Astrakhanskaya st., 83, Saratov, 410012 Russia
Saratov State University
Boev Y. I., Strelkova G. I., Anishchenko V. S.
Estimating dimensions of chaotic attractors using Poincaré recurrences
2015, Vol. 11, No. 3, pp. 475-485
The local theory of Poincaré recurrences is applied to estimate pointwise and information dimensions of chaotic attractors in two-dimensional nonhyperbolic and hyperbolic maps. It is shown that the local pointwise dimension can be defined by calculating the mean recurrence times depending on the return vicinity size. The values of pointwise, information, capacity, and Lyapunov dimensions are compared. It is also analyzed how the structure of attractors can affect the calculation of the dimensions.
Anishchenko V. S., Vadivasova T. E., Strelkova G. I.
Self-sustained oscillations of dynamical and stochastic systems and their mathematical image — an attractor
2010, Vol. 6, No. 1, pp. 107-126
In the present paper autonomous and nonautonomous oscillations of dynamical and stochastic systems are analyzed in the framework of common concepts. The definition of an attractor is introduced for a nonautonomous system. The definitions of self-sustained oscillations and a self-sustained oscillatory system is proposed, that generalize A.A.Andronov’s concept introduced for autonomous systems with one degree of freedom.