Application of a center manifold theory to a reaction-diffusion system of collective motion of camphor disks and boats. (English) Zbl 1340.37085

Ei, Shin-Ichiro; Ikeda, Kota; Nagayama, Masaharu; Tomoeda, Akiyasu

Math. Bohem. 139, No. 2, 363-371 (2014).

Summary: Unidirectional motion along an annular water channel can be observed in an experiment even with only one camphor disk or boat. Moreover, the collective motion of camphor disks or boats in the water channel exhibits a homogeneous and an inhomogeneous state, depending on the number of disks or boats, which looks like a kind of bifurcation phenomena. In a theoretical research, the unidirectional motion is represented by a traveling wave solution in a model. Hence it suffices to investigate a linearized eigenvalue problem in order to prove the destabilization of a traveling wave solution. However, the eigenvalue problem is too difficult to analyze even if the number of camphor disks or boats is 2. Hence we need to make a reduction on the model. In the present paper, we apply the center manifold theory and reduce the model to an ordinary differential system.

MSC:

37L10	Normal forms, center manifold theory, bifurcation theory for infinite-dimensional dissipative dynamical systems
35C07	Traveling wave solutions
70K50	Bifurcations and instability for nonlinear problems in mechanics
34K18	Bifurcation theory of functional-differential equations
35K57	Reaction-diffusion equations

Keywords:

center manifold theory; bifurcation; traveling wave solution; collective motion