By Nils Berglund

Stochastic differential equations play an more and more vital function in modeling the dynamics of a big number of platforms within the traditional sciences, and in technological purposes. This booklet is aimed toward complicated undergraduate and graduate scholars, and researchers in arithmetic, physics, the typical sciences, and engineering. It offers a brand new positive method of the quantitative description of recommendations to structures of stochastic differential equations evolving on well-separated timescales. the strategy, which mixes recommendations from stochastic research and singular perturbation concept, permits the domain names of focus for regular pattern paths to be made up our minds, and gives distinctive estimates at the transition percentages among those domains.

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

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**Extra resources for Noise-induced phenomena in slow-fast dynamical systems: a sample-paths approach**

**Example text**

Roughly speaking, (t− , t+ ) are the points furthest apart on the real axis that can be connected by a path of constant Re Ψ , satisfying certain regularity assumptions. 13. Assume that a(Yt ) = t − t and ω(Yt ) = ω0 > 0. 73) is constant on hyperbolas centred in (t , −iω0 ). We have Π(t) = 2t − t, and the buﬀer times are given by t± = t ± ω0 . 67) is usually not analytic, even when the original system is, because of lack of regularity of centre manifolds. This is why the analysis in [Ne˘ı87, Ne˘ı88] does not use centre manifolds, but keeps track of all original variables.

It is in fact possible to give a rather complete classiﬁcation of the scaling laws that may occur for one-dimensional bifurcations. Such a classiﬁcation involves two steps: Firstly, analyse the power-law behaviour of slow manifolds near a bifurcation point; secondly, study the equation governing the distance between solutions of the slow–fast system and the slow manifold. The ﬁrst step is best carried out with the help of the bifurcation point’s Newton polygon. 46) where f is of class C r for some r 2, with all derivatives uniformly bounded in a neighbourhood N of the origin.

A saddle–node bifurcation occurs if ∂xx f (0, 0) = 0 and ∂y f (0, 0) = 0 . 10) hold. This implies in particular that the slow manifold exists for y < 0. 11) guarantees that trajectories starting near the stable slow manifold are driven towards the bifurcation point. 12) where the slow manifold consists of a stable branch M− = {(x, y) : x = √ √ −y, y < 0} and an unstable branch M+ = {(x, y) : x = − −y, y < 0}. √ The linearisation of f at these branches is given by ∓2 −y. 11), a qualitatively similar behaviour holds in a neighbourhood of the bifurcation point (0, 0).