Existence Of Nontrivial Periodic Solutions Research Articles

Non-smooth dynamics of a SIR model with nonlinear state-dependent impulsive control.

The classic SIR model is often used to evaluate the effectiveness of controlling infectious diseases. Moreover, when adopting strategies such as isolation and vaccination based on changes in the size of susceptible populations and other states, it is necessary to develop a non-smooth SIR infectious disease model. To do this, we first add a non-linear term to the classical SIR model to describe the impact of limited medical resources or treatment capacity on infectious disease transmission, and then involve the state-dependent impulsive feedback control, which is determined by the convex combinations of the size of the susceptible population and its growth rates, into the model. Further, the analytical methods have been developed to address the existence of non-trivial periodic solutions, the existence and stability of a disease-free periodic solution (DFPS) and its bifurcation. Based on the properties of the established Poincaré map, we conclude that DFPS exists, which is stable under certain conditions. In particular, we show that the non-trivial order-1 periodic solutions may exist and a non-trivial order-$ k $ ($ k\geq 1 $) periodic solution in some special cases may not exist. Moreover, the transcritical bifurcations around the DFPS with respect to the parameters $ p $ and $ AT $ have been investigated by employing the bifurcation theorems of discrete maps.

Dynamics and Optimal Control of a Monod–Haldane Predator–Prey System with Mixed Harvesting

This paper investigates the dynamics and optimal control of the Monod–Haldane predator–prey system with mixed harvesting that combines both continuous and impulsive harvestings. The periodic solution of the prey-free is studied and the local stability condition is obtained. The boundedness of solutions, the permanence of the system, and the existence of nontrivial periodic solution are studied. With the change of parameters, the system appears with a stable nontrivial periodic solution when the prey-free periodic solution loses stability. Numerical simulations show that the system has complex dynamical behaviors via bifurcation diagrams. Further, the maximum yield problem of the harvested system is studied, which is transformed into a nonlinear programming problem and solved by the method of combined multiple shooting and collocation.

On dynamics of an HIV pathogenesis model with full logistic target cell growth and cure rate

The objective of this paper is to discuss the dynamics of an HIV pathogenesis model with full logistic target cell growth of uninfected T cells and cure rate of infected T cells. Local and global dynamics of both infection-free and infected equilibrium points are rigorously established. It is found that if basic reproduction number R 0 ≤1, the infection is cleared from T cells and if R 0 >1, the HIV infection persists. Also, we have carried out numerical simulations to verify the results. The existence of non-trivial periodic solution is also studied by means of numerical simulation. Therefore, we find a parameter region where infected equilibrium point is globally stable to make the model biologically significant. From the overall study, it is found that proliferation of T cells cannot be ignored during the study of HIV dynamics for better results and we can focus on a treatment policy which can control the parameters of the model in such a way that the basic reproduction number remains less than or equal to one.

Existences of periodic solutions to the generalized Swift–Hohenberg equation on symmetry-breaking model

This paper deals with the generalized Swift-Hohenberg equation, proposed by Stoop et al. (2015), on the symmetry-breaking model that describes wrinkling morphology and pattern selection in curved elastic bilayer materials. More precisely, with help of the Mountain Pass theorem, we establish the existence of nontrivial periodic solutions to this equation.

Analysis of a vector-borne disease model with impulsive perturbation and reinfection

We formulate a mathematical model for vector-borne disease with impulsive perturbation based on indoor residual spraying. The dynamical properties are studied theoretically and numerically. It is shown that with consideration of impulsive spraying at fixed times, there exists a disease-free periodic solution that is locally stable when the threshold $${\mathcal {R}}_0$$ is less than unity, otherwise the disease is uniformly persistent if $${\mathcal {R}}_0>1$$. Furthermore, the bifurcation analysis is performed, revealing the possible existence of nontrivial periodic solution bifurcated from the disease-free periodic solution at $${\mathcal {R}}_0=1$$. In addition to simulations of parameter sensitivity, when implementing impulsive spraying once the number of infected humans exceeds a threshold level, the effectiveness of such state-dependent control is also conducted numerically.

Bifurcation of Nontrivial Periodic Solutions for a Food Chain Beddington–DeAngelis Interference Model with Impulsive Effect

In this paper, a food chain Beddington–DeAngelis interference model with impulsive effect is studied. The trivial periodic solution is locally asymptotically stable if the release rate or the release period is suitable. Conditions for permanence of the model are obtained. The existence of nontrivial periodic solutions and semi-trivial periodic solutions are established when the trivial periodic solution loses its stability under different conditions.

