Stochastic dynamical systems

In this part we will explain a method that we extensively used to describe stochastic dynamical systems. It is based on the dynamics of the moments of a distribution. We applied it successfully to a variety of globally coupled systems. Advantages of the method are simple applicability and quick numerical investigations. Let us consider a globally coupled stochastic system that is described by the following set of Langevin equations ... 

There is essentially no need to use maps based on Newtonian deterministic dynamics. There are many alternative dynamics models which leave invariant the canonical measure, going well beyond those considered so far. To illustrate this with one intriguing example, we explore here the stochastic dynamical system in 21S + 1 variables defined by... 

Stochastic differential equations some concepts and comments Let us consider a stochastic dynamic system ... 

Stochastic dynamic systems can be classified according to the very nature of/. Arnold Kliemann (1981) summarised the qualitative behaviour of x both for linear and nonlinear systems (for a condensed survey see Arnold (1981). The term linear is not specific here, since / can be linear either in state or in noise, even in both. In applications it is assumed very often that the forcing function has a systematic or deterministic part, and a term due to the rapidly varying, highly irregular random effects ... 

Broeck, C. V. D. Nicolis, G. 1993. Noise-Induced Sensitivity to the Initial Conditions in Stochastic Dynamical Systems, Phys. Rev. E 48, 4845-4846. 

Santilldn M, Qian H (2011) Irreversible thermodynamics in multiscale stochastic dynamical systems. Phys Rev E 83 041,130... 

Sensitivity of First-Excursion Probabilities for Nonlinear Stochastic Dynamical Systems... 

Jensen HA, Sepulveda JC (2011) On the effects of vibration control devices in the reliability-based design of stochastic dynamical systems. In Proceedings of the 8th international conference on structural dynamics (EURODYN 2001), Leuven, 4-6 July 2011 Kelly JM (1986) Aseismic base isolation review and bibliography. Soil Dyn Earthq Eng 5(4) 202-216 Kramer SL (2003) Geotechnical earthquake engineering. Prentice Hall, Upper Saddle River, Englewood Cliffs, New Jersey, NJ, USA... 

The very restrictive conditions that make available exact solutions for the stochastic dynamical systems motivated the development of approximate solution techniques. Methods such as the equivalent linearization have been developed to generate first-order approximate solutions. These techniques can also be applied in some cases to single-degree-of-freedom nonlinear oscillator with hysteretic behavior. 

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