昨年度，講演者の研究指導のもと，斎藤丈士氏と伊原亮輔氏は，それぞれ， 「力学系における中心多様体の数値計算」と「複数のグラフに依存する結合振動子系 の連続極限」に関する研究を行い，研究成果を特別研究論文と修士論文にまとめた． その後，彼らと共同研究を継続し，大きな進展があったので，それらについて報告する． 前半は斎藤丈士氏との，後半は伊原亮輔氏との共同研究成果について説明する．
We propose a numerical approach for computing center manifolds of equilibria in ordinary differential equations. Near the equilibria, the center manifolds are represented as graphs of functions satisfying certain partial differential equations (PDEs). We use a Chebyshev spectral method for solving the PDEs numerically to compute the center manifolds. We illustrate our approach for three examples: A two-dimensional system, the Hénon-Heiles system (a two-degree-of-freedom Hamiltonian system) and a three-degree-of- freedom Hamiltonian system which have one-, two- and four-dimensional center manifolds, respectively. The obtained results are compared with polynomial approximations and other numerical computations.
The continuum limit provides a useful tool for analyzing coupled oscillator networks. Recently, Medvedev (Comm. Math. Sci., 17 (2019), no. 4, pp. 883–898) gave a mathematical foundation for such an approach when the networks are defined on a single graph which may be dense or sparse, directed or undirected, and deterministic or random. In this paper, we consider coupled oscillator networks depending on multiple graphs, and extend his results to show that the continuum limit is also valid in this situation. Especially, we prove that the initial value problem (IVP) of the corresponding continuum limit has a unique solution under general conditions and that the solution becomes the limit of those to the IVP of the networks in some adequate meanings.