## Noise-Induced Phenomena in Slow-Fast Dynamical Systems: A by Nils Berglund PhD, Barbara Gentz PhD (auth.)

By Nils Berglund PhD, Barbara Gentz PhD (auth.)

Stochastic differential equations play an more and more vital function in modeling the dynamics of a giant number of structures within the typical sciences, and in technological functions. This publication is geared toward complicated undergraduate and graduate scholars, and researchers in arithmetic, physics, the normal sciences, and engineering. It provides a brand new positive method of the quantitative description of suggestions to platforms of stochastic differential equations evolving on well-separated timescales. the tactic, which mixes recommendations from stochastic research and singular perturbation idea, permits the domain names of focus for usual pattern paths to be made up our minds, and gives detailed estimates at the transition chances among those domains.

In addition to the distinctive presentation of the set-up and mathematical effects, purposes to difficulties in physics, biology, and climatology are mentioned. The emphasis lies on noise-induced phenomena equivalent to stochastic resonance, hysteresis, excitability, and the aid of bifurcation delay.

Nils Berglund joined the examine crew "Classical and Quantum Dynamics" on the Centre de body Théorique (CNRS) in Marseille-Luminy in 2001. He teaches within the arithmetic division of the Université du Sud Toulon-Var.

Barbara Gentz joined the learn workforce "Interacting Random structures" on the Weierstrass Institute for utilized research and Stochastics (WIAS) in Berlin in 1998. She teaches within the Institute of arithmetic on the Technical collage in Berlin.

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A) Solutions track the stable branch x− (y) at a distance growing up to order ε1/3 . They jump after a delay of order ε2/3 . (b) For multidimensional slow variables y, a saddle–node bifurcation corresponds to a fold in the slow manifold. 15) there is an unstable slow manifold M+ = {(x, y) ∈ N : x = x+ (y), y < 0} , √ where x+ (y) = − −y [1 + Oy (1)]; • there are no other slow manifolds in N ; • g(x, y) > 0 in N . 16) M− being uniformly asymptotically stable for negative y, bounded away from 0, there is an adiabatic manifold associated with it which, in this (1 + 1)– dimensional setting, is simply a particular solution of the system.

27) 26 2 Deterministic Slow–Fast Systems where the last line follows from the implicit-function theorem, applied to the equation f (x (y), y) = 0. 9 (Overdamped motion of a particle in a potential – continued). 3. 29) ¯(y, ε) where ∂yy U (y) denotes the Hessian matrix of U . The dynamics of x = x on the adiabatic manifold is thus governed by the equation y−x ε ¯(y, ε)∇U (x) = −∂y x x˙ = = − 1l + ε∂xx U (x) + O(ε2 ) ∇U (x) . 30) This is indeed a small correction to the limiting equation x˙ = −∇U (x).

27) for t τ and yt < 0. We remark in passing that the corresponding argument for the lower bound in particular shows that zt > 0. Now choosing c1 = (3KM/c− )2/3 , we ﬁnd that for all t τ satisfying yt √ −c1 ε2/3 , we also have zt c− −yt /3M , so that actually t < τ whenever yt −c1 ε2/3 . 19). The lower bound is obtained in a similar way. 28) holds. 29) which is a small perturbation of a solvable Riccati equation. 29) without the error term yields the linear second order equation ϕ (s) = −sϕ(s), whose solution can be expressed in terms of Airy functions.