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Dynamics of self-replicating patterns in the one-dimensional Gray-Scott model. (English) Zbl 0987.34031

Summary: The author studies the self-replicating pattern (SRP) that is observed in the one-dimensional Gray-Scott model from a global bifurcational view point. It is shown that the existence of the hierarchy structure of the limiting points of stationary Turing patterns causes SRP of static type as an aftereffect. The main difficulty lies in the fact that SRP is a real transient phenomenon and it can not be captured as an invariant set in a function space. The aftereffect is the reflection of the fact that each element of the hierarchy structure is connected by unstable manifolds.

MSC:

34C23 Bifurcation theory for ordinary differential equations
35B32 Bifurcations in context of PDEs
35K57 Reaction-diffusion equations
37G10 Bifurcations of singular points in dynamical systems
34C30 Manifolds of solutions of ODE (MSC2000)
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