Almost periodic dynamics of a delayed three-level food-chain model

This paper is concerned with the dynamics of a three-level food-chain model with time delays. By means of Mawhin's continuation theorem of coincidence degree theory, we investigate the existence of positive almost periodic solutions in a three-level food-chain model with time delays. A set of sufficient conditions for the existence of at least one positive almost periodic solution of the model is obtained. In addition, the global asymptotical stability of the almost periodic solution of the above model is also studied. Finally, two examples and simulations are also given to illustrate the main results in this paper.     

Multistability of HNNs with almost periodic stimuli and continuously distributed delays

In this article, we investigate multistability of Hopfield neural networks (HNNs) with almost periodic stimuli and continuously distributed delays. By employing the theory of exponential dichotomy and Schauder's fixed point theorem, sufficient conditions are gained for the existence of 2 ^N almost periodic solutions which lie in invariant regions. Meanwhile, we derive some new criteria for the networks to converge toward these 2 ^N almost periodic solutions and the domain of attraction is also given. The obtained results are new, general and improve corresponding results existing in previous literature.     

Positive periodic solutions for the neutral ratio-dependent predator–prey model

By using a continuation theorem based on coincidence degree theory, we obtain some new sufficient conditions for the existence of positive periodic solutions for the neutral ratio-dependent predator–prey model with Holling type II functional response.     

Existence criteria of periodic oscillations in cyclic gene regulatory networks

In this paper, we consider the existence conditions of periodic oscillations in large-scale cyclic gene regulatory networks. Using the Poincaré–Bendixson theorem for cyclic systems, we first show that the local instability of an equilibrium point implies the existence of periodic oscillations. We then derive the graphical and its equivalent analytic criteria for the existence of periodic oscillations based on local instability analysis. These criteria have a remarkable feature that they can be applied systematically to a large-scale cyclic gene regulatory network consisting of any number of genes. The latter part of this paper is devoted to analyze the relation between an equilibrium point and the biochemical parameters. This leads to an analytic existence criterion that explicitly takes the dependence of the equilibrium state on biochemical parameters into account, which is often overlooked in nonlinear system analyses. In particular, the novel physical quantities that are essential for determining the existence of periodic oscillations are obtained based on the rigorous analytic criterion.     

Existence and exponential stability of periodic solutions for a class of Cohen–Grossberg neural networks with bounded and unbounded delays

This paper is concerned with existence and global exponential stability of periodic solutions for a class of Cohen–Grossberg neural networks with bounded and unbounded delays. By the continuation theorem of coincidence degree theory and differential inequality techniques, we deduce some sufficient conditions ensuring existence as well as global exponential stability of periodic solution. These conditions in our results are milder and less restrictive than that of previous known criteria since the hypothesis of boundedness and differentiability on the activation function are dropped. The theoretical analysis are verified by numerical simulations.     

Existence and stability of periodic solution of a predator–prey model with state-dependent impulsive effects

According to integrated pest management for pests, we investigate the dynamic behavior of a class predator–prey system with state-dependent impulsive effects by releasing natural enemies and spraying pesticide at different thresholds. Using the Poincaré map and the properties of the Lambert W function, we prove the sufficient conditions for the existence and stability of semi-trivial solutions and positive periodic solutions. Numerical results are carried out to illustrate the feasibility of our main results.     

Cyclomonotonicity and stabilizability properties of solutions ofthe difference periodic Riccati equation

The Kalman filter associated with a discrete-time linear T-periodic system is tested. The problem considered is that of selecting an initial covariance matrix such that the periodic filter based on the first T values of the Kalman filter gain is stabilizing. Sufficient conditions are given that hinge on the cyclomonotonicity of the solution of the periodic Riccati equation. Potential applications are found in filter design, quasi-linearization techniques for the periodic Riccati equation, and the design of receding-horizon control strategies for periodic and multirate systems. When specialized to time-invariant systems, the results give rise to new sufficient conditions for the cyclomonotonicity of the solutions of the time-invariant Riccati equation and the existence of periodic stabilizing feedback     

Control of periodic sampling systems subject to actuator saturation

In this paper, the control problem of linear systems with periodic sampling period subject to actuator saturation is considered via delta operator approach. Using periodic Lyapunov function, sufficient conditions of local stabilization for periodic sampling systems are given. By solving an optimization problem, we derive the periodic feedback control laws and the estimate of the domain of attraction. As the saturation function sat(·) belongs to the sector [0,1], sufficient conditions are derived by constructing saturation‐dependent Lyapunov functions to ensure that the periodic sampling system is globally asymptotically stable. A numerical example is given to illustrate the theoretical results proposed in this paper. Copyright © 2014 John Wiley & Sons, Ltd.     

The existence of positive periodic solutions of a class of lotka-volterra type impulsive systems with infinitely distributed delay

In this paper, the existence of positive periodic solutions of a class of periodic Lotka-Volterra type impulsive systems with distributed delays is studied. By using the continuation theorem of coincidence degree theory, a set of easily verifiable sufficient conditions are obtained, which improve and generalize some existing results.     

The numerical simulation of periodic solutions for a predator–prey system

In this article, the existence and global stability of periodic solutions for a semi-ratio-dependent predator–prey system with Holling IV functional response and time delays are investigated. Using coincidence degree theory and Lyapunov method, sufficient conditions for the existence and global stability of periodic solutions are obtained. A numerical simulation is given to illustrate the results.     

