Space–Time Chaos,Critical Phenomena,and Bifurcations of Solutions of Infinite-Dimensional Systems of Ordinary Differential Equations

We study infinite-dimensional systems of ordinary differential equations having applications in some popular and important physical problems. The appearance of infinite-dimensional space–time chaos is considered, namely, the bifurcations and critical phenomena that occur in the phase space of the systems and explain some physical problems are described.     

A new mechanism of particle acceleration and rotation numbers

All-Union Scientific-Research Institute of Electroenergetics. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 82, No. 2, pp. 257–267, February, 1990.     

Infinite-dimensional strange attractors and bifurcations in adynamical system with infinitely many degrees of freedom

A system in which infinitely many degrees of freedom play an essential role is considered. Infinite-dimensional strange attractors that lead to spacetime chaos are constructed, and bifurcations of spatially homogeneous solutions are described.All-Union Research Institute of Electrical Energy. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 92, No. 1, pp. 85–91, July, 1992.     

On Generalized Relativistic Billiards in External Force Fields

In this Letter, we study generalized relativistic billiards: as a particle reflects from the boundary of the domain, its velocity is transformed as if the particle underwent an elastic collision with a moving wall, considered within the framework of the special theory of relativity. Inside the domain, the particle moves under the influence of some gravitational and nongravitational force fields.We study both periodic and 'monotone' action of the boundary. We prove that under some general conditions the invariant manifold in the velocity phase space of the generalized billiard, where the point velocity equals the velocity of light, is an exponential attractor, and for an open set of initial conditions the particle energy tends to infinity.     

The nonautonomous function-theoretic center problem

We study the linearizability and stability of a nonautonomous dynamical system in the neighborhood of a neutral fixed point. Our results generalize the classical results of Schröder and Siegel in the case when the linear part of the mapping is an irrational rotation, well known results in the rational case and the fundamental result on the representation of the system as a translation in the neighborhood of a fixed point at infinity.     

The Standard Map and a Probabilistic Conjecture

Statistical properties of trajectories of a class of dynamical systems, which includes the standard map, are studied. In particular a well known conjecture is stated and its natural probabilistic analogue is defined in the framework of ergodic theory of random transformations. The main result is a proof of a probabilistic analogue of this conjecture.