Order and chaos in nonlinear media with diffusion. I (Q1416175)

This paper is the first of the three parts which address the problem related to properties of nonlinear media where stable self-supporting structures are formed. The authors suggest possible approach to its solution through the application of special local methods to singularly perturbed problems. More precisely, they study auto-wave processes in systems with small diffusion coefficients revealing a number of characteristic features of such systems. Special attention is paid to the buffer phenomenon of higher order attractors and diffusion chaos in parabolic systems of reaction-diffusion type, as well as in similar hyperbolic systems. In the first part, the principal result is an extension of the bifurcation theorem of Turing-Prigogine to a system of parabolic equations with a small diffusion. This theorem allows the authors to reveal all characteristic features of the dynamics of stationary dissipative structures with the principal characteristic being unbounded growth of their number under the decrease of diffusion coefficients with other parameters fixed. The authors develop special methods for studying higher-order regimes in singularly perturbed systems with diffusion, which also turn out to be useful for solving other problems like, for instance, the problem of parametric stimulation of auto-waves. Furthermore, these techniques are applied for the study of the parametric buffer phenomenon in hyperbolic boundary-value problems. The authors also address the problem of the existence and stability of spatially inhomogeneous cycles for a class of nonlinear telegraph equations with a Neumann boundary condition on a closed interval in the smooth setting and under the assumption of smallness of diffusion, higher-mode self-oscillations for a number of specific problems arising in radiophysics and mechanics, and dynamics of a nonlinear wave equation in a plane domain.