# Stochastic dynamical systems

A **stochastic dynamical system** is a dynamical system subjected to the effects of noise. Such effects of fluctuations have been of interest for over a century since the seminal work of Einstein (1905). Fluctuations are classically referred to as "noisy" or "stochastic" when their suspected origin implicates the action of a very large number of variables or "degrees of freedom". For example, the action of many water molecules on the motion of a large protein can be seen as noise. In principle the equations of motion for such high-dimensional dynamics can be written and studied analytically and numerically. However, it is possible to study a system subjected to the action of the large number of variables by coupling its deterministic equations of motion to a "noise" that simple mimics the perpetual action of many variables.

The coupling of noise to nonlinear deterministic equations of motion can lead to non-trivial effects (Schimansky-Geier 1985; Horsthemke 1985; Haenggi, Talkner and Borkovec, 1990; Haenggi and Marchesoni 2005). For example,

noise can stabilize unstable equilibria and shift bifurcations. i.e. the parameter value at which the dynamics change qualitatively (Arnold 2003). Noise can lead to transitions between coexisting deterministic stable states or *attractors* . More interestingly still, noise can induce new stable states that have no deterministic counterpart. At the very least, noise excites internal modes of oscillation in both linear and nonlinear systems. In the latter case, it can even enhance the response of a nonlinear system to external signals (Jung, 1993; Gammaitoni et al. 1998; Lindner et al. 2004).

It is often thought that the action of noise merely amounts to a blurring of trajectories of the deterministic system. That is indeed the case for "observational" or "measurement" noise. However, in nonlinear systems where noise acts as a driving force, noise can drastically modify the deterministic dynamics. We discuss these issues using a basic level of description which couples a stochastic process to a deterministic equation of motion: the stochastic differential equation (SDE).

Source: www.scholarpedia.orgCategory: Forex

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