Oscillatory phenomena in biological systems
Nonlinear oscillations and synchronizations play important roles on biological systems. Patterns of synchronized oscillations are known to be affected by the structure of interacting units. In particular, phase oscillators under noise are affected by network structures [K1]. Besides, non-oscillatory units can exhibit oscillations through interactions among each other: For instance, both the period and the stability of oscillations emerging from gene regulatory networks depends on network structures[K2]. Also, spatially extended excitable systems can exhibit oscillations by introducing long-range interactions[K3]. We are interested in studying the interplay between network structures and oscillatory dynamics.
Inverse problems for PDEs and applications
A lot of phenomena are modeled by PDEs. We are studying inverse problems on identifying those ingredients of interest which cannot be observed directly by partial data of solutions to PDEs. Since inverse problems are usually ill-posed in the sense that either of existence, uniqueness or stability is not satisfied, they are much more difficult than forward problems of solving PDEs. Hence, based on properties of forward problems, we pay close attention to dealing with inverse problems e.g. by means of proving conditional well-posedness by special analysis methods and developing numerical reconstruction methods represented by regularization. Especially, except for the fundamental elliptic, parabolic and hyperbolic equations, recently we are interested in the inverse problems for fractional evolution equations e.g. with half order derivatives, which is a kind of nonlocal models [R1].
[K1] Yasuaki Kobayashi and Hiroshi Kori, Synchronization failure caused by interplay between noise and network heterogeneity, Chaos 26, 094805(2016).
[K2] Yasuaki Kobayashi, Hiroyuki Kitahata and Masaharu Nagayama, Sustained dynamics of a weakly excitable system with nonlocal interactions, Physical Review E 96, 022213 (2017).
[K3] Yasuaki Kobayashi, Tatsuo Shibata, Yoshiki Kuramoto and Alexander S. Mikhailov, Evolutionary design of oscillatory gene networks, European Physical Journal B 76, 167 (2010).
[R1] Y. Liu, Z. Li and M. Yamamoto, “Inverse problems of determining sources of the fractional partial differential equations”, in: Handbook of Fractional Calculus with Applications. Volume 2: Fractional Differential Equations, De Gruyter, Berlin, 2019, 411-430.