Proc. R. Soc. Lond. A 455 (1999) 4205-4222. doi:10.1098/rspa.1999.0498

Destabilization by noise of tranverse perturbations to heteroclinic cycles: a simple model and an example from dynamo theory

J.R. Gog(1), I. Oprea(2,3), M.R.E. Proctor(1) and A.M. Rucklidge(1)

(1) Department of Applied Mathematics and Theoretical Physics,
University of Cambridge, Cambridge, CB3 9EW, UK

(2) Faculty of Mathematics, University of Bucharest,
Str. Academiei 14, Sector 1, Bucharest, Romania

(3) Present address: Department of Mathematics,
Arizona State University, Tempe AZ 85287-1804, USA

Abstract. We show that transverse perturbations from structurally stable heteroclinic cycles can be destabilized by surprisingly small amounts of noise, even when each individual fixed point of the cycle is stable to tranverse modes. A condition that favours this process is that the linearization of the dynamics in the tranverse direction be characterized by a non-normal matrix. The phenomenon is illustrated by a simple two-dimen\-sional switching model and by a simulation of a convectively driven dynamo.

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Movies relevant to this paper

Simulations (1.6MB) of the 44-mode ODE truncation, with Ra=6100, Q=0, sigma=1.2, zeta=0.025, k=2pi and noise=1.0e-12. The movies shows contours of the `stream function' (left) and `flux function' (right) solid lines denote fluid travelling clockwise, dashed lines denote fluid travelling anticlockwise. The fluid executes a stucturally stable hetereoclinic cycle from rolls in one direction to orthogonal rolls and back, while the magnetic field grows erratically.