Asymmetric Oscillators and Twist Mappings
Asymmetric Oscillators and Twist Mappings
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DOI：
10.1112/jlms/53.2.325
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发表时间：
1996-04
影响因子：
1.2
通讯作者：
R. Ortega
中科院分区：
文献类型：
--
作者：
R. Ortega
In this paper we consider some aspects of the dynamics of the differential equation x"+ ax+-bx-= l+ p (t),(1.1) where x+= max (x, 0), x~= max (—x, 0), a and b are positive constants (a# b) and p (t) is a small 1-periodic function. This equation models the motion of a particle subjected to an asymmetric restoring force and appeared, after separation of variables, as a simplified version of the model of the suspension bridge of Lazer and McKenna [13]. In this model the asymmetry is due to the fact that the cables exert no restoring force when they are not tight. The same equation had been previously considered by Fucik [6] and Dancer [4] in their investigation of boundary value problems associated to equations with'jumping nonlinearities'. After these works, the Dirichlet, Neumann and periodic problems for (1.1) have been the subject of several papers [7, 8, 12,...]. In this paper we study other properties of the solutions of this equation that are also relevant for the physical model and obtain results on boundedness and existence of certain recurrent solutions. We first prove that if p {t) is smooth and small enough then every solution of the equation is bounded. This result is in contrast with the well-known phenomenon of linear resonance that occurs in the case a= b=(2nn) 2 for n e N. In such case unbounded solutions often exist even if p (t) is small. We notice that the smallness of p {t) is essential in our result, since otherwise unbounded solutions can also exist in the asymmetric case. After the boundedness result and assuming again that p is small, we find families of quasiperiodic and subharmonic solutions with large amplitude. This class of solutions resembles the structure of the closed orbits of the autonomous equation (p= 0) in a neighbourhood of infinity.
