= Solution
By the <law of mass action>, the net changes in the intermediate concentrations are
$$
\dot X=k_1A-(k_2B+k_4)X+k_3X^2Y,\qquad\dot Y=k_2BX-k_3X^2Y.
$$
The autocatalytic reaction increases $X$ by one and decreases $Y$ by one; its reaction-rate convention is $k_3X^2Y$. With the specified common concentration scale and time scale, choose
$$
\boxed{\alpha=\frac{k_4}{k_1A},\qquad a=\frac{k_3(k_1A)^2}{k_4^3},\qquad b=\frac{k_2B}{k_4}.}
$$
Substitution gives the nondimensional <Brusselator> system. Adding its two steady equations gives $u=1$, and then the second gives $v=b/a$. Its <Jacobian matrix> there is
$$
J=\begin{pmatrix}b-1&a\\-b&-a\end{pmatrix},\qquad\operatorname{tr}J=b-1-a,\quad\det J=a>0.
$$
The equilibrium is asymptotically stable for $b<1+a$ and unstable for $b>1+a$. At $b_c=1+a$ the <eigenvalues> are $\pm i\sqrt a$, crossing the imaginary axis with nonzero speed $\frac d{db}\operatorname{Re}\lambda=1/2$.
The nonlinear <Hopf bifurcation> is nondegenerate. At the threshold, put $x=u-1$ and $y=-\sqrt a[(u-1)+(v-b/a)]$. The translated equations are
$$
x'=-\sqrt a\,y+(1-a)x^2-2\sqrt a\,xy-ax^3-\sqrt a\,x^2y,\qquad y'=\sqrt a\,x.
$$
The <Hopf coefficient of the Brusselator> in the planar radial normalization is $-(a+2)/8<0$: the cubic contribution is $-6a/16$ and the quadratic contribution is $[-2\sqrt a\,2(1-a)]/(16\sqrt a)$. Consequently the bifurcation is supercritical, producing small stable oscillations on the unstable-equilibrium side. Their limiting period is
$$
\boxed{P_\tau\longrightarrow\frac{2\pi}{\sqrt a},\qquad P_t\longrightarrow\frac{2\pi}{k_4\sqrt a}.}
$$
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