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    S. Astakhov

    Astrakhanskaya st., 83, 410012, Saratov, Russia
    Saratov State University


    Anishchenko V. S., Semenova N. I., Astakhov S. V., Boev Y. I.
    The concept of a local fractal dimension has been introduced in the framework of the average Poincaré recurrence time numerical analysis in an $\varepsilon$-vicinity of a certain point. Lozi and Hénon maps have been considered. It has been shown that in case of Lozi map the local dimension weakly depends on the point on the attractor and its value is close to the fractal dimension of the attractor. In case of a quasi attractor observed in both Hénon and Feugenbaum systems the local dimension significantly depends on both the diameter and the location of the $\varepsilon$-vicinity. The reason of this strong dependency is high non-homogenity of a quasi-attractor which is typical for non-hyperbolic chaotic attractors.
    Keywords: Poincaré recurrence, attractor dimension
    Citation: Anishchenko V. S., Semenova N. I., Astakhov S. V., Boev Y. I.,  Poincaré recurrences time and local dimension of chaotic attractors, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 3, pp.  449-460
    Anishchenko V. S., Astakhov S. V., Boev Y. I., Kurths J.
    Statistical properties of Poincaré recurrences in a two-dimensional map with chaotic non-strange attractor have been studied in numerical simulations. A local and a global approaches were analyzed in the framework of the considered problem. It has been shown that the local approach corresponds to Kac’s theorem including the case of a noisy system in certain conditions which have been established. Numerical proof of theoretical results for a global approach as well as the Afraimovich–Pesin dimension calculation are presented.
    Keywords: Poincaré recurrence, attractor dimension, Afraimovich–Pesin dimension
    Citation: Anishchenko V. S., Astakhov S. V., Boev Y. I., Kurths J.,  Poincaré recurrences in a system with non-strange chaotic attractor, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 1, pp.  29-41
    Anishchenko V. S., Astakhov S. V., Vadivasova T. E., Feoktistov A. V.
    The effect of synchonization has been studied in a system of two coupled Van der Pol oscillators under external harmonic force. The bifurcation analysis has been carried out using the phase approach. The mechanisms of complete and partial synchronization have been established. The main type of bifurcation described in the paper is the saddle-node bifurcation of invariant curves that corresponds to the saddle-node bifurcation of two-dimensional tori in the complete system of differential equations for the dynamical system under study. We illustrate the bifurcational mechanisms obtained from numerical experiment by the results of physical experiment. The synchronization phenomenon in the vicinity of resonances on a torus with winding numbers 1 : 1 and 1 : 3 is considered in the physical experiment.
    Keywords: limit cycle, torus, saddle-node bifurcation, synchronization
    Citation: Anishchenko V. S., Astakhov S. V., Vadivasova T. E., Feoktistov A. V.,  Numerical and experimental study of external synchronization of two-frequency oscillations, Rus. J. Nonlin. Dyn., 2009, Vol. 5, No. 2, pp.  237-252

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