0
2013
Impact Factor

Leonid Bunimovich

Atlanta, GA 30332-0160, USA
ABC Math Program and School of Mathematics, Georgia Institute of Technology

Publications:

 Afraimovich V. S., Bunimovich L. A., Moreno S. V. Dynamical networks: continuous time and general discrete time models 2010, vol. 15, no. 2-3, pp.  127-145 Abstract Dynamical networks are characterized by 1) their topology (structure of the graph of interactions among the elements of a network); 2) the interactions between the elements of the network; 3) the intrinsic (local) dynamics of the elements of the network. A general approach to studying the commulative effect of all these three factors on the evolution of networks of a very general type has been developed in [1]. Besides, in this paper there were obtained sufficient conditions for a global stability (generalized strong synchronization) of networks with an arbitrary topology and the dynamics which is a composition (action of one after another) of a local dynamics of the elements of a network and of the interactions between these elements. Here we extend the results of [1] on global stability (generalized strong synchronization) to the case of a general dynamics in discrete time dynamical networks and to general dynamical networks with continuous time. Keywords: global stability, topological pressure, topological Markov chain, dynamical networks Citation: Afraimovich V. S., Bunimovich L. A., Moreno S. V.,  Dynamical networks: continuous time and general discrete time models, Regular and Chaotic Dynamics, 2010, vol. 15, no. 2-3, pp. 127-145 DOI:10.1134/S1560354710020036
 Bunimovich L. A. Criterion of Absolute Focusing for Focusing Component of Billiards 2009, vol. 14, no. 1, pp.  42-48 Abstract We show that a focusing component $\Gamma$ of the boundary of a billiard table is absolutely focusing iff a sequence of convergents of a continued fraction corresponding to any series of consecutive reflections off $\Gamma$ is monotonic. That is, if $\Gamma$ is absolutely focusing this implies monotonicity of curvatures of the wave fronts in the series of reflections off $\Gamma$ and therefore explains why and how the absolutely focusing components may generate hyperbolicity of billiards. Keywords: billiards, continued fractions, dispersing, focusing, defocusing, absolute focusing Citation: Bunimovich L. A.,  Criterion of Absolute Focusing for Focusing Component of Billiards, Regular and Chaotic Dynamics, 2009, vol. 14, no. 1, pp. 42-48 DOI:10.1134/S1560354709010055
 Bunimovich L. A. Absolute Focusing and Ergodicity of Billiards 2003, vol. 8, no. 1, pp.  15-28 Abstract We show that absolute focusing is a necessary condition for a focusing component to be a part of the boundary of a hyperbolic billiard. A sketch of the proof of a general theorem on hyperbolicity and ergodicity of two-dimensional billiards with all three (focusing, dispersing and neutral) components of the boundary is given. The example of a simply connected domain (container) is given, where a system of $N$ elastically colliding balls is ergodic for any $1 \leqslant N <\infty$. Citation: Bunimovich L. A.,  Absolute Focusing and Ergodicity of Billiards, Regular and Chaotic Dynamics, 2003, vol. 8, no. 1, pp. 15-28 DOI:10.1070/RD2003v008n01ABEH000223