Diffusion instability of a uniform cycle bifurcating from a separatrix loop

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{d ₁,d ₂},d ₁,d ₂ > 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.