# Noisy heteroclinic networks

@article{Bakhtin2007NoisyHN,
title={Noisy heteroclinic networks},
author={Yuri Bakhtin},
journal={Probability Theory and Related Fields},
year={2007},
volume={150},
pages={1-42}
}
• Yuri Bakhtin
• Published 23 December 2007
• Mathematics
• Probability Theory and Related Fields
We consider a white noise perturbation of dynamics in the neighborhood of a heteroclinic network. We show that under the logarithmic time rescaling the diffusion converges in distribution in a special topology to a piecewise constant process that jumps between saddle points along the heteroclinic orbits of the network. We also obtain precise asymptotics for the exit measure for a domain containing the starting point of the diffusion.

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## References

SHOWING 1-10 OF 16 REFERENCES

### Noisy heteroclinic networks

• Environmental Science
• 2003
The influence of small noise on the dynamics of heteroclinic networks is studied, with a particular focus on noise-induced switching between cycles in the network. Three different types of switching

### The exit problem for small random perturbations of dynamical systems with a hyperbolic fixed point

We consider the Markov diffusion process ξ∈(t), transforming when ɛ=0 into the solution of an ordinary differential equation with a turning point ℴ of the hyperbolic type. The asymptotic behevior as

### Robust heteroclinic cycles

SummaryOne phenomenon in the dynamics of differential equations which does not typically occur in systems without symmetry is heteroclinic cycles. In symmetric systems, cycles can be robust for

### Diffusion Processes Depending on a Small Parameter

In this paper we consider a random disturbance of a system of ordinary differential equations which can be written in vector form as follows: \[ x( t ) = a ( {t,x} ), x( 0 ) = x_0 ,\quad t \in [

### Random Perturbations of Dynamical Systems

• Mathematics, Physics
• 1984
1.Random Perturbations.- 2.Small Random Perturbations on a Finite Time Interval.- 3.Action Functional.- 4.Gaussian Perturbations of Dynamical Systems. Neighborhood of an Equilibrium Point.-

### Introduction to the Modern Theory of Dynamical Systems

self-contained comprehensive exposition of the theory of dynamical systems as a core mathematical discipline closely intertwined with all areas of develop the interested fundamental tools and

### Limit Theorems for Stochastic Processes

• Mathematics
• 1987
I. The General Theory of Stochastic Processes, Semimartingales and Stochastic Integrals.- II. Characteristics of Semimartingales and Processes with Independent Increments.- III. Martingale Problems

### Convergence of Probability Measures

• P. M. Lee
• Mathematics
The Mathematical Gazette
• 1970
Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

### Dynamics of sequential decision making.

• Computer Science
Physical review letters
• 2006
A new class of dynamical models that are described by ordinary differential equations with a finite number of possibilities at the decision points, and also include rules solving this uncertainty are introduced.