Lipschitz continuous ODE solution intersecting a hyperplane infinitely often in finite time I'm trying to prove that a solution of globally Lipschitz continuous system of ODEs cannot intersect any hyperplane infinitely many times in a finite amount of time. So for example, something like a spiral which converges to it's centre in finite time is not possible, because the centre is an equilibrium and therefore the solution for that point is not unique (either a constant trajectory which stays in the equilibrium, or the spiral itself). Also, the solution can't blow up to infinity because the system is globally Lipschitz.
My intuition so far is that even in more complex cases, this follows from the fact that Lipschitz continuous equations have unique solutions, which would break if the trajectory converges "too fast" to a specific point - like in the case of the spiral above. But I can't quite grasp how to show this for trajectories which do not converge to an equilibrium. For example, take the 2-dimensional above mentioned spiral, but set it into a 3-dimensional system such that $z' = 1$. The spiral still converges to it's centre in finite time, but the centre is not an equilibrium any more. How do I show that the solution is still not unique? Or is it also just "obvious" consequence of Picard–Lindelöf? :) 
 A: The statement is not correct. You’ll find below counterexamples.
Consider
$$\begin{array}{l|rcl}
f : & \mathbb R \times \mathbb R^2 & \longrightarrow & \mathbb R^2 \\
    & (t, (x,y)) & \longmapsto & (1, 3 t^2 \sin(1/t) - t \cos(1/t))\end{array}$$
$f$ is continuous in $t$. $f$ is also independent of $(x,y)$ and therefore uniformly Lipschitz continuous in $(x,y)$.
Consequently, Picard–Lindelöf theorem applies to the IVP
$$(x^\prime(t),y^\prime(t)) = f(t,(x,y)) \text{ and } (x(0),y(0))=(0,0)$$
The map $t \mapsto (t, t^3 \sin(1/t))$ is a (unique) solution. However this solution intersects infinitely many times the hyperplane $y=0$ in the neighborhood of $t=0$.
And if you prefer an autonomous ODE, you can use
$$\begin{array}{l|rcl}
g : & \mathbb R \times \mathbb R^2 & \longrightarrow & \mathbb R^2 \\
    & (t, (x,y)) & \longmapsto & (1, 5 x^4 \sin(1/x) - x^3 \cos(1/x))\end{array}$$
$t \mapsto (t,t^5 \sin(1/t))$ is the unique solution of the IVP
$$(x^\prime(t),y^\prime(t)) = g(t,(x,y)) \text{ and } (x(0),y(0))=(0,0).$$
It intersects infinitely many times the hyperplane $y=0$ in the neighborhood of $t=0$.
