DETERMINISTIC CHAOS IN AN INTERPHASE LAYER OF A LIQUID–VAPOR SYSTEM

A second-order nonlinear differential equation with an aftereffect for the density of a thin homogeneous layer on a liquid and vapor interface is considered. The acts of evaporation and condensation of molecules, which are regarded as periodic "impacts", excite the layer. The mentioned NDE is integrated over a finite time interval to find a 2D (two-dimensional) mapping whose numerical solution describes the chaotic dynamics of density and pressure in time. The algorithms of constructing bifurcation diagrams, Lyapunov's exponents and Kolmogorov's entropy for systems with first-order, second-order phase transitions and Van der Waals' systems were elaborated. This approach allows to associate such concepts as phase transition, deterministic chaos and nonlinear processes. It also allows to answer a question whether deterministic chaos occurs in systems with phase transitions and how fast the information about starting conditions is lost within them.

Global and exponential attractors for nonlinear reaction–diffusion systems in unbounded domains

We study the long-time behaviour of solutions of autonomous and non-autonomous reaction-diffusion equations in unbounded domains of R3. It is shown that, under appropriate assumptions on the nonlinear interaction function and on the external forces, these equations possess compact global (uniform) attractors in the corresponding phase space. Estimates for Kolmogorov's ε-entropy of these attractors in terms of Kolmogorov's entropy of the external forces are given. Moreover, (infinite-dimensional) exponential attractors with the same entropy estimate as that of the corresponding global (uniform) attractor are also constructed.