Dynamical behavior of a stochastic SVIR epidemic model with vaccination

In this paper, we investigate the dynamical behavior of SVIR models in random environments. Firstly, we show that if R0s<1, the disease of stochastic autonomous SVIR model will die out exponentially; if R̃0s>1, the disease will be prevail. Moreover, this system admits a unique stationary distribution and it is ergodic when R̃0s>1. Results show that environmental white noise is helpful for disease control. Secondly, we give sufficient conditions for the existence of nontrivial periodic solutions to stochastic SVIR model with periodic parameters. Finally, numerical simulations validate the analytical results.

Nontrivial periodic solutions of second order singular damped dynamical systems

Assuming that the linear equation $x'' + h(t)x' + a(t)x = 0$ has a positive Green's function, we study the existence of nontrivial periodic solutions of second order damped dynamical systems \[ x'' + h(t)x' + a(t)x = f(t, x) + e(t), \] where $h$, $a\in \C(\!(\R/T\Z),\R)$, $e\! =\! (e_1,\ldots, e_N\!)^T\!\! \in \C(\!(\R/T\Z),\R^N\!)$, $N \ge 1$, and the nonlinearity $f = (f_1,\ldots, f_N)^T\in\C((\R=T\Z)\times\R^N\setminus\{0\},\R^N)$ has a repulsive singularity at the origin. We consider a very general singularity and do not need any kind of strong force condition. The proof is based on a nonlinear alternative principle of Leray-Schauder. Recent results in the literature are generalized and improved.

Impulsive predator-prey model

In this paper we considered a predator-prey model with state-dependent impulse. We determine periodic solutions for the model without impulses and we prove the existence of nontrivial periodic solution in the case of impulse depending on the state of the model.

Mathematical model for pulsed chemotherapy

A pulsed chemotherapeutic treatment model is investigated in this work. We prove the existence of nontrivial periodic solutions by the mean of Lyapunov-Schmidt bifurcation method of a cancer model. In this model we consider the case of application of two drugs, the first one P with continuous effect, it appears in the differential equations, and the second one T with instantaneous effects expressed by impulse equations. The existence of bifurcated nontrivial periodic solutions are discussed with respect to the competition parameter values.

Nonsynchronous bifurcation of SIRS epidemic model with birth pulse and pulse vaccination

In this paper, we construct an SIRS epidemic model with birth pulse and pulse vaccination at the different fixed moment. The stability of the infection-free periodic solution is obtained by using the Poincare map. The existence of nontrivial periodic solution bifurcated from the infection-free periodic solution is discussed by means of the bifurcation theory. It is shown that once a threshold is reached, a nontrivial periodic solution emerges via a supercritical bifurcation. Furthermore, some numerical simulations are given, which are in good accordance with the theoretical results.

New periodic solutions of singular Hamiltonian systems with fixed energies

By using the variational minimizing method with a special constraint and the direct variational minimizing method without constraint, we study second-order Hamiltonian systems with a singular potential and , which may have an unbounded potential well, and prove the existence of non-trivial periodic solutions with a prescribed energy. Our results can be regarded as complements of the well-known theorems of Benci-Gluck-Ziller-Hayashi and Ambrosetti-Coti Zelati and so on. MSC:35A15, 47J30.

Existence and multiplicity of periodic solutions for some second order Hamiltonian systems

In this paper, we study the existence of nontrivial periodic solutions for the second order Hamiltonian systems $ \ddot u(t)+\nabla F(t,u(t))=0$, where $F(t,x)$ is either nonquadratic or superquadratic as $|u|\mathbb{R}ightarrow \infty$. Furthermore, if $F(t,x)$ is even in $x$, we prove the existence of infinitely many periodic solutions for the general Hamiltonian systems $ \ddot u(t)+A(t)u(t)+\nabla F(t,u(t))=0$, where $A(\cdot)$ is a continuous $T$-periodic symmetric matrix. Our theorems mainly improve the recent result of Tang and Jiang [X.H. Tang, J. Jiang, Existence and multiplicity of periodic solutions for a class of second-order Hamiltonian systems, Comput. Math. Appl. 59 (2010) 3646-3655].

Bifurcation of nontrivial periodic solutions for a Beddington–DeAngelis interference model with impulsive biological control

In this paper, a Beddington–DeAngelis interference model with impulsive biological control is studied. The pest-free periodic solution is local asymptotically stable if the impulsive control rate is larger than a critical value or the release period is smaller than another critical value. Conditions for permanence of the model are established. The existence of nontrivial periodic solution is established when the pest-free periodic solution loses its stability.

