Quantifying noisy attractors: from heteroclinic to excitable networks
SIAM Journal on Applied Dynamical Systems
Society for Industrial and Applied Mathematics
Attractors of dynamical systems may be networks in phase space that can be heteroclinic (where there are dynamical connections between simple invariant sets) or excitable (where a perturbation threshold needs to be crossed to a dynamical connection between “nodes”). Such network attractors can display a high degree of sensitivity to noise both in terms of the regions of phase space visited and in terms of the sequence of transitions around the network. The two types of network are intimately related—one can directly bifurcate to the other. In this paper we attempt to quantify the effect of additive noise on such network attractors. Noise increases the average rate at which the networks are explored, and can result in “macroscopic” random motion around the network. We perform an asymptotic analysis of local behaviour of an escape model near heteroclinic/excitable nodes in the limit of noise η → 0 + as a model for the mean residence time T near equilibria. The heteroclinic network case has T proportional to − ln η while the excitable network has T given by a Kramers’ law, proportional to exp(B/η2 ). There is singular scaling behaviour (where T is proportional to 1/η) at the bifurcation between the two types of network. We also explore transition probabilities between nodes of the network in the presence of anisotropic noise. For low levels of noise, numerical results suggest that a (heteroclinic or excitable) network can approximately realise any set of transition probabilities and any sufficiently large mean residence times at the given nodes. We show that this can be well modelled in our example network by multiple independent escape processes, where the direction of first escape determines the transition. This suggests that it is feasible to design noisy network attractors with arbitrary Markov transition probabilities and residence times.
We thank many people for stimulating conversations that contributed to the development of this paper: in particular Chris Bick, Nils Berglund, Mike Field, John Terry, Ilze Ziedins. We thank the London Mathematical Society for support of a visit of CMP to Exeter, and the University of Auckland Research Council for supporting a visit of PA to Auckland during the development of this research. PA gratefully acknowledges the financial support of the EPSRC via grant EP/N014391/1.
This is the author accepted manuscript. The final version is available from the Society for Industrial and Applied Mathematics via the DOI in this record.
Vol. 15 (4), pp. 1989–2016