Theoretical models of physical systems with rough chaos

Abstract

The purpose of this review is to present in a unified manner the latest results on mathematical modeling of rough hyperbolic chaos in systems of various physical nature. Main research Methods are the numerical solution of systems of differential equations and partial differential equations, numerical extraction of the phase of oscillatory processes or spatial patterns, calculating of Lyapunov exponents, and studying the mutual arrangement of the stable and unstable manifolds of chaotic trajectories, the calculation of Gaussian curvature of surfaces. These procedures allow to reveal the typical attributes of rough hyperbolic chaos. Results reproduce the studies of already known phenomena, however, their qualitative explanation and quantitative confirmation are given in in a more detailed form, in accordance with the development of ideas about them. Conclusion. Methodologically the proposed review article may be interesting for undergraduate and graduate students in terms of studying the principles of construction and analysis of systems with chaotic behavior.

Keywords

• hyperbolic chaos
• surfaces of negative curvature
• geodesic flow
• turing patterns
• smale –williams attractor