Leonid Bunimovich

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


Afraimovich V. S., Bunimovich L. A., Moreno S. V.
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, nos. 2-3, pp. 127-145
Bunimovich L. A.
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
Bunimovich L. A.
Absolute Focusing and Ergodicity of Billiards
2003, vol. 8, no. 1, pp.  15-28
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

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