Strange attractors and Dynamical Models

One of the remarkable achievements in science in the 20th century is the discovery of dynamical chaos. Using this paradigm, many problems in modern science and engineering can be realistically simulated and analyzed via the tools of nonlinear dynamics. The quest for understanding and explaining chaos in real physical systems has led to the creation of new mathematical techniques. Indeed, there is historical precedence; namely, the quest for understanding oscillations in weakly nonlinear and quasilinear systems has led to Poincare's theory of limit cycles and Lyapunov's stability theory. However, many modern problems associated with systems involving high energies, powers, velocities, etc. must be modeled by multidimensional and strongly nonlinear differential equations (ordinary, partial etc.). The study of such systems has spawned numerous new concepts and terminologies: e.g. hyperbolic sets, symbolic dynamics, homoclinic and heteroclinic orbits, global bifurcations, s trange at t ractors , entropy (topological and metric), Lyapunov's exponents, dimension, capacity, etc. In addition, dynamical chaos has also been characterized and studied by statistical methods, including an extensive application of numerical and experimental analysis of correlation functions and power spectrums. In the creation of new mathematical techniques for studying dynamical chaos, the methods of qualitative theory, bifurcation theory and the theory of strange att ractors have played a particularly important role. In particular, strange a t t ractors have appeared as a mathematical image of dynamical chaos. Strange a t t ractors in finite-dimensional dynamical systems can be divided into three main classes: hyperbolic, Lorenz-type and quasiattractors (abbreviation of