The Current Model of Cardiac Arrhythmia Based on an Autonomous Dynamical System with A Smale - Williams Hyperbolic Attractor
Journal Title: Biomedical Journal of Scientific & Technical Research (BJSTR) - Year 2018, Vol 10, Issue 5
Abstract
Autonomous hyperbolic attractors were introduced: by Smale and Anosov, Alekseev, Williams, Sinai, Ruelle, and other Neuhausen, about 70 years ago. Traditional examples of uniformly hyperbolic attractors are discrete-time geometric models such as the Smale - Williams attractor or the Plykin attractor presented in Figure 1 [1]. Initially, they were expected to be adequate for many real-world situations of chaotic behavior, such as hydrodynamic turbulence, etc. As time passed, it became clear that the early hyperbolic theory was too narrow to include most chaotic systems of interest to applications. So, the efforts of mathematicians were redirected to generalizations of the theory corresponding to wider classes of systems. For example, we developed the notion of nonuniform hyperbolic attractors, partially hyperbolic systems, quasihyperbolicity or singular hyperbolic attractor, quasi-attractor, etc. Uniformly hyperbolic attractor is presented in Figure 2, is an attractive object in the phase space of a dissipative dynamic system consisting exclusively of saddle trajectories.Their stable and unstable manifolds have the same dimension for all trajectories on the attractor; they should not touch; only intersections with non-zero angles are allowed. The hyperbolic nature of the attractors can be verified using the cone criterion. The Figure 3 below shows this for a discrete time system (iterated map). Cones of expanding and compressing infinitesimal perturbation vectors must exist at each point of the region containing the attractor, which smoothly depends on the position. The image of the expanding cone shall be placed inside the expanding cone at the point of the image, and the prototype of the connecting cone shall be placed inside the dosing cone at the point of Providence. For flows, the same considerations apply in terms of the Poan- care map. Geometric constructions of hyperbolic attractors of the Smale-Williams Attractor. The mathematical theory of chaos, based on a strict axiomatic Foundation, deals with strange attractors of hyperbolic type Figure 4. In such an attractor, all orbits belonging to it in the phase space of the saddle system, with stable and unstable varieties (invariant sets composed of trajectories approaching the original in forward or reverse time) intersect transversally, i.e. without touching.Unfortunately, known physical systems, such as simple chaos generators, nonlinear oscillators with periodic action and others, do not belong to the class of systems with hyperbolic attractors. Chaos in them is usually associated with the so-called quasi-tractor, which, along with chaotic trajectories includes stable orbits of a large period (not distinguishable in solving equations on the computer because of the narrowness of the regions of attraction). Hyperbolic strange attractors are robust (structurally stable). This means the insensitivity of the nature of movements and the relative position of trajectories in the phase space with respect to the variation of the equations of the system. In contrast to the hyperbolic attractor, quasi-attractors are characterized by a sensitive dependence of the dynamic's details on the parameters. This is obviously undesirable for potential applications of chaos, such as communication systems, signal masking, etc. Thus, from both a fundamental and applied point of view, it is interesting to implement hyperbolic chaos in physical systems.
Authors and Affiliations
Sergey Belyakin
Keywords
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- EP ID EP592327
- DOI 10.26717/BJSTR.2018.10.002005
- Views 157
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