= Weak focus of an exponential polynomial planar system
{title2=$\dot x=ye^{xy},\quad\dot y=1-x^4-y^2$}
This real planar <dynamical system> has a <saddle point> at $(-1,0)$ and a repelling <weak focus> at $(1,0)$. The latter has purely imaginary linearized <eigenvalues> $\pm2i$, so its <linearization> has closed elliptical orbits. Nonlinear terms nevertheless produce outward drift.
Set $X=2(x-1)$, $Y=y$, $\tau=2t$ and $r^2=X^2+Y^2$. The cubic expansion is
$$
X_\tau=Y+Y^2+\tfrac12XY^2+\tfrac12Y^3+O(r^4),\qquad
Y_\tau=-X-\tfrac34X^2-\tfrac12Y^2-\tfrac14X^3+O(r^4).
$$
The linear rotation acts on scalar functions by $L=Y\partial_X-X\partial_Y$. For $H_2=r^2/2$, its cubic derivative is $XY^2-3X^2Y/4-Y^3/2$. The homogeneous correction $H_3=7X^3/12+XY^2/2+Y^3/3$ satisfies $LH_3$ equal to the negative of that derivative. The remaining quartic derivative of $H_2+H_3$ is $3X^2Y^2/2-X^3Y$. With
$$
H_4=\tfrac14X^4-\tfrac3{16}X^3Y+\tfrac3{16}XY^3,
$$
direct differentiation gives
$$
\frac d{d\tau}(H_2+H_3+H_4)=\tfrac3{16}r^4+O(r^5)>0
$$
for all sufficiently small nonzero $r$. Meanwhile $H_2+H_3+H_4=r^2/2+O(r^3)>0$. In reversed time it is a strict <Lyapunov function>, proving asymptotic stability of the reversed system and hence repulsion forward in time. The angular speed stays close to $-1$, so the nearby nonzero trajectories spiral rather than approaching along a fixed direction. This explicit calculation illustrates why <a centre in the linearization need not be a nonlinear centre>.
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