Is there a system of ordinary differential equations (over the reals) of the form
$$ \begin{align} \dot{x}(t) &= f(x(t)) \\ x(0) &= x_0 \end{align} $$
with the following properties:
- $f(0) = 0$
- $f$ is continuously differentiable everywhere
- $f$ does not have any singularities (implied by the second property)
- $x(t)$ has infinitly many singularities
For example, the system
$$ \begin{align} \dot{x}(t) &= x(t)^2 \\ x(0) &= 1 \end{align} $$
satisfies the first three properties but its solution is
$$ x(t) = \frac{1}{1 - t} $$
so it has only one singularity at $t = 1$ and so it fails the fourth property.
Another example (taken from the comment of Lutz Lehmann) is the system
$$ \begin{align} \dot{x}(t) &= x(t)^2 + 1 \\ x(0) &= 0 \end{align} $$
which satisfies the last three properties as its solution
$$ x(t) = \tan(t) $$
has infinitley many singularities, namely $\frac{\pi}{2}(2 k + 1)$ for $k \in \mathbb{Z}$. However, it fails the first property because $f(0) \neq 0$ (in fact the system has no equilibrium at all).
Is there a system with the above properties which has solutions with infinitely many singularities?