Brusselator

The Brusselator is a two-species autocatalytic reaction model. In the parametrizationits positive homogeneous equilibrium is $(u_{\ast },v_{\ast })=(\alpha ,\beta /\alpha )$.

For $u^{\prime }=1-(b+1)u+a{u}^{2}v$, $v^{\prime }=bu-a{u}^{2}v$, the equilibrium $(1,b/a)$ has a Hopf bifurcation at $b=1+a$. Coordinates $x=u-1$, $y=-\sqrt{a}[(u-1)+(v-b/a)]$ give the threshold equations $x^{\prime }=-\sqrt{a}y+(1-a)x^2-2\sqrt{a}xy-a{x}^3-\sqrt{a}{x}^{2}y$, $y^{\prime }=\sqrt{a}x$. In the usual planar radial normalization the cubic coefficient is $-(a+2)/8<0$, proving a nondegenerate supercritical bifurcation. The leading small-amplitude angular frequency is $\sqrt{a}$.

For the spatially homogeneous Brusselator with parameters $r,s>0$, set $\alpha =r/(1+s)$. The quadrilateralis a trapping region. The four inward-pointing tests use $\dot{x}\ge 0$ on $x=\alpha$, $\dot{y}\ge 0$ on $y=0$, $\dot{y}\le 0$ on $y=s/\alpha$, and $\dot{x}+\dot{y}=r-x\le 0$ on the sloping edge.

The spatially homogeneous Brusselator has its unique positive equilibrium at $(r,s/r)$. Its Jacobian there has determinant $r^2$ and trace $s-1-r^2$. If $s-1>r^2$, the equilibrium is a repeller inside the Brusselator trapping region; the Poincaré-Bendixson theorem then supplies a periodic orbit.

When $u$ and $v$ have diffusivities $D$ and $1$, respectively, and the homogeneous Brusselator equilibrium is stable, a Turing instability occurs precisely whenAt onset,

On a specified finite domain, only its boundary-compatible wavenumbers can grow. Solve the Brusselator mode determinant for its neutral value of $b$ and minimize over those nonzero modes. Minimizing over all continuous $k$ instead gives the usual threshold and $k_c^2=a/\sqrt{D_u{D}_v}$. The finite-mode minimum must still precede the homogeneous threshold to describe a Turing instability.

For the Brusselator, the stationary spatial threshold is $b_T=(1+a\sqrt{D_u/D_v}{)}^2$ and the homogeneous oscillatory threshold is $b_H=1+a^2$. A genuine Turing instability appears first only if $b_T<b_H$, which rearranges to the displayed diffusivity-ratio bound. Equality gives simultaneous stationary spatial and homogeneous oscillatory neutral modes. Otherwise homogeneous loss of stability occurs first.