The linear periodic output regulation problem

The problem of asymptotic output regulation for linear systems driven by time-varying, T-periodic exosystems is considered in this paper. Necessary and sufficient condition for its solvability based on the existence of periodic solutions of differential Sylvester equations are derived. These conditions constitute a generalization to the periodic case of the celebrated algebraic regulator equations of Francis. A general algorithm for the synthesis of an error-feedback regulator is given. For the case of minimum-phase systems, it is shown that the regulator design can be carried out without the knowledge of the Floquet decomposition of the exosystem, thus extending significantly the applicability of the general result. The more challenging issue of robust regulation by error feedback is also addressed, and solved under a stronger observability condition.     

Existence conditions and properties for the maximal periodic solution of periodic Riccati difference equations

The existence and properties of the maximal symmetric periodic solution of the periodic Riccati difference equation, is analysed for the optimal filtering problem of linear periodic discrete-time systems. Special emphasis is given to systems not necessarily reversible and subject only to a detectability assumption. Necessary and sufficient conditions for the existence and uniqueness of periodic non-negative definite solutions of the periodic Riccati difference equation which gives rise to a stable filter are also established. Furthermore, the convergence of non-negative definite solutions of the Riccati equation is investigated.     

Periodic oscillation of discrete-time bidirectional associative memory neural networks

In this paper, a discrete-time bidirectional associative memory neural networks model is considered. By employing the theory of coincidence degree and using Halanay-type inequality technique we give some sufficient conditions ensuring the existence and globally exponential stability of periodic solutions for the discrete-time bidirectional neural networks. An example with the numerical simulations is provided to show the correctness of our analysis.     

Global attractivity and oscillation in a nonlinear periodic delay difference equation

The existence of a periodic solution for a class of nonlinear delay difference equations with periodic coefficients is established. Some sufficient conditions for the global attractivity and oscillation about the periodic solutions are also obtained.     

On the existence of periodic motion and the maximum sector bounds for absolute stability of nonlinear nonstationary systems

In this contribution we consider the absolute stability problem. We derive necessary and sufficient conditions for the existence of periodic motion using an operator approach. The results yield an efficient algorithm to approximate the maximum sector bounds for absolute stability numerically.     

The difference periodic Ricati equation for the periodic predictionproblem

Gives a comprehensive treatment of several important aspects of the discrete-time periodic Riccati equation (DPRE) arising from the prediction problem for linear discrete-time periodic systems. The authors analyze the symmetric periodic positive semidefinite (SPPS) solution of the DPRE under appropriate assumptions of stabilizability and detectability of the periodic system. Among the results obtained are necessary and sufficient conditions for the existence and uniqueness of the SPPS solution and the stability of the resulting closed-loop system. Some of these results can be seen as extensions of the corresponding results for the time-invariant case; however, a number of them contain contributions to the time-invariant case as well. The paper also gives a numerical algorithm based on an iterative linearization procedure for computing the SPPS solution. The algorithm is a periodic version of Kleinman's algorithm for the time-invariant case     

Existence and multiplicity of positive periodic solutions for a class of higher-dimension functional differential equations with impulses

This paper deals with the existence of multiple periodic solutions for n-dimensional functional differential equations with impulses. By employing the Krasnoselskii fixed point theorem, we obtain some easily verifiable sufficient criteria which extend previous results.     

On control of chaos: Higher periodic orbits

In this paper, we consider control and simultaneous stabilization of chaotic dynamical systems onto higher periodic orbits. Our aim is to control chaos using local linear state, feedback control. After analysing the applicability of such a control, we present numerical techniques for constructing effective controllers. The control is achieved using small, bounded perturbations. We also give a sufficient condition for stabilizing control around higher periodic orbits. The methods proposed are shown to be effective for some examples even in the presence of relatively small random dynamical noise.     

Algorithms and complexity concerning the preemptive scheduling of periodic,real-time tasks on one processor

We investigate the preemptive scheduling of periodic, real-time task systems on one processor. First, we show that when all parameters to the system are integers, we may assume without loss of generality that all preemptions occur at integer time values. We then assume, for the remainder of the paper, that all parameters are indeed integers. We then give, as our main lemma, both necessary and sufficient conditions for a task system to be feasible on one processor. Although these conditions cannot, in general, be tested efficiently (unless P=NP), they do allow us to give efficient algorithms for deciding feasibility on one processor for certain types of periodic task systems. For example, we give a pseudo-polynomial-time algorithm for synchronous systems whose densities are bounded by a fixed constant less than 1. This algorithm represents an exponential improvement over the previous best algorithm. We also give a polynomial-time algorithm for systems having a fixed number of distinct types of tasks. Furthermore, we are able to use our main lemma to show that the feasibility problem for task systems on one processor is co-NP-complete in the strong sence. In order to show this last result, we first show the Simultaneous Congruences Problem to be NP-complete in the strong sense. Both of these last two results answer questions that have been open for ten years. We conclude by showing that for incomplete task systems, that is, task systems in which the start times are not specified, the feasibility problem is ₂ ^p -complete.This work was supported in part by National Science Foundation Grant No. CCR-8711579. Some of these results were presented at the 15th Symposium on Mathematical Foundations of Computer Science, 1990.