attractor merging intermittency [13] and in-out intermittency [19] recently discovered) in these models. This may be taken as an possible indication that more than one type of intermittency may occur in solar and stellar dynamos [20].…”

Compressible magnetoconvection in three dimensions: pattern formation in a strongly stratified layer

“…As R is progressively decreased this attractor approaches two or more invariant sets, and trajectories spend longer and longer in their neighbourhoods, until a pair of these sets eventually becomes attracting. This bursting appears to be a form of on-off or in-out intermittency (Platt, Spiegel & Tresser 1993;Ashwin, Covas & Tavakol 1999). Such behaviour is often associated with the presence of symmetry-invariant subspaces and structurally stable heteroclinic orbits; studies of shearing instabilities in two-and three-dimensional convection or magnetoconvection provide specific examples of such orbits Matthews et al 1996).…”