All of my question's notation and a large part of my question comes from this post.
So Arnold proves that the phase curves to the Lotka-Volterra equations
$$\frac{dx}{dt} = kx-axy, \frac{dy}{dt} = -ly+bxy,$$
where all constants are positive, are precisely the lines $p(x)+q(y) = C$ where $p(x)=bx-l \log x$ and $q(x) = ay-k \log y$. I am having trouble proving the following:
Prove that the period of oscillation of Lotka-Volterra equation reaches infinity logarithmically as the equilibrium point reaches the origin.
On the above linked post, I read a comment from the user Did as following:
For every starting point $(x_0,y_0)$ in the positive quadrant, $x'(t)⩽kx(t)$ for every $t⩾0$ hence $x(t)⩽x_0e^{kt}$. In particular, $x(t)⩽l/b$ as long as $x_0e^{kt}⩽l/b$. Every solution curve in the positive quadrant must enter the domain $x⩾l/b$ hence the period $T$ of the solution curve passing by $(x_0,y_0)$ solves $x_0e^{kT}⩾ℓ/b$. This proves that $T→∞$ when $x_0→0$, uniformly on $y_0$. Similarly, $T→∞$ when $y_0→0$, uniformly on $x_0$. The limit $(x_0,y_0)→(0,0)$ is not required.
Why does "every solution curve in the positive quadrant must enter the domain $x⩾ℓ/b$"? Also, why does $T$ approach infinity logarithmically? Solving for $T$ alone in terms of $x_0e^{kT} \geq l/b$ we have that $T \geq \frac{\log l - \log bx_0}{k}$ so we do have that $T$ grows at least on a logarithmic scale, but we do not know (for instance ) whether $T$ grows exponentially. Thanks.
The system has an equilibrium point at $x_*=\frac{l}{b}$ and $y_*=\frac{k}{a}$. Any solution inside the positive quadrant circles around this point.
Solutions that come very close to the coordinate axes will come closest to the origin at $(x_0,y_0)$. So they have to follow a segment close to the $y$ axis from $y_*$ to $y_0$ with almost constant $x$ and then close to the $x$ axis from $x_0$ to $x_*$. Then of course you get the curved bow that closes the loop on the opposite side of the equilibrium $(x_*,y_*)$.
So in first approximation, the closed curve contains the segments $(x_0,y_0e^{-lt})$ for $-(\ln(y_*)-\ln(y_0))/l<t<0$ and $(x_0e^{kt},y_0)$ for $0<t<(\ln(x_*)-\ln(x_0))/k$.
For $x>x_*$ the solution will drift slowly away from the $x$ axis. Inserting the approximation into the second equation gives as a slightly refined approximation $$ \ln(y(t)/y_0)=-l(t-T)+(x_0/k)(e^{kt}-e^{kT}),~~~ t\ge T. $$ There is thus a small first segment where the exponential growth has to outgrow the falling linear term, so that the right side becomes positive and growing. And then a second segment where the doubly exponential factor magnifies the infinitesimal $y_0$ to appreciable values. The first segment is again over an about logarithmic time span, while the second is somewhat doubly logarithmic.
While this does not easily extend to the part away from the axes, the trend is more to logarithmic time spans.