ON THE STABILITY OF PERIODIC SOLUTIONS OF PIECEWISE SMOOTH PERIODIC DIFFERENTIAL EQUATIONS

In this paper,we address the stability of periodic solutions of piecewise smooth periodic differential equations.By studying the Poincarémap,we give a sufficient condition to judge the stability of a periodic solution.We also present examples of some applications.

TIME PERIODIC SOLUTIONS TO THE EVOLUTIONARY OSEEN MODEL FOR A GENERALIZED NEWTONIAN INCOMPRESSIBLE FLUID

In this paper we study a nonstationary Oseen model for a generalized Newtonian incompressible fluid with a time periodic condition and a multivalued,nonmonotone friction law.First,a variational formulation of the model is obtained;that is a nonlinear boundary hemivariational inequality of parabolic type for the velocity field.Then,an abstract first-order evolutionary hemivariational inequality in the framework of an evolution triple of spaces is investigated.Under mild assumptions,the nonemptiness and weak compactness of the set of periodic solutions to the abstract inequality are proven.Furthermore,a uniqueness theorem for the abstract inequality is established by using a monotonicity argument.Finally,we employ the theoretical results to examine the nonstationary Oseen model.

Bifurcations, Analytical and Non-Analytical Traveling Wave Solutions of (2 + 1)-Dimensional Nonlinear Dispersive Boussinesq Equation

For the (2 + 1)-dimensional nonlinear dispersive Boussinesq equation, by using the bifurcation theory of planar dynamical systems to study its corresponding traveling wave system, the bifurcations and phase portraits of the regular system are obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of analytical and non-analytical solutions of the singular system are given by using singular traveling wave theory. For certain special cases, some explicit and exact parametric representations of traveling wave solutions are derived such as analytical periodic waves and non-analytical periodic cusp waves. Further, two-dimensional wave plots of analytical periodic solutions and non-analytical periodic cusp wave solutions are drawn to visualize the dynamics of the equation.

Dynamics and Exact Solutions of (1 + 1)-Dimensional Generalized Boussinesq Equation with Time-Space Dispersion Term

We study exact solutions to (1 + 1)-dimensional generalized Boussinesq equation with time-space dispersion term by making use of improved sub-equation method, and analyse the dynamical behavior and exact solutions of the sub-equation after constructing the nonlinear transformation and constraint conditions. Accordingly, we obtain twenty families of exact solutions such as analytical and singular solitons and singular periodic waves. In addition, we discuss the impact of system parameters on wave propagation.

Diversity of Rogue Wave Solutions to the (1+1)-Dimensional Boussinesq Equation

A periodically homoclinic solution and some rogue wave solutions of (1+1)-dimensional Boussinesq equation are obtained via the limit behavior of parameters and different polynomial functions. Besides, the mathematics reasons for different spatiotemporal structures of rogue waves are analyzed using the extreme value theory of the two-variables function. The diversity of spatiotemporal structures not only depends on the disturbance parameter u0 but also has a relationship with the other parameters c0, α, β.

Existence of Periodic Solutions for Higher Order Autonomous Birkhoff Systems

Qualitative methods of ordinary differential equation and Liapunov center theorem were used to study the existence of periodic solutions for higher order autonomous Birkhoff systems. For higher order autonomous Birkhoff systems, the character of the characteristic roots of the Fréchet derivative C was obtained. Furthermore the existence theorem of periodic solutions was obtained by using Liapunov center theorem, and an example was presented to illustrate the results.

On the Existence of Periodic Solutions of a Class Nonlinear Differential Systems

One method to show the existence of ω-periodic system is given. This method is based on the ultimately boundedness of the solution of the systems. By using comparing theorem and discussing some one dimensional equations the main results are obtained.

Exponential stability and existence of periodic solutions for a class of recurrent neural networks with delays

Both the global exponential stability and the existence of periodic solutions for a class of recurrent neural networks with continuously distributed delays （RNNs） are studied. By employing the inequality α∏k=1^m bk^qk≤1/r ∑qkbk^r＋1/rα^r（α≥0,bk≥0,qk〉0,with ∑k=1^m qk=r-1,r≥1, constructing suitable Lyapunov r k=l k=l functions and applying the homeomorphism theory, a family of simple and new sufficient conditions are given ensuring the global exponential stability and the existence of periodic solutions of RNNs. The results extend and improve the results of earlier publications.

Existence of Periodic Solutions for Generalized Liénard-Typed Functional Differential Equations

The existence of periodic solutions for a kind of generalized Liénard typed functional differential equation is studied. By means of the continuation theorem of coincidence degree theory, existence criteria are established for the existence of periodic solutions and some previous results are extended.

