Transverse instability for non-normal parameters

Abstract

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.

15 figures, submitted to Nonlinearity, the full paper available at http://www.maths.qmw.ac.uk/~eoc

References in corpus (2)

• Non-normal parameter blowout bifurcation: an example in a truncated mean field dynamo model
• Cycling chaos: its creation, persistence and loss of stability in a model of nonlinear magnetoconvection

Cited by in corpus (9)

• Extreme events in dynamical systems and random walkers: A review
• Neural Network Forecast of the Sunspot Butterfly Diagram
• In--out intermittency in PDE and ODE models
• Characteristics of in-out intermittency in delay-coupled FitzHugh-Nagumo oscillators
• Multiple forms of intermittency in PDE dynamo models
• Burst Mechanisms in Hydrodynamics
• Dynamics of Axisymmetric Truncated Dynamo Models
• Mean Field Dynamos with Algebraic and Dynamic alpha-Quenchings
• Extreme events in the Lienard system with asymmetric potential: An in-depth exploration