Periodic Differential Systems Research Articles

A periodic delay differential system for mosquito control with Wolbachia incompatible insect technique

Wolbachia incompatible insect technique (IIT) depends on releasing male-only Wolbachia-infected mosquitoes to reduce/eradicate wild type mosquito populations. Mathematical models can be utilized to help investigate the altered mosquito population dynamics under IIT and propose the appropriate releasing strategies. Incorporating seasonal temperature variations and larval competition, a delayed periodic stage structured model is formulated, which is essentially different from existing models since additional density-dependent death rate due to larval competition introduces a density-dependent survival probability of larvae. This novel feature brings new challenges to mathematical analysis and further model extension in studying the dynamics of mosquito borne diseases (MBDs). By assuming continuous proportional releasing strategies, threshold dynamics in terms of the basic offspring number R0 are established by employing the dynamical system approach. The mosquito population size is shown to either go extinction if R0<1 or stabilize at a periodic pattern when R0>1. Then, we numerically explore the seasonal effects on R0 and the impact of larval competition and released amount of Wolbachia-infected males under two levels of mosquito control objectives, i.e., population eradication and reduction. Overestimates of R0 are observed if seasonality is ignored. The ratio of released Wolbachia-infected males to wild type males (the overflooding ratio, pw) and larval competition are shown to play positive roles in reducing both the required time for population eradication and ultimate adult peak abundance. In particular, the initial population size is negatively correlated with the eradication time, which can be shorten to a minimum of 5 years. In summary, IIT alone may be difficult to eradicate/reduce the mosquito population. Other conventional mosquito control approaches such as spraying larvicides should be used in combination to efficiently control mosquitoes.

A novel computing stochastic algorithm to solve the nonlinear singular periodic boundary value problems

In this work, a class of singular periodic nonlinear differential systems (SP-NDS) in nuclear physics is numerically treated by using a novel computing approach based on the Gudermannian neural networks (GNNs) optimized by the mutual strength of global and local search abilities of genetic algorithms (GA) and sequential quadratic programming (SQP), i.e. GNNs-GA-SQP. The stimulation of offering this numerical computing work comes from the aim of introducing a consistent framework that has an effective structure of GNNs optimized with the backgrounds of soft computing to tackle such thought-provoking systems. Two different problems based on the SPNDS in nuclear physics will be examined to check the proficiency, robustness and constancy of the GNNs-GA-SQP. The outcomes obtained through GNNs-GA-SQP are compared with the true results to find the worth of designed procedures based on the multiple trials.

A Siegel theorem for periodic difference systems

In this paper, we extend the Siegel's theorem for analytically reducing periodic difference system to a linear one under certain small-divisor condition.

Stochastic Periodic Solution and Permanence of a Holling–Leslie Predator-Prey System with Impulsive Effects

Considering the environmental effects, a Holling–Leslie predator-prey system with impulsive and stochastic disturbance is proposed in this paper. Firstly, we prove that existence of periodic solution, the mean time boundness of variables is found by integral inequality, and we establish some sufficient conditions assuring the existencle of periodic Markovian process. Secondly, for periodic impulsive differential equation and system, it is different from previous research methods, by defining three restrictive conditions, we study the extinction and permanence in the mean of all species. Thirdly, by stochastic analysis method, we investigate the stochastic permanence of the system. Finally, some numerical simulations are given to illustrate the main results.

Modeling seasonal variation for mosquito-borne disease in the tropical monsoon environment

Mosquitoes play an important role in the spread of mosquito-borne diseases. Considering the sensitivity of mosquitoes’ aquatic stage to the seasonal shift, in this paper, we present a seasonally forced mosquito-borne epidemic model by incorporating mosquitoes’ aquatic stage (eggs, larvae, and pupae) and seasonal shift factor, which is a periodic discontinuous differential system. Firstly, some sufficient conditions for the existence and uniqueness of a disease-free solution are obtained. Further, we define the basic reproduction number mathcal{R}_{0}, and obtain the stability of the disease-free solution when mathcal{R}_{0} is less than one. And, if mathcal{R}_{0} is greater than one, the mosquito-borne disease is uniformly persistent and the model admits a positive periodic solution. Finally, some numerical simulations are given to illustrate the main theoretical results. In addition, simulation results also imply that ignoring the effects of seasonal succession can overestimate or underestimate mosquito-borne disease trends.

Basic reproduction ratios for almost periodic compartmental models with time delay

The theory of basic reproduction ratio R0 is established for a large class of almost periodic and time-delayed compartmental population models. We first present some dynamical properties for linear almost periodic functional differential systems. By using the product space and evolution semigroup approach, we then prove that R0−1 has the same sign as the exponential growth bound of an associated linear system. As an application, we apply the developed theory to an almost periodic SEIR model with an incubation period and obtain a threshold result on its global dynamics in terms of R0. Finally, we present some numerical simulations. Numerical simulations indicate that prolonging the length of incubation period is beneficial for the control of the disease. In addition, a simple model shows that the basic reproduction ratio may be underestimated or overestimated if an almost periodic coefficient is approximated by a periodic one.

