Diffusion instability of a uniform cycle bifurcating from a separatrix loop

A. Yu. Kolesov

Matematičeskie zametki, Tome 63 (1998) no. 5, pp. 697-708

Résumé

We consider the boundary value problem $$ \frac{\partial u}{\partial t} =D\frac{\partial^2u}{\partial x^2}+F(u,\mu), \qquad\frac{\partial u}{\partial x}\Big|_{x=0} =\frac{\partial u}{\partial x}\Big|_{x=\pi}=0. $$ Here $u\in\mathbb R^2$, $D=\operatorname{diag}\{d_1,d_2\}$, $d_1,d_2>0$, and the function $F$ is jointly smooth in $(u,\mu)$ and satisfies the following condition: for $0\mu\ll1$ the boundary value problem has a homogeneous (independent of $x$) 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.