Mathematical Sciences: Nonlocal Bifurcations and Strange Attractors

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9404199 Afraimovich/Chow Depending on parameters, real systems with dissipation and energy pumping can manifest slow nonessential changes as well as abrupt jumps in their dynamical behavior. In order to explain, describe, and predict phenomena of such a kind in a specific applied system, people need to know a mathematical classification of transitions from simple to complex behavior and a theory of evolution of stationary (established) motions during the changes of parameters. The theory of nonlocal bifurcations of strange attractors is an adequate mathematical tool to study changes in conduct of dissipative systems which allows one to control and, in principal, to govern their behavior. Stationary motions (regimes) of real systems correspond to attractors in their mathematical models. Simple regimes correspond to simple attractors while the complex ones correspond to so called strange attractors. Mathematical image of the onset to complex behavior is a bifurcation of a rising of a strange attractor; mathematical image of evolution of complex behavior is a scenarium (or a chain of bifurcations) of evolution of a strange attractor. In the proposed work we are going to study bifurcations leading to appearance of strange attractors and to describe scenaria of their evolution. We propose to apply the developed mathematical technique to such systems as coupled oscillators, laser systems, circuit systems of electrical engineering and others. In particular, we want to study the problem of stochastic synchronization, i.e., similar behavior of dissipatively coupled dissipative individual subsystems. The phenomenon of stochastic synchronization is interesting, for example, for the problem of secure communications. It also plays a fundamental role in the explanation of deterministic behavior of nonequilibrium media. We are going to describe mechanisms of the occurrence of stochastic synchronization in the language of nonlocal bifurcation theory and strange attractors. The problem of appearance and evolution of strange attractors during the changes of parameters in one-parametrical families of smooth vector fields is very important from the mathematical point of view. It also plays a fundamental role in studying of specific dissipative systems from applications. In the proposed work we are going to study the nonlocal codimension one bifurcations on the boundary of the Morse-Smale systems which may lead to the birth of strange attractors and, also scenaria (i.e., chains of bifurcations) which can be responsible for characteristics of strange attractors. In the first problem, we propose to classify behavior of homoclinic and heteroclinic trajectories at the bifurcation moment, single out situations related to appearance of strange attractors and describe the arising attractors in terms of symbolic dynamics. In the second problem, we want to study scenaria of appearance of new positive Lyapunov exponents in strange attractors and investigate their crises which are related to nontransversal intersections of stable and unstable manifolds. We are going to apply expected results to investigate some specific systems (coupled oscillators, laser systems and others) in the form of dissipatively coupled dissipative individual subsystems. We propose to describe mechanisms of occurrence of stochastic synchronization phenomenon in such systems. For identical subsystems, stability of spatially-homogeneous solutions implies stochastic synchronization. For different individual subsystems, a theory of stochastical synchronization based on bifurcations of strange attractors needs to be developed.