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.