Efficient active chatter mitigation for boring operation by electromagnetic actuator using optimal fractional order PDλ controller

Abstract Due to the factor of limited rigidity of boring bars, small depths of cutting are normally applied for chatter free machining. Stability lobe diagrams can truly represent that limit. However, due to the presence of manufacturing inaccuracies like ovality and eccentricity, these limits are considerably changed. Hence, considering these during modelling the cutting process, stability analysis and controller design is a novel idea, hence implemented in the present work. Dynamic modelling of boring process is presented in detail using a 3-DOF model. Dynamics of such systems is represented using delay differential equations with time periodic coefficients. The system stability is enhanced with active control techniques. The closed loop system considering fractional order PDλ in the loop is a nonlinear time periodic delay differential equation system. Any systematic controller synthesis process for such systems is rarely available in literature. It is observed that a fractional order PDλ controller designed by using combination of pseudo spectral and response optimization techniques is highly efficient in terms of the requirement of low amplitude of the peak force and simplicity of implementation. Transient vibrations can also be quenched in a limited period of time by using this controller. Different control techniques available in literature (H∞ Loop shaping and PD control) are tested and compared to the proposed controller for enhancing the material removal rates and surface finish of the workpiece. By using the active chatter control, the chatter can be efficiently reduced and the material removal rate can be increased. The critical depth of cut is increased from 0.2 mm (open loop) to 0.6 mm (closed loop) with a limited actuator size in the case study.

Asymptotically almost periodic solutions of limit periodic difference systems with coefficients from commutative groups

We study the behaviour of solutions of limit periodic difference systems over (infinite) fields with absolute values. The considered systems are described by the coefficient matrices that belong to commutative groups whose boundedness is not required. In particular, we are interested in special systems with solutions which vanish at infinity or which are not asymptotically almost periodic. We obtain a transparent condition on the matrix groups which ensures that the special systems form a dense subset in the space of all considered systems, i.e. that, in any neighbourhood of any considered limit periodic system, there exists asystem which have non-asymptotically almost periodic or vanishing solutions. The presented results improve and extend known ones.

Exponential Attractivity in a Delayed Almost Periodic Differential Neoclassical Growth System

This paper investigates a delayed coupled almost periodic differential neoclassical growth system. By using the theory of dichotomy and differential inequality techniques, a new set of sufficient conditions is derived to guarantee the existence and exponential attractivity of almost periodic solutions for the addressed system. In addition, an example is given to exhibit the efficiency of the theoretical results. The obtained results are essentially new and extend previously known results.

Dynamics of epidemic models with asymptomatic infection and seasonal succession.

In this paper, we consider a compartmental SIRS epidemic model with asymptomatic infection and seasonal succession, which is a periodic discontinuous differential system. The basic reproduction number R0 is defined and evaluated directly for this model, and uniform persistence of the disease and threshold dynamics are obtained. Specially, global dynamics of the model without seasonal force are studied. It is shown that the model has only a disease-free equilibrium which is globally stable if R0 ≤ 1, and as R0 > 1 the disease-free equilibrium is unstable and there is an endemic equilibrium, which is globally stable if the recovering rates of asymptomatic infectives and symptomatic infectives are close. These theoretical results provide an intuitive basis for understanding that the asymptomatically infective individuals and the seasonal disease transmission promote the evolution of the epidemic, which allow us to predict the outcomes of control strategies during the course of the epidemic.

Analysis of an age structured model for tick populations subject to seasonal effects

We investigate an age-structured hyperbolic equation model by allowing the birth and death functions to be density dependent and periodic in time with the consideration of seasonal effects. By studying the integral form solution of this general hyperbolic equation obtained through the method of integration along characteristics, we give a detailed proof of the uniqueness and existence of the solution in light of the contraction mapping theorem. With additional biologically natural assumptions, using the tick population growth as a motivating example, we derive an age-structured model with time-dependent periodic maturation delays, which is quite different from the existing population models with time-independent maturation delays. For this periodic differential system with seasonal delays, the basic reproduction number R0 is defined as the spectral radius of the next generation operator. Then, we show the tick population tends to die out when R0<1 while remains persistent if R0>1. When there is no intra-specific competition among immature individuals due to the sufficient availability of immature tick hosts, the global stability of the positive periodic state for the whole model system of four delay differential equations can be obtained with the observation that a scalar subsystem for the adult stage size can be decoupled. The challenge for the proof of such a global stability result can be overcome by introducing a new phase space, based on which, a periodic solution semiflow can be defined which is eventually strongly monotone and strictly subhomogeneous.

