Hopf dances near the tips of Busse balloons (Q428433)

This paper deals with stationary periodic patterns related with reaction-diffusion systems, in one spatial dimension, of the form NEWLINE\[NEWLINE\left\{\begin{aligned} \epsilon^2U_t&=U_{xx}-\epsilon^2\mu U+U^{\alpha_1}V^{\beta_1},\\ V_t&=\epsilon^2V_{xx}-V+U^{\alpha_2}V^{\beta_2}, \end{aligned}\right.\tag{1}NEWLINE\]NEWLINE with \(0<\epsilon\ll1\), \(\mu>0\), \(\alpha_2<0\), \(\beta_1>1\), \(\beta_2>1\), \(d=(\alpha_1-1)(\beta_2-1)-\alpha_2\beta_1>0\), known as the generalized Gierer-Meinhardt model.The associated Busse balloon is the region, in the wave number-parameter space, for which stable periodic patterns exist. The Hopf dance, near the tip of the Busse balloon is, first, graphically presented in the case of the Gray-Scott model (Sections 2--3). The results are generalized to system (1) in Sections 4--5. In the last section, possible extensions to larger classes of systems are discussed.