Colored noise influence on the system evolution

We present a picture of phase transitions of the system with colored multiplicative noise. Considering the noise amplitude as the power-law dependence of the stochastic variable $x^a$ we show the way to phase transitions disorder-order and order-disorder. The governed equations for the order parameter and one-time correlator are obtained and investigated in details. The long-time asymptotes in the disordered and ordered domains on the phase portrait of the system are defined.Comment: 12 pages, 8 figures, LaTeX. Submitted to EPJ-

Rare Events Statistics in Reaction--Diffusion Systems

We develop an efficient method to calculate probabilities of large deviations from the typical behavior (rare events) in reaction--diffusion systems. The method is based on a semiclassical treatment of underlying "quantum" Hamiltonian, encoding the system's evolution. To this end we formulate corresponding canonical dynamical system and investigate its phase portrait. The method is presented for a number of pedagogical examples.Comment: 12 pages, 6 figure

The determination of the topological structure of skin friction lines on a rectangular wing-body combination

A short tutorial in the application of topological ideas to the intepretation of oil flow patterns is presented. Topological concepts such as critical points, phase portraits, topological stability, and indexing are discussed. These concepts are used in an ordered procedure to construct phase portraits of skin friction lines with oil flow patterns for a wing-body combination and two angles of attack. The relationship between the skin friction phase portrait and planar cuts of the velocity field is also discussed

Asymptotic integration of nonlinear systems of differential equations whose phase portrait is foliated on invariant tori

We consider the class of autonomous systems $\dot x=f(x)$, where $x \in {\bf R}^{2n}$, $f \in C^1({\bf R}^{2n})$ whose phase portrait is a Cartesian product of $n$ two-dimensional {\em centres}. We also consider perturbations of this system, namely $\dot x=f(x)+g(t,x)$, where $g \in C^1({\bf R}\times{\bf R}^{2n})$ and $g$ is asymptotically small, that is $g\Rightarrow 0$ as $t\to +\infty$ uniformly with respect to $x$. The rate of decrease of $g$ is assumed to be $t^{-p}$ where $p>1$. We prove under this conditions the existence of bounded solutions of the perturbed system and discuss their convergence to solutions of the unperturbed system. This convergence depends on $p$. Moreover, we show that the original unperturbed system may be reduced to the form $\dot r=0$, $\dot\theta=A(r)$, and taking $r\in {\bf R}^m_{+}$, $\theta\in {\bf T}^n$, where ${\bf T}^n$ denotes the $n$-dimensional torus, we investigate the more general case of systems whose phase portrait is foliated on invariant tori. We notice that integrable Hamiltonian systems are of the same nature. We give also several examples, showing that the conditions of our theorems cannot be improved