Dynamical stochastic models of single neurons and neural networks often take the form of a system of n>2 coupled stochastic differential equations. We consider such systems under the assumption that third and higher order central moments are relatively small. In the general case, a system of 1 2 n(n13) ~generally! nonlinear coupled ordinary differential equations holds for the approximate means...

The evolution equations of Einstein's theory and of Maxwell's theory|the latter used as a simple model to illustrate the former| are written in gauge covariant rst order symmetric hyperbolic form with only physically natural characteristic directions and speeds for the dynamical variables. Quantities representing gauge degrees of freedom the spatial shift vector i (t; x j) and the spatial scala...

the study of the dynamic behavior of a rigid body with one fixed point (gyroscope) has a long history. a number of famous mathematicians and mechanical engineers have devoted enormous time and effort to clarify the role of dynamic effects on its movement (behavior) – stable, periodic, quasi-periodic or chaotic. the main objectives of this review are: 1) to outline the characteristic features of...

The hierarchy of integrable equations are considered. The dynamical approach to theory of nonlinear waves is proposed. The special solutions (nonlinear waves) of considered equations are derived. We use powerful methods of computer algebra such as differential resultant and others.

This harmony that human intelligence believes it discovers in nature —does it exist apart from that intelligence? No, without doubt, a reality completely independent of the spirit which conceives it, sees it or feels it, is an impossibility. A world so exterior as that, even if it existed, would be forever inaccessible to us. But what we call objective reality is, in the last analysis, that whi...

This article surveys some recent applications of ideas from dynamical systems theory to understand the qualitative behavior of solutions of dissipative partial differential equations with a particular emphasis on the two-dimensional Navier-Stokes equations.

Based on scaling arguments the governing equations for turbulent flows are classified. The similarity for stratified and rotating flows is characterized and the conditions for a hydrostatic assumption are shown for several flow regimes. For stratified rotating flow a scale analysis of the turbulent stresses exhibits different classes of second order closure. The complete sets of the governing e...

This manuscript provides an introduction to ordinary differential equations and dynamical systems. We start with some simple examples of explicitly solvable equations. Then we prove the fundamental results concerning the initial value problem: existence, uniqueness, extensibility, dependence on initial conditions. Furthermore we consider linear equations, the Floquet theorem, and the autonomous...

We use a modification of the parameterization method to study invariant manifolds for difference equations. We establish existence, regularity, smooth dependence on parameters and study several singular limits, even if the difference equations do not define a dynamical system. This method also leads to efficient algorithms that we present with their implementations. The manifolds we consider in...