THE NEW METHOD FOR THE SEARCHING PERIODIC SOLUTIONS OF PERIODIC DIFFERENTIAL SYSTEMS

In the paper we are giving the new method for searching periodic solutions of periodic differential systems. For this we construct a differential system with the same Reflecting Function as the Reflecting Function of the given system and with a known periodic solution. Then the initial data of the periodic solutions of this two systems coincide. In such a way the problem of existance periodic solutions goes to the Cauchy problem.

Averaging methods of arbitrary order, periodic solutions and integrability

In this paper we provide an arbitrary order averaging theory for higher dimensional periodic analytic differential systems. This result extends and improves results on averaging theory of periodic analytic differential systems, and it unifies many different kinds of averaging methods.Applying our theory to autonomous analytic differential systems, we obtain some conditions on the existence of limit cycles and integrability.For polynomial differential systems with a singularity at the origin having a pair of pure imaginary eigenvalues, we prove that there always exists a positive number N such that if its first N averaging functions vanish, then all averaging functions vanish, and consequently there exists a neighborhood of the origin filled with periodic orbits. Consequently if all averaging functions vanish, the origin is a center for n=2.Furthermore, in a punctured neighborhood of the origin, the system is C∞ completely integrable for n>2 provided that each periodic orbit has a trivial holonomy.Finally we develop an averaging theory for studying limit cycle bifurcations and the integrability of planar polynomial differential systems near a nilpotent monodromic singularity and some degenerate monodromic singularities.

Robust H 2 / H ∞ control for periodic stochastic difference systems with mutiplicative noise

The aim of this study is to investigate the discrete-time periodic stochastic mixed H 2/H ∞ control problem with state and external disturbance dependent noises. Under the assumption of 𝒦 N -exact observability, the authors present a sufficient condition for the existence of mixed H 2/H ∞ control for stochastic periodic difference systems, which transforms the H 2/H ∞ controller design into solving four coupled matrix-valued equations.

Floquet Theory for Stability of Differential Algebraic Equations.

Motivated by a great useful of some types of non autonomous differential algebraic&#x0D; equation systems ( so called strangeness free ) and its applied in different scientific&#x0D; fields, we present several new results for studying such systems by classical Floquet&#x0D; Theory, which we extended from linear periodic ordinary differential equation systems&#x0D; into linear periodic differential algebraic equation systems. For both systems we&#x0D; investigate that they have the same Floquet exponents. The relation between monodromy&#x0D; matrices of both systems is also presented. Classification of solution according to the&#x0D; nature of Floquet exponent is established. Then according to these results, we study the&#x0D; stability and bifurcation phenomenon of our differential algebraic equation systems.

Nash-type equilibria and periodic solutions to nonvariational systems

Abstract The paper deals with variational properties of fixed points for contraction-type operators. Under suitable conditions, the unique fixed point of a vector-valued operator is a Nash-type equilibrium of the corresponding energy functionals. This is achieved by an iterative scheme based on Ekeland's variational principle. An application to periodic solutions for second order differential systems is given to illustrate the theory.

On the Minimum Differential Feedback for Time-Correlated MIMO Rayleigh Block-Fading Channels

In this paper, we consider a general multiple input multiple output (MIMO) system with channel state information (CSI) feedback over time-correlated Rayleigh block-fading channels. Specifically, we first derive the closed-form expression of the minimum differential feedback rate to achieve the maximum erdodic capacity in the presence of channel estimation errors and quantization distortion at the receiver. With the feedback-channel transmission rate constraint, in the periodic feedback system, we further investigate the relationship of the ergodic capacity and the differential feedback interval, and we find by theoretical analysis that there exists an optimal differential feedback interval to maximize ergodic capacity. Finally, analytical results are verified through simulations in a practical periodic differential feedback system using Lloyd's quantization algorithm.

Almost periodic skew-symmetric differential systems

We analyse solutions of almost periodic skew-symmetric homogeneous linear differential systems. We prove that in any neighbourhood of such a system there exists an almost periodic skew-symmetric system which does not possess any non-trivial almost periodic solution.

A second order analysis of the periodic solutions for nonlinear periodic differential systems with a small parameter

We deal with nonlinear T -periodic differential systems depending on a small parameter. The unperturbed system has an invariant manifold of periodic solutions. We provide the expressions of the bifurcation functions up to second order in the small parameter in order that their simple zeros are initial values of the periodic solutions that persist after the perturbation. In the end two applications are done. The key tool for proving the main result is the Lyapunov–Schmidt reduction method applied to the T -Poincaré–Andronov mapping.

Time symmetries and in-period transformations

A theorem has been proved which gives us an opportunity to investigate the properties of in-period transformations of periodic differential systems.