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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: continuous time and general discrete time models
2010, vol. 15, no. 2-3, pp.  127-145
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
Bunimovich L. A.
Criterion of Absolute Focusing for Focusing Component of Billiards
2009, vol. 14, no. 1, pp.  42-48
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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