No abstract

No abstract

No abstract

We present numerical verification of hyperbolic nature for chaotic attractor in a system of two coupled non-autonomous van der Pol oscillators (Kuznetsov, Phys. Rev. Lett., 95, 144101, 2005). At certain parameter values, in the four-dimensional phase space of the Poincaré map a toroidal domain (a direct product of a circle and a three-dimensional ball) is determined, which is mapped into itself and contains the attractor we analyze. In accordance with the computations, in this absorbing domain the conditions of hyperbolicity are valid, which are formulated in terms of contracting and expanding cones in the tangent spaces (the vector spaces of the small state perturbations).Mathematical theory of chaotic dynamics based on a rigorous axiomatic foundation exploits a concept of hyperbolicity [1][2][3][4][5][6][7][8].An orbit in phase space of a dynamical system is called hyperbolic if there are trajectories approaching exponentially the original orbit, and those departing from it in a similar manner. Moreover, an arbitrary small perturbation of a state on the original orbit must admit representation via a linear combination of the growing and the decaying perturbations.In dissipative systems contracting the space volume the attractors may occur, which consist exclusively of the hyperbolic orbits. These are attractors with strong chaotic properties, like existence of the well-defined invariant SRB-measure, a possibility of description in terms of Markov partitions and symbolic dynamics, positive metric and topological entropy etc. Such hyperbolic (or, more definitely, uniformly hyperbolic) attractors are robust or structurally stable, that means insensitivity of the type of dynamics and of the phase space structure in respect to slight variations of functions and parameters in the evolutionary equations.Although the basic statements of the hyperbolic theory were formulated 40 years ago, no convincing examples of physical systems were introduced with uniform hyperbolic attractors. In textbooks and reviews on nonlinear dynamics, such attractors are represented by artificial mathematical constructions, like Plykin attractor and SmaleWilliams solenoid [1][2][3][4][5][6][7][8]. For realistic systems, in which the chaotic dynamics is mathematically proved, like the Lorenz model [9,10], the strange attractors do not relate to 1

No abstract