Example of a suspension bridge ODE model exhibiting chaotic dynamics: A topological approach

Using an elementary phase-plane analysis combined with some recent results on topological horseshoes and fixed points for planar maps, we prove the existence of infinitely many periodic solutions as well as the presence of chaotic dynamics for a simple second order nonlinear ordinary differential equation arising in the study of Lazer-McKenna suspension bridges model.     

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.     

Symmetric excited states for a mean-field model for a nucleon

In this paper, we consider a stationary model for a nucleon interacting with the ω and σ mesons in the atomic nucleus. The model is relativistic, and we study it in a nuclear physics nonrelativistic limit. By a shooting method, we prove the existence of infinitely many solutions with a given angular momentum. These solutions are ordered by the number of nodes of each component.     

Multiple periodic points of the Poincaré map of Lagrangian systems on tori

In this paper, suppose , A is positive definite and symmetric, and both A and V are and 1-periodic in all of their variables. We prove that the Poincaré map (i.e. the time-1-solution map) of the Lagrangian system possesses infinitely many periodic points on produced by contractible integer periodic solutions. Received July 23, 1997; in final form December 17, 1998     

Chaotic dynamics in periodically forced asymmetric ordinary differential equations

Using a topological approach, we prove the existence of infinitely many periodic solutions and the presence of chaotic dynamics for the periodically forced second order ODE u^″+bu⁺−au⁻=p(t). The choice of the equation is motivated by the studies about the Dancer-Fu?ik spectrum and the Lazer-McKenna suspension bridge model.     

一类二阶Hamilton系统的周期解

本文研究一类含非定线性项的二阶Ｈａｍｉｌｔｏｎ系统多周期解问题．在位势函数满足超二次齐次条件下，利用临界点理论中对称型越山定理，证明了系统存在无穷多个给定周期的周期解．     

Unbounded Solutions of Almost Periodically Forced Pendulum-Type Equations

In this paper, we prove that the forced pendulum-type equation , where with 1-periodicity in x satisfies the conditions: and , possesses infinitely many unbounded solutions on a cylinder S ¹×R for any almost periodic function p(t) with nonvanishing mean value. Received October 7, 1997, Accepted June 9, 1999     

Well/Ill Posedness for the Euler-Korteweg-Poisson System and Related Problems

We consider a general Euler-Korteweg-Poisson system in R ³, supplemented with the space periodic boundary conditions, where the quantum hydrodynamics equations and the classical fluid dynamics equations with capillarity are recovered as particular examples. We show that the system admits infinitely many global-in-time weak solutions for any sufficiently smooth initial data including the case of a vanishing initial density - the vacuum zones. Moreover, there is a vast family of initial data, for which the Cauchy problem possesses infinitely many dissipative weak solutions, i.e. the weak solutions satisfying the energy inequality. Finally, we establish the weak-strong uniqueness property in a class of solutions without vacuum. In this paper we show that, even in presence of a dispersive tensor, we have the same phenomena found by De Lellis and Székelyhidi.     

Existence and infinity of periodic solutions of some second order differential equations with isochronous potentials

In this paper, we deal with the existence and infinity of periodic solutions of differential equations, $$x^{\prime\prime}+f(x^{\prime})+V^{\prime}(x)+g(x)=p(t),$$ where V is a 2??/n-isochronous potential. When f, g are bounded, we give sufficient conditions to ensure the existence of periodic solutions of this equation. We also prove that the given equation has infinitely many 2??-periodic solutions under resonant conditions by using the topological degree approach.     

Infinitely Many Noncollision Periodic Solutions for Planar Keplerian-Like 2-Body Problems

Using the implicit rotational invariant property for the potential and the variational functional, we prove the existence of infinitely many noncollision periodic solutions for planar Keplerian-like 2-body problems.AMS Mathematics Subject Classification: 34C15, 34C25, 58F, 70F10     

The existence of quasiperiodic solutions for coupled Duffing-type equations

In this paper, we prove the existence of infinitely many quasiperiodic solutions for a class of coupled Duffing-type equations via KAM theorem. Moreover, the set of quasiperiodic solutions is of infinitely Lebesgue measure in the phase space.     

