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.     

Digitization of nonautonomous control systems

Methods of the theory of nonautonomous differential equations are used to study the extent to which the properties of local null controllability and local feedback stabilizability are preserved when a control system with time-varying coefficients is digitized, e.g., approximated by piecewise autonomous systems on small time subintervals.     

Digitization of nonautonomous control systems

Methods of the theory of nonautonomous differential equations are used to study the extent to which the properties of local null controllability and local feedback stabilizability are preserved when a control system with time-varying coefficients is digitized, e.g., approximated by piecewise autonomous systems on small time subintervals.     

Estimation of solutions of nonautonomous systems

We study the problem of using the direct Lyapunov method to get estimates for solutions of a system of ordinary differential system in general form. Theorems on asymptotic stability and behavior of solutions are proved.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 2, pp. 242–248, February, 1995.     

Morse decompositions of nonautonomous dynamical systems

The global asymptotic behavior of dynamical systems on compact metric spaces can be described via Morse decompositions. Their components, the so-called Morse sets, are obtained as intersections of attractors and repellers of the system. In this paper, new notions of attractor and repeller for nonautonomous dynamical systems are introduced which are designed to establish nonautonomous generalizations of the Morse decomposition. The dynamical properties of these decompositions are discussed, and nonautonomous Lyapunov functions which are constant on the Morse sets are constructed explicitly. Moreover, Morse decompositions of one-dimensional and linear systems are studied.

     

Extinction in nonautonomous competitive Lotka-Volterra systems

It is well known that for the two species autonomous competitive Lotka-Volterra model with no fixed point in the open positive quadrant, one of the species is driven to extinction, whilst the other population stabilises at its own carrying capacity. In this paper we prove a generalisation of this result to nonautonomous systems of arbitrary finite dimension. That is, for the species nonautonomous competitive Lotka-Volterra model, we exhibit simple algebraic criteria on the parameters which guarantee that all but one of the species is driven to extinction. The restriction of the system to the remaining axis is a nonautonomous logistic equation, which has a unique solution that is strictly positive and bounded for all time; see Coleman (Math. Biosci. 45 (1979), 159-173) and Ahmad (Proc. Amer. Math. Soc. 117 (1993), 199-205). We prove in addition that all solutions of the -dimensional system with strictly positive initial conditions are asymptotic to .

     

Extinction of species in nonautonomous Lotka-Volterra systems

A nonautonomous th order Lotka-Volterra system of differential equations is considered. It is shown that if the coefficients satisfy certain inequalities, then any solution with positive components at some point will have all of its last components tend to zero, while the first one will stabilize at a certain solution of a logistic equation.

     

Multiple periodic solutions for nonautonomous Hamiltonian systems

In this paper, by the use of minimax method, we obtain some existence and multiplicity theorems for periodic solutions of nonautonomous Hamiltonian systems with bounded nonlinearity of the type:¶ J [(x)\dot] + ?H(t, x) + e(t) = 0. J \dot x + \nabla H(t, x) + e(t) = 0.     

Spline approximations for linear nonautonomous delay systems

We develop a semi-discrete approximation framework for linear nonautonomous nonhomogeneous functional differential equations of retarded type. The approximation schemes are constructed and convergence results are obtained through the approximation of an associated abstract evolution operator. Computer implementation of the schemes is outlined and a spline-based method included in the framework is constructed. Extensions of the semi-discrete methods to schemes incorporating full discretization and difference equation approximation are also discussed. Numerical results for several examples demonstrating the feasibility of the schemes are presented.     

The extinction in nonautonomous prey-predator Lotka-Volterra systems

1.IntroductionConsiderthenonautonomousprey-predatorLotka-Volterrasystem:whereal30andxZ20;hi(t)andail(t)(i,j=1,2)arecontinuousandboundedfunctionsdefinedon[0,co).Weassumetbxoughoutthispapera'j(t)20forallt20andi,j~1,2andthatthereealstpositiveconstantsfisuchthatTheextinctionofperiodicpredator-preyLotka-Volterrasystemhasbeenstudiedin[l].Inthispaper,ourpurposeistostudytheextinctionofgeneralnonautonomouspredatorpreyLotka-Volterrasystems.Weshallestablishsomesufficientconditionsandnecessary.Thisres…     

Converse Lyapunov theorems for nonautonomous discrete-time systems

This paper presents converse Lyapunov theorems for exponential stability of nonautonomous discrete-time systems with disturbances and free of disturbances, respectively. It is shown that Lyapunov functions exist for discrete-time systems if the systems are exponentially stable. Moreover, in the periodic case, we explicitly construct a Lyapunov function for systems with disturbances. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 61, Optimal Control, 2008.     

Asymptotic stability for nonautonomous dissipative wave systems

We study the problem of asymptotic stability, as time tends to infinity, of solutions of dissipative wave systems, governed by time-dependent nonlinear damping forces and subject to the action of strongly nonlinear potential energies. In the final part of the paper we also consider the case of polyharmonic wave operators and higher-order dissipation terms. © 1996 John Wiley & Sons, Inc.