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

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 transverse modes. A condition that favours this process is that the linearization of the dynamics in the transverse direction be characterized by a non–normal matrix. The phenomenon is illustrated by a simple two–dimensional switching model and by a simulation of a convectively driven dynamo.

[1]  E. Lorenz.  Deterministic nonperiodic flow , 1963 .

[2]  A Model for Magnetohydrodynamic Convection Relevant to the Solar Dynamo Problem , 1976 .

[3]  E. Knobloch,et al.  Oscillatory and steady convection in a magnetic field , 1981, Journal of Fluid Mechanics.

[4]  Ruby Krishnamurti,et al.  Large-scale flow in turbulent convection: a mathematical model , 1986, Journal of Fluid Mechanics.

[5]  P. Holmes,et al.  Random perturbations of heteroclinic attractors , 1990 .

[6]  Proctor,et al.  Noise and slow-fast dynamics in a three-wave resonance problem. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[7]  Farrell,et al.  Variance maintained by stochastic forcing of non-normal dynamical systems associated with linearly stable shear flows. , 1994, Physical review letters.

[8]  Solar and Planetary Dynamos: Testing for Dynamo Action , 1994 .

[9]  Melbourne,et al.  Asymptotic stability of heteroclinic cycles in systems with symmetry , 1995, Ergodic Theory and Dynamical Systems.

[10]  Björn Sandstede,et al.  Forced symmetry breaking of homoclinic cycles , 1995 .

[11]  The three-dimensional development of the shearing instability of convection , 1996 .

[12]  I. Stewart,et al.  From attractor to chaotic saddle: a tale of transverse instability , 1996 .

[13]  Analysis of the shearing instability in nonlinear convection and magnetoconvection , 1996 .

[14]  Simulating the Kinematic Dynamo Forced by Heteroclinic Convective Velocity Fields , 1997 .

[15]  Lectures on solar and planetary dynamos , 1997 .

[16]  Lloyd N. Trefethen,et al.  Pseudospectra of Linear Operators , 1997, SIAM Rev..

[17]  Arnd Scheel,et al.  Transverse bifurcations of homoclinic cycles , 1997 .

[18]  P. Matthews,et al.  Dynamo action in simple convective flows , 1999, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[19]  P. Chossat,et al.  Generalized Heteroclinic Cycles in Spherically Invariant Systems and Their Perturbations , 1999 .

[20]  P. Roberts,et al.  Convection-driven dynamos in a rotating plane layer , 2000, Journal of Fluid Mechanics.