近可积哈密尔顿系统的辛算法及其稳定性

近可积哈密尔顿系统的拟周期解的研究被庞加莱称为动力学的基本问题，其数值计算自然是计算数学和科学计算的一个基本而重要的课题。冯康于上世纪八十年代在中国开创并系统发展的哈密尔顿系统辛几何算法和动力系统几何算法引发了大量的后续研究，取得了丰富的理论和应用成果，形成了一个具有重要国际影响的研究方向。数值结果表明辛算法在计算哈密尔顿系统拟周期解和不变换面时具有显著的优势。申请人等在可积系统的情形进行了系统的理论研究，阐明了一些基本问题，取得了一些突破性结果。但是对于近可积系统，辛算法虽然比非辛算法有明显优势，但是其长时间收敛性和稳定性受扰动影响比较明显。本项目拟就近可积系统这类特殊但重要的系统从算法构造、数值实验和理论分析几方面进行深入研究。

The study of quasi-periodic solutions of nearly integrable Hamiltonian systems was called a fundamental problem of dynamics by Poincare. Numerical computation of such solutions is a significantly important subject in computational mathematics and scientific computing. In 1980's Feng Kang initiated and developed systematically the symplectic algorithms of Hamiltonian systems as well as geometric numerical integration of dynamical systems, which has stimulated a lot of subsequent studies, forming an internationally recognized research direction with tremendous impact on the time-integration of dynamical problems. Numerical results showed that symplectic methods are remarkably superior to non-symplectic ones in computing quasi-periodic solutions and invariant tori of Hamiltonian systems. The applicant and some other researchers have conducted some systematic theoretical studies for integrable systems, including formulating some basic problems and obtaining some breakthrough results. For nearly integrable systems, however, symplectic methods performs not as good as integrable ones in the convergence and stability of long time simulations subject to the influence by perturbations. The project devotes to a deeper study in both the construction of numerical methods with computer experiments and theoretical analysis specifically for nearly integrable systems.