Morse equation of attractors for nonsmooth dynamical systems

This paper is concerned with a Morse theory of attractors for finite-dimensional nonsmooth dynamical systems described by differential inclusions with upper semi-continuous righthand sides. We first show that all open attractor neighborhoods of an attractor share the same homotopy type. Then based on this basic fact we introduce the concept of homology index for Morse sets and establish Morse inequalities and Morse equation by using smooth Morse–Lyapunov functions.     

Weak shadowing for discrete dynamical systems on nonsmooth manifolds

In J. Math. Anal. Appl. 189 (1995) 409-423, Corless and Pilyugin proved that weak shadowing is a C⁰ generic property in the space of discrete dynamical systems on a compact smooth manifold M. In our paper we give another proof of this theorem which does not assume that M has a differential structure. Moreover, our method also works for systems on some compact metric spaces that are not manifolds, such as a Hilbert cube (or generally, a countably infinite Cartesian product of manifolds with boundary) and a Cantor set.     

Investigation of economical difference systems for dynamical problems in elasticity theory in a class of nonsmooth solutions

Using an explicit central difference scheme, an economical difference system is constructed for the first boundary-value problem of the dynamical theory of elasticity for velocities and stresses. Estimates of the rate of convergence of the difference scheme is obtained under weak hypotheses on the smoothness of solutions to the differential problem.Translated fromVychislitel'naya i Prikladnaya Matematika, No. 69, pp. 16–23, 1989.     

Periodic solutions for N-body type dynamical systems

We prove the existence of infinitely many T-periodic solutions of any assigned period T for a class of dynamical systems which contains the N-body one. We also show a sub class for which such solutions are not of simultaneous collision.     

Second-order dynamical systems of Lagrangian type with dissipation

We give a coordinate-independent version of the smallest set of necessary and sufficient conditions for a given system of second-order ordinary differential equations to be of Lagrangian form with additional dissipative forces.     

Involutive distributions and dynamical systems of second-order type

We investigate the existence of coordinate transformations which bring a given vector field on a manifold equipped with an involutive distribution into the form of a second-order differential equation field with parameters. We define associated connections and we give a coordinate-independent criterion for determining whether the vector field is of quadratic type. Further, we investigate the underlying global bundle structure of the manifold under consideration, induced by the vector field and the involutive distribution.     

Entropy formula of Pesin type for noninvertible random dynamical systems

In this paper we consider random dynamical systems (abbreviated henceforth as RDS's) generated by compositions of random endomorphisms (maybe noninvertible and with singularities) of class of a compact manifold. Entropy formula of Pesin type is proved for such RDS's under some absolute continuity conditions on the associated invariant measures. Received October 17, 1997; in final form January 5, 1998     

A cohomological index of Fuller type for set-valued dynamical systems

In the paper we develop the theory of a cohomological index of the Fuller type detecting periodic orbits of a set-valued dynamical system generated by a differential inclusion or a differential equation without the uniqueness of solutions. The theory presented is applied to establish a general result on the existence of bifurcation of periodic orbits from an equilibrium point of a differential inclusion.     

Completely integrable nonlinear dynamical systems of the Langmuir chains type

The solution of the Cauchy problem for semi-infinite chains of ordinary differential equations, studied first by O. I. Bogoyavlenskii in 1987, is obtained in terms of the decomposition in a multidimensional continuous fraction of Markov vector functions (the resolvent functions) related to the chain of a nonsymmetric operator; the decomposition is performed by the Euler-Jacobi-Perron algorithm. The inverse spectral problem method, based on Lax pairs, on the theory of joint Hermite-Padé approximations, and on the Sturm-Liouville method for finite difference equations is used. Translated fromMatematicheskie Zametki, Vol. 62, No. 4, pp. 588–602, October, 1997. Translated by A. M. Chebotarev     

Nonsymmetric systems on nonsmooth planar domains

We study boundary value problems, in the sense of Dahlberg, for second order constant coefficient strongly elliptic systems. In this class are systems without a variational formulation, viz. the nonsymmetric systems. Various similarities and differences between this subclass and the symmetrizable systems are examined in nonsmooth domains.

     

Existence of stable equilibrium point for dynamical systems of Volterra type

This paper presents sufficient conditions for the existence of a nonnegative and stable equilibrium point of a dynamical system of Volterra type, (1) $\text{(}\text{d}\text{dt}\text{) x}{}_{\mathrm{i}}\text{(t) = ?x}{}_{\mathrm{i}}\text{(t)[f}{}_{\mathrm{i}}\text{(x}{}_1\text{(t),…, x}{}_{\mathrm{n}}\text{(t)) ? q}{}_{\mathrm{i}}\text{], i = 1,…, n}$, for every q = (q₁,…, q_n)^T?Rⁿ. Results of a nonlinear complementarity problem are applied to obtain the conditions. System (1) has a nonnegative and stable equilibrium point if (i) f(x) = (f₁(x),…,f_n(x))^T is a continuous and differentiable M-function and it satisfies a certain surjectivity property, or (ii), f(x) is continuous and strongly monotone on R₊₀ⁿ.     

Fuzzy dynamical systems

A fuzzy dynamical system on an underlying complete, locally compact metric state space X is defined axiomatically in terms of a fuzzy attainability set mapping on X. This definition includes as special cases crisp single and multivalued dynamical systems on X. It is shown that the support of such a fuzzy dynamical system on X is a crisp multivalued dynamical system on X, and that such a fuzzy dynamical system can be considered as a crisp dynamical system on a state space of nonempty compact fuzzy subsets of X. In addition fuzzy trajectories are defined, their existence established and various properties investigated.     

A fast-slow dynamical systems theory for the Kuramoto type phase model

We present a fast-slow dynamical systems theory for the Kuramoto type phase model. When the order parameters are frozen, the fast system consists of independent oscillator equations, whereas the slow system describes the evolution of order parameters. We average out the slow system over the fast manifold to derive a weak form of an amplitude-angle coupled system for the evolution of Kuramoto?s order parameters. This yields the slow evolution of order parameters to be constant values which gives a rigorous proof to Kuramoto?s original assumption in his self-consistent mean-field theory.     

Reconstruction of the spectral type of limiting distributions in dynamical conflict systems

We establish the conditions of reconstruction of pure spectral types (pure point, pure absolutely continuous, or pure singularly continuous spectra) in the limiting distributions of dynamical systems with compositions of alternative conflict. In particular, it is shown that the point spectrum can be reconstructed starting from the states with pure singularly continuous spectra. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 6, pp. 771–784, June, 2007.