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Noise and O,,1... amplitude effects on heteroclinic cycles Emily Stone

Summary: Noise and O,,1... amplitude effects on heteroclinic cycles
Emily Stone
Department of Mathematics and Statistics, Utah State University, Logan, Utah 84322-3900
Dieter Armbruster
Department of Mathematics, Arizona State University, Tempe, Arizona 85287-1804
Received 5 March 1998; accepted for publication 1 February 1999
The dynamics of structurally stable heteroclinic cycles connecting fixed points with
one-dimensional unstable manifolds under the influence of noise is analyzed. Fokker-Planck
equations for the evolution of the probability distribution of trajectories near heteroclinic cycles are
solved. The influence of the magnitude of the stable and unstable eigenvalues at the fixed points and
of the amplitude of the added noise on the location and shape of the probability distribution is
determined. As a consequence, the jumping of solution trajectories in and out of invariant subspaces
of the deterministic system can be explained. 1999 American Institute of Physics.
S1054-1500 99 01602-X
The relevance of a mathematical model to the physical
situation it is supposed to represent depends critically on
the sensitivity of the model to external noise. The effects
of noise on mathematical models must be well under-
stood, and in many cases must be introduced explicitly,
which creates a stochastic model. There are many types


Source: Armbruster, Dieter - Department of Mathematics and Statistics, Arizona State University


Collections: Mathematics