Mathematical analysis to coupled oscillators system with a conservation law Isamu Ohnishi (collaborated with T. Miyaji) Dept. of Math. and Life Sciences, Graduate School of Science, Hiroshima University isamu_o@math.sci.hiroshima-u.ac.jp AbstractF We are interested in bifurcation structure of stationary solution for 3-component reaction diffusion system with a conservation law in the following: u/t = E (D1 u) + f(u, v) + w, v/t = E (D2v) + g(u, v), w/t = (D3 w) - f(u, v) - w, where the functions f(u, v) and g(u, v) are chosen in such forms that the local oscillator du/dt = f(u, v), dv/dt = g(u, v) can undergo the supercritical Hopf bifurcation. Obviously, the total amount of u + w is conserved under homogeneous Neumann (no-flux) boundary condition and some natural and appropriate conditions. The system describes the time-evolution of (u, v, w), which may obtain some spatio-temporal oscillation solutions. We explain the mechanism heuristically in the following: We note that if w does not exist, then this two component system is a coupled oscillators system with diffusion coupling. This system has temporally oscillation solutions homogeneously spatially, but does not have any spatially structural solution. But, there is a kind of interesting phenomena which undergoes Turing-Hoph bifurcation. A typical case is described by 3 component system. The characteristic property is that the diffusion rates are quite different between the first variable u and the third variable w. Our objective is that we prove such a "wave bifurcation" in this system mathematically rigorously in the system, and understand how many structural varieties this system has from the viewpoint of bifurcation of stationary solutions. Note that this three component system has a mass conservation law, and its freedom of degree is less than the usual three component system in some sense.