Two-state intermittency near a symmetric interaction of saddle-node and Hopf bifurcations: a case study from dynamo theory
Ashwin, Peter; Rucklidge, Alastair M.; Sturman, Rob
Date: 2004
Journal
Physica D
Publisher
Elsevier
Publisher DOI
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Abstract
We consider a model of a Hopf bifurcation interacting as a codimension 2 bifurcation with a saddle-node on a limit cycle, motivated by a low-order model for magnetic activity in a stellar
dynamo. This model consists of coupled interactions between a saddle-node and two Hopf bifurcations,
where the saddle-node bifurcation is assumed ...
We consider a model of a Hopf bifurcation interacting as a codimension 2 bifurcation with a saddle-node on a limit cycle, motivated by a low-order model for magnetic activity in a stellar
dynamo. This model consists of coupled interactions between a saddle-node and two Hopf bifurcations,
where the saddle-node bifurcation is assumed to have global reinjection of trajectories.
The model can produce chaotic behaviour within each of a pair of invariant subspaces, and also
it can show attractors that are stuck-on to both of the invariant subspaces. We investigate the
detailed intermittent dynamics for such an attractor, investigating the effect of breaking the
symmetry between the two Hopf bifurcations, and observing that it can appear via blowout
bifurcations from the invariant subspaces.
We give a simple Markov chain model for the two-state intermittent dynamics that reproduces
the time spent close to the invariant subspaces and the switching between the different
possible invariant subspaces; this clarifes the observation that the proportion of time spent near
the different subspaces depends on the average residence time and also on the probabilities of
switching between the possible subspaces.
Mathematics and Statistics
Faculty of Environment, Science and Economy
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