Periodic solutions of the N-body problem

This paper proves the existence of six new classes of periodic solutions to the N-body problem by small parameter methods. Three different methods of introducing a small parameter are considered and an appropriate method of scaling the Hamiltonian is given for each method. The small parameter is either one of the masses, the distance between a pair of particles or the reciprocal of the distances between one particle and the center of mass of the remaining particles. For each case symmetric and non-symmetric periodic solutions are established. For every relative equilibrium solution of the (N ? 1)-body problem each of the six results gives periodic solutions of the N-body problem. Under additional mild non-resonance conditions the results are roughly as follows. Any non-degenerate periodic solutions of the restricted N-body problem can be continued into the full N-body problem. There exist periodic solutions of the N-body problem, where N ? 2 particles and the center of mass of the remaining pair move approximately on a solution of relative equilibrium and the pair move approximately on a small circular orbit of the two-body problems around their center of mass. There exist periodic solutions of the N-body problem, where one small particle and the center of mass of the remaining N ? 1 particles move approximately on a large circular orbit of the two body problems and the remaining N ? 1 bodies move approximately on a solution of relative equilibrium about their center of mass. There are three similar results on the existence of symmetric periodic solutions.     

Periodic Solutions of Classical Hamiltonian Systems without Palais-Smale Condition

In this paper, we study the existence of periodic solutions for classical Hamiltonian systems without the Palais-Smale condition. We prove that the information of the potential function contained in a finite domain is sufficient for the existence of periodic solutions. Moreover, we establish the existence of infinitely many periodic solutions without any symmetric condition on the potential function V.     

Existence of periodic solutions for a predator-prey system with sparse effect and functional response on time scales

In this paper, we systematically investigates the existence of periodic solutions of a predator-prey system with sparse effect and Beddington-DeAngelis or Holling III functional response on time scales. By using a continuation theorem based on coincidence degree theory, we obtain sufficient criteria for the existence of periodic solutions for the systems. Moreover, when the time scale T is chosen as R or Z, the existence of the periodic solutions of the corresponding continuous and discrete models follows. Therefore, the methods are unified to provide the existence of the desired solutions for the continuous differential equations and discrete difference equations.     

New explicit solutions for the Klein-Gordon equation with cubic nonlinearity

In this paper, we investigate Klein-Gordon equation with cubic nonlinearity. All explicit expressions of the bounded travelling wave solutions for the equation are obtained by using the bifurcation method and qualitative theory of dynamical systems. These solutions contain bell-shaped solitary wave solutions, kink-shaped solitary wave solutions and Jacobi elliptic function periodic solutions. Moreover, we point out the region which these periodic wave solutions lie in. We present the relation between the bounded travelling wave solution and the energy level h. We find that these periodic wave solutions tend to the corresponding solitary wave solutions as h increases or decreases. Finally, for some special selections of the energy level h, it is shown that the exact periodic solutions evolute into solitary wave solution.     

Limit periodic motions in some systems with aftereffect under resonance condition

Volterra-type integrodifferential equations and their solutions are considered which, when the time increases without limit, exponentially tend to periodic modes. In the critical case of stability, when the characteristic equation has a pair of pure imaginary roots and the remaining roots have negative real parts, the problem of the existence of limit periodic solutions with resonance, caused by coincidence between the periodic part of the limit external periodic perturbation and the natural frequency of the linearized system, is solved. It is shown that, if the right-hand side of the equation is an analytic function and the existence of limit periodic solutions is determined by terms of the (2m + 1)-th order, these solutions are represented by power series in the arbitrary initial values of the non-critical variables and the parameter μ^1/(2m+1), where μ is a small parameter, characterizing the magnitude of the maximum external periodic perturbation. The amplitude equations are presented.     

Existence of periodic solutions of second order nonlinear p-Laplacian difference equations

By using the critical point theory, the existence of periodic solutions to second order nonlinear p-Laplacian difference equations is obtained. The main approach used is a variational technique and the saddle point theorem. The problem is to solve the existence of periodic solutions of second order nonlinear p-Laplacian difference equations.     

Periodic solutions for p-Laplacian neutral functional differential equation with multiple deviating arguments

By using the theory of coincidence degree, we study a kind of periodic solutions to p-Laplacian neutral functional differential equation with multiple deviating arguments, a new result on the existence of periodic solutions is obtained.     

Periodic solutions with nonconstant sign in Abel equations of the second kind

The study of periodic solutions with constant sign in the Abel equation of the second kind can be made through the equation of the first kind. This is because the situation is equivalent under the transformation x?x−1, and there are many results available in the literature for the first kind equation. However, the equivalence breaks down when one seeks for solutions with nonconstant sign. This note is devoted to periodic solutions with nonconstant sign in Abel equations of the second kind. Specifically, we obtain sufficient conditions to ensure the existence of a periodic solution that shares the zeros of the leading coefficient of the Abel equation. Uniqueness and stability features of such solutions are also studied.     

Periodic solutions for Hamiltonian systems under strong constraining forces

We consider periodic solutions of Hamiltonian systems in Euclidean spaces whose motion is constrained to a submanifold M. We prove that under some nondegeneracy assumptions, periodic solutions persist when the constraint is replaced by a strong restoring potential.     

