= Solution
The change of time is $d\tau/dt=u$, which is positive in the half-plane under consideration. Dividing the two approximate equations by $u$ gives
$$
\frac{du}{d\tau}=\mu+u-v,\qquad \frac{dv}{d\tau}=\lambda u.
$$
Set $w=v-\mu$. The inhomogeneous system becomes $(u,w)'=(u-w,\lambda u)$, with <eigenvalues> $(1\pm\sqrt{1-4\lambda})/2$. For $\lambda>1/4$, write $\omega=\sqrt{4\lambda-1}/2$. Integrating with the prescribed initial data gives
$$
\boxed{u(\tau)=e^{\tau/2}\left[h\cos(\omega\tau)+\frac{\mu+h/2-v_0}{\omega}\sin(\omega\tau)\right],}
$$
and
$$
\boxed{v(\tau)=\mu+e^{\tau/2}\left[(v_0-\mu)\cos(\omega\tau)+
\frac{\lambda h-(v_0-\mu)/2}{\omega}\sin(\omega\tau)\right].}
$$
These expressions also verify both initial values and their initial <derivatives>. For fixed $\mu>0$ and small $h,v_0$ on the outgoing branch, the sine coefficient in $u$ is positive and dominates the initial cosine term. The first return to small $u$ is near the first positive zero, $\omega\tau\simeq\pi$. Hence
$$
\boxed{\tau_{\rm excursion}\simeq\frac{\pi}{\omega}=\frac{2\pi}{\sqrt{4\lambda-1}}.}
$$
The factor is a half-turn of the growing spiral, not its full rotation period. After this excursion the approximate trajectory approaches the small-$u$, large-$v$ region. In the full system the term $-\lambda v$, omitted during the large excursion, allows the motion to descend along that region and return.
As $\lambda\downarrow1/4$, the rescaled excursion time diverges and the factor $e^{\tau/2}$ produces arbitrarily large excursions. At the critical value the matrix is a nontrivial Jordan block, and the limiting formula is
$$
u(\tau)=e^{\tau/2}[h+(\mu+h/2-v_0)\tau].
$$
For the small outgoing initial data this is positive and unbounded: the <unstable manifold> no longer turns back. Below $1/4$ the two <eigenvalues> are real and positive, so there is again no oscillatory return. This <quadratic escape desingularization> explains \b[loss of the <periodic orbit> at infinity], together with the unstable-manifold escape, as $\lambda$ decreases through $1/4$. It is not a finite homoclinic <bifurcation> to the origin. The divergent $\tau$ travel time must not be identified with the physical period without integrating $dt=d\tau/u$; the time change becomes singular near $u=0$.
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