Subharmonic solutions with prescribed minimal period for some second-order impulsive differential equations

This paper uses critical point theory and variational methods to investigate the subharmonic solutions with prescribed minimal period for a class of second-order impulsive differential equations. The conditions for the existence of subharmonic solutions are established. Our results rectify some known results in the literature and will allow us, in the future, to deal with the subharmonic solutions for more extensive impulsive problems.     

Multiplicity results for periodic solutions with prescribed minimal periods to discrete Hamiltonian systems

By making use of minimax theory and geometrical index theory, some results on the existence and multiplicity of subharmonic solutions with prescribed minimal period to discrete Hamiltonian systems are obtained.     

Multiple Periodic Solutions with Minimal Period for Second Order Discrete System

By making use of Clark duality, minimax theory and geometrical index theory, some results on the existence and multiplicity of subharmonic solutions with prescribed minimal period to second order subquadratic discrete system are obtained.     

一类超线性Hill型对称碰撞方程的周期运动

本文考虑一类超线性Hill型对称碰撞方程的对称碰撞周期解的存在性、重性和分布问题.通过坐标变换的方法把碰撞相平面转化为全平面进行研究,在一类关于时间映射的超线性条件下证明有外力方程无穷多个对称碰撞调和解和对称碰撞次调和解的存在性;同时研究在没有外力时方程的对称碰撞周期解的稠密性分布.本文还给出对称碰撞方程对称碰撞周期解存在的充分条件.     

Applications of Clark duality to periodic solutions with minimal period for discrete Hamiltonian systems

By making use of Clark duality, perturbation technique and dual least action principle, some results on the existence of subharmonic solutions with minimal period to second-order subquadratic discrete Hamiltonian systems are obtained.     

A global solution curve for a class of periodic problems,including the relativistic pendulum

Using continuation methods, we study the global solution structure of periodic solutions for a class of periodically forced equations, generalizing the case of relativistic pendulum. We obtain results on the existence and multiplicity of periodic solutions. Our approach is suitable for numerical computations, and in fact we present some numerically computed bifurcation diagrams illustrating our results.     

The minimal period problem for the classical forced pendulum equation

Considered in this paper is the existence/nonexistence of periodic solutions with prescribed minimal periods to the classical forced pendulum equation,

     

Subharmonic Solutions of a Pendulum Under Vertical Anharmonic Oscillations of the Point of Suspension

We prove the existence of subharmonic solutions in the dynamics of a pendulum whose point of suspension executes a vertical anharmonic oscillation of small amplitude.     

Symmetric period solutions with prescribed minimal period for even autonomous semipositive Hamiltonian systems

We study some monotonicity and iteration inequality of the Maslov-type index i-1of linear Hamiltonian systems.As an application we prove the existence of symmetric periodic solutions with prescribed minimal period for first order nonlinear autonomous Hamiltonian systems which are semipositive,even,and superquadratic at zero and infinity.This result gives a positive answer to Rabinowitz’s minimal period conjecture in this case without strictly convex assumption.We also give a different proof of the existence of symmetric periodic solutions with prescribed minimal period for classical Hamiltonian systems which are semipositive,even,and superquadratic at zero and infinity which was proved by Fei,Kim and Wang in 2001.     

Subharmonic solutions of systems of difference equations with periodic perturbations Part I: Existence

We consider the existence of subharmonic solutions of systems of difference equations with periodic perturbations. The theory of coincidence degree, coupled with some detailed a priori estimates, is applied to show that the perturbed system admits subharmonic solutions near a hyperbolic periodic orbit of the unperturbed system if the perturbation is sufficiently small.     

The iteration formula of the Maslov-type index theory with applications to nonlinear Hamiltonian systems

In this paper, the iteration formula of the Maslov-type index theory for linear Hamiltonian systems with continuous, periodic, and symmetric coefficients is established. This formula yields a new method to determine the minimality of the period for solutions of nonlinear autonomous Hamiltonian systems via their Maslov-type indices. Applications of this formula give new results on the existence of periodic solutions with prescribed minimal period for such systems, and unify known results under various convexity conditions.

     

Existence of periodic and subharmonic solutions for second order functional difference equations

In this paper, by using the critical point theory, the existence of periodic and subharmonic solutions to a class of second order functional difference equations is obtained. The main approach used in our paper is variational technique and the Saddle Point Theorem. The problem is to solve the existence of periodic and subharmonic solutions of second order forward and backward difference equations.     

Period two implies chaos for a class of ODEs

We extend a result of J. Andres and K. Pastor concerning scalar time-periodic first order ordinary differential equations without uniqueness, by proving that the existence of just one subharmonic implies the existence of large sets of subharmonics of all given orders. Since these periodic solutions must coexist with complicated dynamics, we might paraphrase T. Y. Li and J. A. Yorke by loosely saying that in this setting even period two implies chaos. Similar results are obtained for a class of differential inclusions.

     

Homoclinic orbits and subharmonics for nonlinear second order difference equations

We show the existence of a nontrivial homoclinic orbit and subharmonic solutions for a class of second order difference equations by applying the “Mountain Pass” theorem relying on Ekeland’s variational principle and the diagonal method, and the homoclinic orbit as the limit of the subharmonics. A completely new way is provided for dealing with the existence of solutions for difference equations.     

SOME RESULTS ON THE MINIMAL PERIOD PROBLEM OF NONCONVEX SECOND ORDER HAMILTONIAN SYSTEMS

1.IntrodnctionandMainResultsInthispaper,weconsidertheminimalperiodproblemforthefollowingautonomoussecondorderHamiltoniansystemswhereNisapositiveinteger.andV'denotesitsgradient.Inthetextofthispaper,wedenotebya.bandtheusualinnerproductandnorminRNrespec...     

Existence of periodic and subharmonic solutions for second-order superlinear difference equations

By critical point theory, a new approach is provided to study the existence and multiplicity results of periodic and subharmonic solutions for difference equations. For secord-order difference equations△2xn-1+f(n,xn)=0some new results are obtained for the above problems when f(t, z) has superlinear growth at zero and at infinityin z.     

A system of coupled pendulii

We study the existence of solutions for a coupled system of n-dimensional pendulum equations under generalized periodic-type conditions. We obtain existence results under appropriate conditions using topological degree methods and a shooting type argument.     

HARMONIC AND SUBHARMONIC BIFURCATION IN THE BRUSSEL MODEL WITH PERIODIC FORCE

1.IntroductionABrusselatorisoneofthebestexaminedmodelchemicalreactionswhichconsistsoffourstepsItisshowninFig.1schematicallyandisrepreselltedbythefollowingsetofequationsffevedFebruary6,1995.*~workissupportedbytheNationalNaturalScienceFOundationmanYuan"TermsinChina.ThemodelweadoptistheoneduetoPrigogine,Lefever,andNicolis(Brusselator)t'.Fig.1.'TheschematicdiagramofBrusselmodel(AdditionalcirculararrowsrepreseDttheexistenceofautocatalysis.)Herexandystandfortheconcentrationsofreferencereacta…     

Periodic and subharmonic solutions for 2nth-order p-Laplacian difference equations

By using the critical point theory, some new criteria for the existence and multiplicity of periodic and subharmonic solutions for 2nth-order p-Laplacian difference equations are obtained. The proof is based on the Linking Theorem in combination with variational technique. Our results generalize and improve the known in the literature results.     

Almost periodic weak solutions to forced pendulum type equations without friction

Summary We obtain sufficient conditions for the existence of almost periodic weak solutions to equations of the frictionless pendulum type with forcing.