Multiple periodic solutions of Hamiltonian systems with prescribed energy

Consider the periodic solutions of autonomous Hamiltonian systems on the given compact energy hypersurface Σ=H⁻¹(1). If Σ is convex or star-shaped, there have been many remarkable contributions for existence and multiplicity of periodic solutions. It is a hard problem to discuss the multiplicity on general hypersurfaces of contact type. In this paper we prove a multiplicity result for periodic solutions on a special class of hypersurfaces of contact type more general than star-shaped ones.     

Time periodic and almost time periodic solutions to rotating stratified fluids subject to large forces

Consider the 3D incompressible Boussinesq equations for rotating stably stratified fluids. It is shown that this set of equations possesses a unique time periodic or almost time periodic solutions for external forces satisfying these properties, which, however, do not necessarily need to be small. An explicit bound on the size of the external force, depending on the buoyancy frequency N, is given, which then allows for the unique existence of time periodic or almost periodic solutions. In particular, the size of the external forces can be taken large with respect to the buoyancy frequency. The approach depends crucially on the dispersive effect of the rotation and the stable stratification.     

Periodic two-phase “Rogue waves”

A family of periodic (in x and z) two-gap solutions of the focusing nonlinear Schrödinger equation is constructed. A condition under which the two-gap solutions exhibit the behavior of periodic “rogue waves” is obtained.     

On the structure of periodic solutions of differential equations

This paper studies the “internal structure” of the periodic solutions of differential equations with the aim of stating when they are constant functions. Yorke [21] and Lasota and Yorke [10] are the first works which show the existence, uńder certain conditions, of a lower bound for the period of non-constant solutions. As applications of the general results proved in Section 1 we obtain a negative solution to an open problem of Browder, the discovery that the periodic solutions ensured by Vidossich [17, Theorem 3.16], are constant functions, and conditions under which the periodic solutions of hyperbolic and parabolic equations are constant functions. Finally, we note that Li [11] applies the results of Section 1 to differential equations with delay.Various result of this paper point out a strong connection between the existence of periodic solutions of small period of x′ = f(x) and the fact that the origin belongs to the range of f. This situation is explored in [19].     

Multiple periodic solutions of delayed predator–prey systems with type IV functional responses

In this paper, we considered a periodic predator–prey system with a type IV functional response, which incorporates the periodicity of the environment. Sufficient conditions for the existence of multiple positive periodic solutions are established by applying the continuation theorem. This is the first time that multiple periodic solutions are obtained by using the theory of coincidence degree. Moreover, unlike other types of functional responses, a type IV functional response declines at high prey densities. Thus the existing arguments for obtaining bounds of solutions to the operator equation Lx=λNx are inapplicable to our case and some new arguments are employed for the first time.     

The extended Jacobian elliptic function expansion method and its application in the generalized Hirota–Satsuma coupled KdV system

In this paper an extended Jacobian elliptic function expansion method, which is a direct and more powerful method, is used to construct more new exact doubly periodic solutions of the generalized Hirota–Satsuma coupled KdV system by using symbolic computation. As a result, sixteen families of new doubly periodic solutions are obtained which shows that the method is more powerful. When the modulus of the Jacobian elliptic functions m→1 or 0, the corresponding six solitary wave solutions and six trigonometric function (singly periodic) solutions are also found. The method is also applied to other higher-dimensional nonlinear evolution equations in mathematical physics.     

Uniqueness of periodic solutions of a nonautonomous density-dependent predator–prey system

In this present paper, we investigate the uniqueness of periodic solutions of a nonautonomous density-dependent and ratio-dependent predator–prey system, where not only the prey density dependence but also the predator density dependence are considered, such that the studied predator–prey system conforms to the realistically biological environment. We start with a sufficient condition for the permanence of the system and then construct a weaker sufficient condition by introducing a specific set, denoted as Γ. Based on this Γ and the Brouwer fixed-point theorem, we obtain the existence condition of positive periodic solutions. Moreover, since the uniqueness of positive periodic solutions can be ensured by global attractiveness, we alternatively introduce a sufficient condition for global attractiveness. Similarly, we also provide a sufficient condition for the uniqueness of non-negative periodic solutions.     

Positive periodic solutions of nonautonomous functional differential systems

Considered is the periodic functional differential system with a parameter, x^′(t)=A(t,x(t))x(t)+λf(t,xt). Using the eigenvalue problems of completely continuous operators, we establish some criteria on the existence of positive periodic solutions. Moreover, we apply the results to a couple of population models and obtain sufficient conditions for the existence of positive periodic solutions, which are compared with existing ones.     

Doubly periodic solutions of the coupled scalar field equations

Some doubly periodic (Jacobi elliptic function) solutions of the coupled Schrödinger–Boussinesq (KdV) equations are presented in closed form. Our approach is to introduce an auxiliary ordinary differential equation and use its Jacobi elliptic function solution to construct doubly periodic solutions of the coupled equations. When the module m→1, these solutions degenerate to the exact solitary wave solutions of the coupled equations.     

Periodic solutions of the evolutionary p-Laplacian with nonlinear sources

This paper is concerned with the evolutionary p-Laplacian with nonlinear and periodic sources. We will give a rather complete characterization, in terms of the parameter p and the exponent q of the source, of whether or not the positive periodic solutions exist.     

Multiple periodic solutions of a class of p-Laplacian

By using the recent generalization of coincidence degree method, the existence of multiple periodic solutions for a class of p-Laplacian is obtained under the existence of strict upper and lower solutions.