Analytical Expression for Low - dimensional Resonance Island in 4 - dimensional Symplectic Map

We study a 2-and a 4-dimensional nearly integrable symplectic maps using a singular perturbation method. Resonance island structures in the 2-and 4-dimensional maps are successfully obtained. Those perturbative results are numerically confirmed. Chaotic Hamiltonian systems with many degrees of freedom are not only of interest in view of the foundation of dynamical system, but also have a lot of applications. Classical Hamiltonian mechanics has been applied to various physical problems, such as chemical reactions, plasma physics and solar systems. 1) Although it is important to study such systems, we cannot understand the global flow generated by a Hamiltonian in the phase space due to the high-dimensionality. For a low-dimensional system, the phase space is visualized and then one overinterprets high-dimensional Hamiltonian flows beyond ones in a 2-dimensional phase space, but such imaginations are not guaranteed. On the other hand, the origin of the global Hamiltonian chaos is an overlap of two resonance islands in the phase space. 1) Then a fundamental problem for the study of Hamiltonian chaos is to analyze a resonance structure. To tackle this problem for systems with many degrees of freedom, we take a singular perturbation method. Using a systematic singular perturbation method, we can show the functional form of an island structure even for high-dimensional systems without information on the geometry of the phase space. In this article, we take a renormalization method which is one of singular perturbation methods. All sorts of renormalization methods remove secular or divergent terms from a naive perturbation series by renormalized integration constants of the non-perturbation solution, and each prescription of the method does not depend on the detail of a given system. 2)–5) One of the reformulated renormalization method which we take here is easily applied to non-chaotic systems and also chaotic maps. 5)–7) Using this reformulated renormalization method, we tackle the standard map defined in 2-dimensional phase and the Froeschlé map defined in 4-dimensional phase space. Both maps are chaotic and are known for the standard models to study Hamiltonian chaos.