Existence of Periodic Solutions of Nonlinear Neutral Differential Equation with Deviating Argument

Existence criteria is established for the periodic solution of the nonlinear neutral delay differential equation x′(t)=f(t,x(t),x(t-τ 1(t)),x′(t-τ 2(t)))+p(t) by means of an abstract continuous theorem of k-set contractive operator and some analysis technique.

Existence of Many and Infinitely Many Periodic Solutions for Some Types of Differential Delay Equations

J.Kaplan and J.Yorke's method is extended to establish the exis- tence of many and infinitely many periodic solutions for the DDEs (t) =±f(x(t-1))±f(x(t-2))and (t)=±f(x(t-1).

EXISTENCE OF POSITIVE PERIODIC SOLUTIONS FOR NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENTS

By means of an abstract continuation theorem, the existence criteria are established for the positive periodic solutions of a neutral functional differential equation d N d t=N(t)[a(t)-β(t)N(t)-b(t)N(t-σ(t))-c(t)N′(t-τ(t))].

EXISTENCE OF POSITIVE PERIODIC SOLUTIONS FOR VOLTERRA INTERGO-DIFFERENTIAL EQUATIONS

This paper deals with the existence of positive periodic solutions for a kind of nonautonomous Volterra intergo-differential equations by employing the Krasnoselskii fixed point theorem. Applying the general theorems established to several biomathematical models, the paper improves some previous results and obtains some new results.

Infinitely many periodic solutions for second-order Hamiltonian systems

The existence of high energy periodic solutions for the second-order Hamiltonian system -ü（t）＋A（t）u（t）=▽F（t,u（t）） with convex and concave nonlinearities is studied, where F（t, u） = F1（t,u）＋F2（t,u）. Under the condition that F is an even functional, infinitely many solutions for it are obtained by the variant fountain theorem. The result is a complement for some known ones in the critical point theory.

EXISTENCE OF PERIODIC SOLUTIONS FOR A DIFFERENTIAL INCLUSION SYSTEMS INVOLVING THE p(t)-LAPLACIAN

We study a nonlinear periodic problem driven by the p（t）-Laplacian and having a nonsmooth potential （hemivariational inequalities）. Using a variational method based on nonsmooth critical point theory for locally Lipschitz functions, we first prove the existence of at least two nontrivial solutions under the generalized subquadratic and then establish the existence of at least one nontrivial solution under the generalized superquadratic.

Existence of two periodic solutions of a ratio-dependent Holling-Taner model with infinite delay and prey harvest

This paper considers a class of ratio-dependent Holling-Taner model with infinite delay and prey harvest, which is of periodic coefficients. By means of the coincidence degree theory, a set of sufficient conditions for the existence of at least two positive periodic solutions of this model is established.

Periodic Solutions of Mixed Type p-Laplacian Equations with Deviating Arguments

Based on Mansevich-Mawhin continuation theorem and some analysis skill,some sufficient conditions for the existence of periodic solutions for mixed type p-Laplacian equation with deviating arguments are established,which are complement of previously known results.

THE PERIODIC SOLUTIONS FOR TIME DEPENDENT AGE-STRUCTURED POPULATION MODELS

In this paper, the existence of periodic solutions for a time dependent age-structured population model is studied. The averaged net reproductive number is introduced as the main parameter to determine the dynamical behaviors of the model. The existence of a global parameterized branch of periodic solutions of the model is obtained by using the contracting mapping theorem in a periodic and continuous function space. The global stability of the trivial equilibrium is studied and a very practical stability criteria for the model is obtained. The dynamics of the linear time-periodic model is similar to that of the linear model.

New Periodic Wave Solutions to Generalized Klein-Gordon and Benjamin Equations

By making use of extended mapping method and auxiliary equation for finding new periodic wave solu tions of nonlinear evolution equations in mathematical physics, we obtain some new periodic wave solutions for generalized Klein-Cordon equation and Benjamin equation, which cannot be found in previous work. This method also can be used to find new periodic wave solutions of other nonlinear evolution equations.

The pth-order periodic solutions for a family of N-coupled nonlinear SchrSdinger equations

By using the solutions of an auxiliary Lame equation, a direct algebraic method is proposed to construct the exact solutions of N-coupled nonlinear Schrodinger equations. The abundant higher-order exact periodic solutions of a family of N-coupled nonlinear Schrodinger equations are explicitly obtained with the aid of symbolic computation and they include corresponding envelope solitary and shock wave solutions.