| 概要 | Abstract
There has been a lot of studies about spatiotemporal patterns and their dynamics to systems of reaction-diffusion equations. Let us consider variations of bifurcations from a stationary solution. Suppose this stationary solution is spatially uniform. Then we linearize the equation about this equilibrium and we know a bifurcation occurs only if the linearized problem has zero or purely imaginary eigenvalues which we call critical eigenvalues. Moreover if this critical eigenvalue is 0 with spatially non-trivial eigenfunction, a stationary bifurcation to non-uniform steady state occurs. One of the typical examples for this is the well-known Turing instability. On the contrary, if the critical eigenvalues are a pair of purely imaginary numbers with spatially non-trivial eigenfunctions, spatially non-trivial oscillations may occur. The so-called wave instability corresponds to this.
Now what can we say about the bifurcations from a spatially non-uniform steady state? Since the linearization about non-uniform steady states is not easy in general, we need to restrict ourselves to some special cases. One of the possibilities where we can predict bifurcations of non-uniform steady states is in the analysis about degenerate instability points. We focus our attention to the case where the system has the triple degeneracy with 0, 1 and 2 modes (0:1:2-mode interaction). We can, in fact, show that this type of triple degeneracy really occurs in 3-component RD system. And we found the case where the 1-mode stationary solution may become unstable with a pair of purely imaginary critical eigenvalues. Moreover we obtained the complete condition for the Hopf bifurcation from the 1-mode stationary solutions. Thus we can conclude that oscillatory solutions may bifurcate from a non-uniform steady state in 3 component RD systems.
This is the joint work with Takashi Okuda in Meterological college.
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