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Mathematical Notes

, Volume 63, Issue 5, pp 614–623 | Cite as

Diffusion instability of a uniform cycle bifurcating from a separatrix loop

  • A. Yu. Kolesov
Article
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Abstract

We consider the boundary value problem
$$\frac{{\partial u}}{{\partial t}} = D\frac{{\partial ^2 u}}{{\partial x^2 }} + F(u,\mu ), \frac{{\partial u}}{{\partial x}} \left| {_{x = 0} = } \right.\frac{{\partial u}}{{\partial x}} \left| {_{x = \pi } = } \right.{\text{0}}{\text{.}}$$
. Hereu ∈ ℝ2,D = diag{d1,d2},d1,d2 > 0, and the functionF is jointly smooth in (u, μ) and satisfies the following condition: for 0 <μ ≪ 1 the boundary value problem has a homogeneous (independent ofx) cycle bifurcating from a loop of the separatrix of a saddle. We establish conditions for stability and instability of this cycle and give a geometric interpretation of these conditions.

Key words

weakly nonlinear parabolic equation reaction-diffusion equation Neumann problem, homogeneous cycle Turing-Prigogine theorem homogeneous medium spatially inhomogeneous structure 

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Copyright information

© Plenum Publishing Corporation 1998

Authors and Affiliations

  • A. Yu. Kolesov
    • 1
  1. 1.Yaroslavl State UniversityUSSR

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