An evolutionary model of chaotic wave processes in complex dynamical systems on the basis of the matrix decomposition theory

Abstract

A general model of the origin and evolution of chaotic wave processes in complex systems based on the propo sed method of matrix decomposition of operators of nonlinear systems is developed. The proposed model shows that the effect of self-organization in complex systems of different physical nature (for example, hydrodynamic, electronic, and physiological ones) is based on the interaction of nonlinear processes of higher orders leading to the stabilization (to a finite value) of the amplitude of the chaotic wave process. Mathematically, this means the synchronous “counteraction” of nonlinear processes of even and odd orders in a general vector-matrix model of a complex system, being in the chaotic mode. The implementation of the vector-matrix decomposition by means of computational experiments shows that the model of L.D. Landau describes the scenario of the occurrence of chaotic modes in complex systems quite well. It is noted that the regime of hard self-excitation of nonlinear oscillations in complex systems leads to the appearance of a chaotic attractor in the state-space. Moreover, the proposed vector-matrix model allows one to find more general conditions for the origin and evolution of chaotic wave processes and, as a result, to explain the appearance of coherent nonlinear phenomena in complex systems.