On Periodic Rational Difference Equations of Order k

The present paper is concerned with scalar differential equations of first order which are limit periodic in the independent variable. Some tools provided by the theories of exponential dichotomies and periodic differential equations are applied to prove that, in a generic sense, the existence of a bounded solution implies the existence of a limit periodic solution.

In this paper, the application of the Magnus expansion on periodic time-delayed differential equations is proposed, where an approximation technique of Chebyshev Spectral Continuous Time Approximation (CSCTA) is first used to convert a system of delayed differential equations (DDEs) into a system of ordinary differential equations (ODEs), whose solution are then obtained via the Magnus expansion. The stability and time response of this approach are investigated on two examples and compared with known results in the literature.

linear periodic impulsive differential equations are considered. The impulse effect instants are assumed to satisfy the average dwell-time condition (the ADT condition). The stability problem is reduced to studying the stability of an auxiliary abstract impulsive differential equation. This is a perturbed periodic impulsive differential equation, which considerably simplifies the construction of a Lyapunov function. Sufficient conditions for the asymptotic stability of abstract linear periodic impulsive differential equations are obtained. It is shown that the ADT conditions lead to less conservative dwell-time estimates guaranteeing asymptotic stability. Bibliography: 24 titles.

Periodic differential difference equations

By means of piecewise continuous functions which are modifications of classifical Lyapunov’s functions, some sufficient conditions for asymptotic stability in the large of solutions of almost periodic impulsive differential equations are presented.

In this paper, the periodic stochastic differential equations are studied. By applying the theory of Lyapunov?s second method, contraction mapping principle and establishing new lemmas, the existence and uniqueness of stochastic periodic solutions to stochastic periodic differential equations are obtained. Moreover, several examples are introduced to illustrate our theoretical results.

In this paper we consider periodic functional differential and integral equations. Massera proved that the existence of a bounded solution implies the existence of a periodic solution for linear (inhomogeneous) ordinary differential equations. Chow proved a similar result for linear functional differential equations with finite delay. We give a new proof for Chow's result that works for linear functional differential equations with infinite delay and also for integral equations. We also give examples that do not satisfy uniform boundedness or uniform ultimate boundedness, the conditions frequently used to prove that there is a periodic solution.

As is proved earlier (the Massera theorem), the first-order scalar periodic ordinary differential equation does not have strongly irregular periodic solutions (solutions with a period incommensurable with the period of the equation). For difference equations with discrete time, strong irregularity means that the equation period and the period of its solution are relatively prime numbers. It is known that in the case of discrete equations, the mentioned result has no complete analog.The purpose of this paper is to investigate the possibility of realizing an analog of the Massera theorem for certain classes of difference equations. To do this, we consider the class of linear difference equations. It is proved that a linear nonhomogeneous non-stationary periodic discrete equation of the first order does not have strongly irregular non-stationary periodic solutions.

By using critical point theory in combination with periodic approximations, we obtain novel sufficient conditions for the existence of nontrivial homoclinic solutions for a class of periodic partial difference equations with sign-changing mixed nonlinearities. Notably, in certain specific cases, a necessary and sufficient condition is also established.

The problem of controlling small state deviations from a periodic path generated by the solution to an autonomous optimal control problem with periodic boundary conditions is considered by solving the associated infinite-time accessary minimum problem. Some of the properties of the resulting solution to the periodic Riccati differential equation with periodic boundary condition are derived using Floquet theorem. Given a controllability condition, the periodic Riccati solution required for the periodic regulator is characterized in terms of the eigenvalues and periodic eigenvectors associated with the monodromy matrix, the transition matrix associated with the perturbed Hamiltonian system evaluated over the period.

SummaryIn this paper, we consider a linear quadratic regulator control problem for spacecraft rendezvous in an elliptical orbit. A new spacecraft rendezvous model is established. On the basis of this model, a linear quadratic regulator control problem is formulated. A parametric Lyapunov differential equation approach is used to design a state feedback controller such that the resulting closed‐loop system is asymptotically stable, and the performance index is minimized. By an appropriate choice of the value of a parameter, an approximate state feedback controller is obtained from a solution to the periodic Lyapunov differential equation, where the periodic Lyapunov differential equation is solved on the basis of a new numerical algorithm. The spacecraft rendezvous mission under the controller obtained will be accomplished successfully. Several illustrative examples are provided to show the effectiveness of the proposed control design method. Copyright © 2014 John Wiley & Sons, Ltd.

This note is an addendum to the results reported in Dragan (2014, IMA J. Math. Control Inf.) concerning the numerical computation of the stabilizing solution of periodic game theoretic Riccati differential equation arising in some stochastic control problems. A globally convergent iterative algorithm is proposed for this purpose. The performances of the proposed algorithm are illustrated on some numerical examples.

Using the linear noise approximation, we approach to the problem of evaluation and prediction of behavior of autonomous oscillatory genetic networks fluctuated by intrinsic noise. In this approach, the state covariance of the networks represents all the information concerning the fluctuation by intrinsic noise, and further, when the genetic network is an autonomous oscillatory system, the state covariance obeys a periodic Lyapunov differential equation that has generally unbounded solutions. Based on the analysis of stability and periodicity properties of the solution to the periodic Lyapunov differential equation, we propose two measures for evaluating intrinsic noise. Circadian clock is a typical example of autonoumous oscillatory genetic networks. Focusing on the simple core model (Gonze and Goldbeter, 2006) for circadian clocks and using the proposed measures, we evaluate robust performances concerning periodicity and amplitude of oscillations in the model fluctuated by intrinsic noise by changing system parameters.

As an application of the Schauder Fixed Point Theorem, an existence theory is developed for second order nonlinear differential equations with Stieltjes derivative related to a left-continuous, nondecreasing function. By the method of lower and upper solutions, under a Nagumo-type assumption we get a very general result which can be further applied to deduce the existence of solutions for second order nonlinear problems in the settings of impulsive differential equations, time scale analysis or generalized differential equations.

Periodic Differential Equations.