Periodic structure of transversal maps on sum-free products of spheres

In this article, we study the periodic structure of transversal maps on the product of spheres of different dimensions. In particular, we give sufficient conditions in order that such maps have infinitely many even and odd periods. Moreover, we also provide sufficient conditions for having non-zero Lefschetz numbers of period m for infinitely many m's. We extend these results to transversal maps on rational exterior spaces of rank 1.     

概周期摆型方程无界解的存在性

对摆型方程x+Gx(x,t)=p(t),其中G(x,t)∈C1(R2)关于变量x是1周期的,并且sup(x,t)∈R2|Gx(x,t)|<+∞,limsupt→∞{supx∈R}=0,p(t)是平均值非零的概周期函数,证明了在柱面S1×R上方程具有无穷多的无界解.     

Almost periodic linear differential equations with non-separated solutions

A celebrated result by Favard states that, for certain almost periodic linear differential systems, the existence of a bounded solution implies the existence of an almost periodic solution. A key assumption in this result is the separation among bounded solutions. Here we prove a theorem of anti-Favard type: if there are bounded solutions which are non-separated (in a strong sense) sometimes almost periodic solutions do not exist. Strongly non-separated solutions appear when the associated homogeneous system has homoclinic solutions. This point of view unifies two fascinating examples by Zhikov-Levitan and Johnson for the scalar case. Our construction uses the ideas of Zhikov-Levitan together with the theory of characters in topological groups.     

On the existence of infinitely many periodic solutions to some problems of n-body type

We prove the existence of infinitely many periodic solutions with prescribed period to a class of problems of n-body type.     

Saari's conjecture for the collinear -body problem

In 1970 Don Saari conjectured that the only solutions of the Newtonian -body problem that have constant moment of inertia are the relative equilibria. We prove this conjecture in the collinear case for any potential that involves only the mutual distances. Furthermore, in the case of homogeneous potentials, we show that the only collinear and non-zero angular momentum solutions are homographic motions with central configurations.

     

Renormalization in the Hénon family,II: the heteroclinic web

We study highly dissipative Hénon maps

$F_{c,b}: (x,y) \mapsto (c-x^2-by, x)$

with zero entropy. They form a region Π in the parameter plane bounded on the left by the curve W of infinitely renormalizable maps. We prove that Morse-Smale maps are dense in Π, but there exist infinitely many different topological types of such maps (even away from W). We also prove that in the infinitely renormalizable case, the average Jacobian b _F on the attracting Cantor set \({\mathcal{O}}_{F}\) is a topological invariant. These results come from the analysis of the heteroclinic web of the saddle periodic points based on the renormalization theory. Along these lines, we show that the unstable manifolds of the periodic points form a lamination outside \({\mathcal{O}}_{F}\) if and only if there are no heteroclinic tangencies.

     

对称超二次二阶哈密尔顿系统的周期解

我们利用Ambrosetti-Rabinowitz对称形式的山路引理证明了给定周期T的对称超二次二阶哈密尔顿系统具有无穷多个反T/2-周期且奇的周期解.     

On periodic Teichmüller disks of modular transformations

We study the periodic Teichmüller disks of modular transformations. Especially, we prove that a parabolic modular transformation has either no periodic Teichmullüller disk or infinitely many periodic Teichmüller disks which can be chosen to cover infinitely many arithmetic Teichmüller curves in the Riemann moduli space M _g . Some related topics are also discussed.     

Variational approaches to p-Laplacian discrete problems of Kirchhoff-type

Critical point results for Kirchhoff-type discrete boundary value problems are exploited in order to prove that a suitable class possesses at least one solution under an asymptotical behaviour of the potential of the nonlinear term at zero, and also possesses infinitely many solutions under some hypotheses on the behaviour of the potential of the nonlinear term at infinity. Some recent results are extended and improved. Some examples are presented to demonstrate the applications of our main results.