Yoshihiro HIRATA

R-lab., Dept. of Phys., Nagoya Univ., 
Furo-cho, Nagoya, 464-01, 
Japon 
yhirata@allegro.phys.nagoya-u.ac.jp
 

Asymptotic expansion of the 2-dimensional unstable manifold in a 4-dimensional symplectic mapping

Abstract: It is a very interesting problem to construct analytical representations of the unstable manifold in symplectic mappings. Recently an analytical method for constructing the unstable manifold of 2-dimensional symplectic mappings is reported in various systems. This method is called ``asymptotic expansions beyond all orders'', which is a kind of singular perturbation methods. The phase space structure of 4- or more-dimensional symplectic mappings is different from that of 2-dimensional symplectic mappings. Furthermore it is difficult to understand global structure of phase space of the high- dimensional systems and it is important to compose analytical approaches. Therefore we have tried to extend and apply the method mentioned above to 4-dimensional symplectic mappings. In the previous paper (http://xxx.lanl.gov/abs/chao-dyn/9802009), we analytically obtained a 1-dimensional sub-manifold of the unstable manifold in a 4-dimensional symplectic mapping. And we showed that the analytically obtained manifold agrees well with numerical trajectories, and discussed the high-dimensionality of the 4-dimensional symplectic mapping. In this paper we shall construct the 2-dimensional unstable manifold in a 4-dimensional symplectic mapping. We will discuss the phase space structure of 4-dimensional symplectic mappings from the view point of the unstable manifold.