| In this paper we investigate the persistence of the lower dimensional invariant tori for the symplectic mapping with parameters. We proved that for most of the parameters, the lower dimensional elliptic invariant tori and lower dimensional hyperbolic invariant tori of the twist symplectic mappings survive under the symplectic perturbation.In 60's of the last century, famous mathematicians Kolmogorov, Arnold and Moser established KAM theory. The celebrated theory is the landmark of the research of the Hamiltonian systems. It gives a reasonable explanation to motion rule of the solar system which has puzzled scientists for a long time and shows a lot of understanding of Hamiltonian systems. Before the establishment of KAM theory, it had been believed that the orbit was ergodic on the energy surfaces for almost all Hamiltonian systems. But according to KAM theory, the orbits of the classical In dimensional nearly integrable Hamiltonian system are only ergodic on n dimensional invariant tori, rather than on the energy surfaces (2n- 1 dimensional). So far, KAM theory has become a powerful toolin studying dynamics properties in nearly integrable Hamiltonian systems and nearly twist mappings and has been applied to many physical problems.The KAM theory of the symplectic mappings is to descibe the dynamics properties of nearly twist symplectic mappings. The so-called nearly twist symplectic mappings are the symplectic ones which perturbat given twist mappings. In the action-angle coordinates the twist mapping split the phase space into the tori, in which the orbits of the mapping are quasi-periodic. According to the classical KAM theory of the symplectic mappings, most of invariant tori of the twist symplectic mapping survive after the small symplectic perturbation if the twist mapping satisfies some conditions. The KAM theory of symplectic mapping has been applied to the numerical computation. For conservative systems Channel and Scovel, Feng Kang, Sanz-Serna and Calvo, Shang Zaijiu etc. proposted and applied it to the symplectic logarithm of the integrable Hamiltonian systems. So the research of KAM theory of symplectic mappings are important and pratical in applications. The previous results for the nearly twist mapping were mainly focused on the existence of the highest dimensional invariant tori. In this paper, we shall investigat the existence of the lower dimensional invarint tori of nearly twist symplectic mapping and shall prove several KAM type theorems.Consider where is bounded closed set in R. In case of unperturbed system, i.e. P0= 0, the mapping (1) becomes to the following twist mappingObviously,this mapping has lower dimensional invariant toriThe types of lower dimensional tori may be different when have different forms. Corresponding to these,there are different dynamics. In order to investigate the persistence of somewhat invariant tori, we need different more explicit analysis and different methods.Firstly, we investigate the persistence of lower dimensional elliptic type invariant tori. AssumeThen the orbits of the mapping (2) rotate around the original point, neither shrinking to the interior nor extending to the exterior. In this case, we call the invariant tori T{y) of (2) the lower dimensional elliptic invariant tori. For the nearly twist symplectic (1), the normal frequency will become a factor that causes the small divisor. Therefore we must consider tangent frequency as well as the normal frequency using KAM iteration. Now the number of variables become much greater than that in the case of highest dimensional invariant tori because of the appearance of normal variables. By carrying out a series of tendons computing, we proved that most of the invariant tori survive underthe small syinplectic perturbation if the twist mappings satisfy some condition. Moreover, the orbits of the mappings on these invariant tori are qusi-periodic, whose frequencies are close to the original ones. We can state this result in the following theorem,Theorem 1 Assume w0= id and the normal fre... |