Time periodic solutions of the diffusive Nicholson blowflies equation with delay

This paper is concerned with the Nicholson blowflies equation with nonlinear diffusion and time delay subject to the homogeneous Dirichlet boundary condition in a bounded domain. We establish the existence of nontrivial periodic solutions of the time-periodic problem under general conditions by constructing a coupled upper–lower solution pair and by applying the Schauder fixed point theorem. The attractivity of the periodic solutions is also discussed by using the monotone iteration method.

Nontrivial periodic motions for resonant type asymptotically linear lattice dynamical systems

In this paper, we consider the following one dimensional lattices consisting of infinitely many particles with nearest neighbor interactionq¨i(t)=Φi−1′(t,qi−1(t)−qi(t))−Φi′(t,qi(t)−qi+1(t)),i∈Z, where Φi(t,x)=−(αi/2)|x|2+Vi(t,x) is T-periodic in t for T>0 and satisfies Φi+N=Φi for some N∈N, qi(t) is the state of the i-th particle. Assume that αi=0 for some i∈Z and Vi′(t,x) denoting the derivative of Vi respect to x is asymptotically linear with x both at origin and at infinity. We would like to point out that this system is resonant both at origin and at infinity and not studied up to now. Based on some new results concerning the precise computations of the critical groups, for a given m∈Z, we obtain the existence of nontrivial periodic solutions satisfying qi+mN(t+T)=qi(t) for all t∈R and i∈Z under some additional conditions.

Variational approach to a class of p-Laplacian systems on time scales

In this paper, we present a recent approach via variational methods and critical point theory to obtain the existence of nontrivial periodic solutions for a class of p-Laplacian systems on time scales. By establishing a proper variational setting, three existence results are obtained. Finally, two examples are presented to illustrate the feasibility and effectiveness of our results.MSC:34N05, 37J45, 34C25.

Existence of periodic solutions in a linear higher order system of difference equations

In this paper we investigate the existence of nontrivial periodic solutions of a higher order system of difference equations. Our framework is based on an earlier periodicity condition of us. We also use the theory of circulant matrices combined by a theorem of Silvester on the computation the determinant of block matrices. An illustrative application is given to show the effectiveness of our framework and to point out the connection between our periodicity results and some known stability conditions.

Periodic solutions for asymptotically linear Hamiltonian system with resonance both at infinity and origin via computation of critical groups

In this paper, we study the existence of nontrivial periodic solutions for asymptotically linear Hamiltonian system which has the same linear term both at infinity and origin. By computation of critical groups at origin and infinity associated with Maslov-type index, we establish the existence for nontrivial periodic solution of the system. Moreover, some conditions employed for the computation of critical groups are weakened.

Periodic Solutions of Second-Order Difference Problem with Potential Indefinite in Sign

We investigate the periodic solutions of second-order difference problem with potential indefinite in sign. We consider the compactness condition of variational functional and local linking at 0 by introducing new number <svg style="vertical-align:-3.27605pt;width:17.112499px;" id="M1" height="16.3125" version="1.1" viewBox="0 0 17.112499 16.3125" width="17.112499" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.162)"><path id="x1D706" d="M529 97q-70 -109 -136 -109q-41 0 -56 94q-23 144 -37 284q-38 -88 -99 -202.5t-93 -156.5q-26 -8 -76 -19l-9 21q71 78 145.5 193t124.5 232q-5 84 -15 128q-12 55 -29.5 75.5t-42.5 20.5q-21 0 -45 -13l-8 24q16 17 46 30t55 13q43 0 70 -46.5t40 -169.5&#xA;q27 -249 51 -392q7 -46 23 -46q24 0 70 60z" /></g> <g transform="matrix(.012,-0,0,-.012,9.45,16.25)"><path id="x2217" d="M471 153q-22 -15 -61 -13q-25 31 -45.5 49.5t-56.5 41.5q4 -71 28 -134q-17 -33 -42 -46q-24 12 -42 46q24 63 28 134q-36 -23 -56.5 -41.5t-45.5 -49.5q-36 -2 -61 13q0 28 19 59q65 10 130 43q-65 33 -130 43q-19 31 -19 59q22 15 61 13q25 -31 45.5 -49.5t56.5 -41.5&#xA;q-4 71 -28 134q17 33 42 46q24 -12 42 -46q-24 -63 -28 -134q36 23 56.5 41.5t45.5 49.5q36 2 61 -13q0 -28 -19 -59q-65 -10 -130 -43q65 -33 130 -43q19 -31 19 -59z" /></g> </svg>. By using Morse theory, we obtain some new results concerning the existence of nontrivial periodic solution.