Stable equilibrium of systems of generalized Volterra type

In this paper, the global stability of systems of the generalized Volterra type:

$\text{dx}{}_{\mathrm{i}}\text{dt}\text{=x}{}_{\mathrm{i}}\text{∑}\text{j=1}\text{n}\text{a}{}_{\mathrm{ij}}\text{x}{}_{\mathrm{j}}\text{+b}{}_{\mathrm{i}}\text{, i=1,…,n.}$

is investigated. It is shown that a stable equilibrium point of (1) must be a solution of corresponding linear complementarity problem. Further, when the structure of (1) is restricted, the known stability condition is proved to be necessary and sufficient for the existence of a stable equilibrium point for every b?Rⁿ. Some structures satisfying these restrictions are given.     

Existence of uniformly stable solutions of nonautonomous discontinuous dynamical systems

We are concerned here with the existence of uniformly Lyapunov stable integrable solution of linear and nonlinear nonautonomous discontinuous dynamical systems.     

Exponentially asymptotically stable dynamical systems

In this paper, we give some sufficient conditions which guarantee that the zero solution of the differential equation with two delayed terms x(t) = a(t)x(t-p(t)) - b(t)x(t-r(t))is uniformly stable     

Existence of backwards-compact pullback attractors for non-autonomous lattice dynamical systems

A pullback attractor is called backward compact if the union of attractors over the past time is pre-compact. We show that this kind of attractor exists for the first-order non-autonomous lattice dynamical system when the external force is backwards tempered and backwards asymptotically tail-null.     

Subharmonics near an equilibrium point for hamitonian systems

This work is devoted to the study of subharmonic solutions near an equilibrium for certain Hamiltonian systems. We impose a weak condition on the Floquet exponents of the linearized system, and a superquadratic condition on the higher order term. This last condition is reduced to the center manifold. This research was supported in part by the Air Force Office of Scientific Research under Grant AFOSR-87-0202.     

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.     

Poisson stable motions of monotone nonautonomous dynamical systems

In this paper, we study the Poisson stability(in particular, stationarity, periodicity, quasiperiodicity, Bohr almost periodicity, almost automorphy, recurrence in the sense of Birkhoff, Levitan almost periodicity, pseudo periodicity, almost recurrence in the sense of Bebutov, pseudo recurrence, Poisson stability) of motions for monotone nonautonomous dynamical systems and of solutions for some classes of monotone nonautonomous evolution equations(ODEs, FDEs and parabolic PDEs). As a byproduct, some of our results indicate that all the trajectories of monotone systems converge to the above mentioned Poisson stable trajectories under some suitable conditions, which is interesting in its own right for monotone dynamics.     

Age-dependent diffusive Lotka–Volterra type systems

In this paper it is shown that the sub-supersolution method works for age-dependent diffusive nonlinear systems with non-local initial conditions. As an application, we prove the existence and uniqueness of positive solutions for a kind of Lotka–Volterra system, as well as the blow-up in finite time in a particular case.     

Existence and genericity problems for dynamical systems with nonzero Lyapunov exponents

This is a survey-type article whose goal is to review some recent results on existence of hyperbolic dynamical systems with discrete time on compact smooth manifolds and on coexistence of hyperbolic and non-hyperbolic behavior. It also discusses two approaches to the study of genericity of systems with nonzero Lyapunov exponents.      

Existence theorems of global solutions for nonlinear Volterra type integral equations in Banach spaces

In this paper, first a new fixed point theorem is established, and then, by the use of it, the existence theorems of global solutions for nonlinear Volterra type integral equations in Banach spaces are investigated. The results obtained in this paper generalize and improve the results corresponding to those obtained by others.     

Existence for a doubly nonlinear Volterra equation

We consider a doubly nonlinear Volterra equation involving a nonsmooth kernel and two possibly degenerate monotone operators. By exploiting an implicit time-discretization procedure, we obtain the existence of a global strong solution and extend to the nonlocal in time situation some former results by Colli [P. Colli, On some doubly nonlinear evolution equations in Banach spaces, Japan J. Indust. Appl. Math. 9 (2) (1992) 181-203].     

Existence of solutions for systems of generalized quasi‐equilibrium problems

In this paper, first of all, we consider a generalized game in choice form with 2 constraints and its corresponding equilibrium in choice. We assert new conditions under which the equilibrium in choice exists. As a consequence, we establish the existence of the equilibrium for generalized abstract economies. Then, we apply the obtained theorems to prove the existence of solutions for systems of quasi‐equilibrium problems. We do this by considering new hypotheses for the properties of the involved correspondences. This approach leads us to results which differ a lot from the ones existing in literature.     

Existence of solution for nonlinear Volterra integral equations

We prove an existence theorem for a class of nonlinear Volterra integral equations.     

Discrete dynamical systems with invariant asymptotically stable toroidal manifold

We obtain conditions for asymptotic stability of quasiperiodic trajectories of discrete dynamical systems in the case of infinite-dimensional Banach space. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 4, pp. 466–471, April, 1999.     

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.     

Existence of periodic solutions for the Lotka-Volterra type systems

In this paper we prove the existence of nonstationary periodic solutions of delay Lotka-Volterra equations. In the proofs we use the S¹-degree due to Dylawerski et al. [G. Dylawerski, K. Geba, J. Jodel, W. Marzantowicz, An S¹-equivariant degree and the Fuller index, Ann. Polon. Math. 63 (1991) 243-280].     

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     

Local stable and unstable sets for positive entropy C1 dynamical systems

Science China Mathematics - For any C1 diffeomorphism on a smooth compact Riemannian manifold that admits an ergodic measure with positive entropy, a lower bound of the Hausdorff dimension for the...     

Assessing alternative control strategies for systems with asymptotically stable equilibrium positions

In a number of optimal control applications, it is possible to arrange control guided only by an analysis of a system’s dynamic properties. These controls are customarily referred to as alternatives to those that satisfy the Pontryagin maximum principle. This work considers autonomous systems of ordinary differential equations with a terminal objective functional that at each fixed value of the control parameter have unique and asymptotically stable equilibrium positions. It is shown that the problem of arranging alternative control can then be reduced to a finite-dimensional problem of mathematical programming. An estimate of the alternative control error in terms of the objective functional is obtained. Sufficient conditions for obtaining this estimate are given. A mathematical model of leukemia therapy is considered as an example.