Positive periodic solutions of competitive kolmogorov diffusion systems with interference constants

In this paper, we consider the positive periodic solutions of multispecies patches connected by discrete diffusion with a within-patch dynamics of periodic Kolmogorov type. A new criterion for existence and globally asymptotic stability of positive periodic solution is established. The results in [1–4] are improved and extended.     

Nonrecursiveness of the integral of a conditionally periodic function

Translated from Matematicheskie Zametki, Vol. 49, No. 6, pp. 138–140, June, 1991.     

Global controllability of conditionally periodic linear system

We use the notation: Rⁿ Is n-dimensional Euclidean space;S _a (x⁰)={x Rⁿ: ¦x-x ⁰ ¦ }; int Q is the interior of setQ Rⁿ. With any linear systemx=A (t)x +B (t) u, x Rⁿ,u R^m, (1)Translated from Matematicheskie Zametki, Vol. 32, No. 2, pp. 169–174, August, 1982.     

Recurrence of the integral of a smooth conditionally periodic function

We prove that the integral of a smooth multifrequency conditionally periodic function with zero mean oscillates. Translated fromMatematicheskie Zametki, Vol. 63, No. 5, pp. 737–748, May, 1998. The author wishes to express his deep gratitude to S. V. Konyagin for fruitful discussions, verification of the results, and the correction of some misprints. This research was supported by the Russian Foundation for Basic Research under grants No. 96-01-00378 and No. 96-15-96072 and by the EFAKOD Foundation.     

Recurrence of the integral of an odd conditionally periodic function

We prove that the integral of a sufficiently smooth odd conditionally periodic function with zero mean and incommensurable frequencies recurs. Furthermore, we obtain the lower and upper bounds for smoothness guaranteeing the recurrence of the integral. Translated fromMatematicheskie Zametki, Vol. 61, No. 4, pp. 570–577, April, 1997. Translated by N. K. Kulman     

Quasi-modes corresponding to the stable type of conditionally periodic motion

Translated from Matematicheskie Zametki, Vol. 55, No. 5, pp 36–42, May, 1994.     

Estimates of convergence rate in Renyi's theorem

Journal of Mathematical Sciences -     

Asymptotic motions and the inversion of the lagrange-dirichlet theorem

The motions of natural mechanical systems which tend to an equilibrium position as time increases without limit are studied. The degenerate case when several frequencies of small oscillations vanish is explained. An existence theorem is proved for asymptotic trajectories on the assumption that the Maclaurin series for the potential energy has the form V₂ + V_m + V_{m + 1} + …(V₈ is a homogeneous form of degree s) and the function V₂ + V_m does not have a local minimum at the equilibrium position. We proved earlier a claim /1, 2/ about the asymptotic motions for the special case when V₂ 0. This theorem is used to solve the question of the existence of asymptotic trajectories in the case of simple and unimodal singularities of the potential energy, for which “canonical” normal forms are known. Similar assertions also hold for the equilibrium positions of gradient dynamic systems. The existence of a trajectory, asymptotic to the equilibrium position, naturally implies that this position is unstable in Lyapunov's sense.     

Recurrence of the integral of a smooth three-frequency conditionally periodic function

We prove V. V. Kozlov's famous conjecture claiming that the integral of an analytic three-frequency conditionally periodic function with zero mean and incommensurable frequencies recurs. For a conditionally periodic function of classC ² onT ⁿ ,n=2, 3, we prove that the integral recurs uniformly with respect to the initial data. Translated fromMatematicheskie Zametki, Vol. 58, No. 5, pp. 723–735, November, 1995.     

Stability of the periodic motions of the vibro-impact systems

In many other papers a new direct method is developed for the study of vibro-impact system stability, a case in which the stability conditions are directly and rapidly obtained. This new method is stated and applied for systems with one or several degrees of freedom whose motion is analysed by Lagrange's equations. Thus, the equations in variations determined from the differential equations are considered, which by applying the boundary conditions lead by integration to the linearized equation in perturbations necessary to write the stability conditions. As example, is analysed in detail a particular case with one degree of freedom realised by the rod-crank mechanism. Also, a general system with two degrees of freedom is completely studied.     

A theorem on the Markov periodic approximation in ergodic theory

One presents a new variant of the theory of periodic approximations of dynamical systems and C^*-algebras, namely the construction for each automorphism of the Lebesgue space of a Markov tower (or adic model) of periodic automorphisms. One gives several examples.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 115, pp. 72–82, 1982.     

Hypoelliptic vector fields and almost periodic motions on the torus Tn

Abstract: In this paper we prove that a hypoelliptic vector fields on the torus Tⁿ may be transformed by a smooth diffeomorphism of torus Tⁿ into a vector field with constant coefficients, which moreover satisfy a Diophantine condition. We also discuss the relation between the hypoelliptic vecttor fields and the almost periodic motions on the torus Tⁿ      

Estimates for periodic solutions of differential equations

A class of infinite delay equations which are per- turbations of finitely delayed equations is considered. Asymptotic estimates are obtained for the solutions from which we get the existence of periodic solutions. We review a few technics for fixed point theorems