  Real vector fields $\dot{z} = f(z)$ in $\mathbb{R}^N$ extend to
$\mathbb{C}^N$, for complex entire $f$. One known consequence are exponentially
small upper bounds \begin{equation*} \label{*} C_\eta \exp(-\eta/\varepsilon)
\tag{*} \end{equation*} on homoclinic splittings under discretizations of step
size $\varepsilon>0$, or under rapid forcings of that period. Here the complex
time extension of $\Gamma(t)$ is assumed to be analytic in the complex
horizontal strip $|\mathrm{Im}\, t|\leq \eta$. The phenomenon relates to
adiabatic elimination, infinite order averaging, invisible chaos, and backward
error analysis.
  However, what if $\Gamma(t)$ itself were complex entire? Then $\eta$ could be
chosen arbitrarily large. We consider connecting orbits $\Gamma(t)$ between
limiting hyperbolic equilibria $f(v_\pm)=0$, for real $t\rightarrow\pm\infty$.
For the linearizations $f'(v_\pm)$, we assume real eigenvalues which are
nonresonant, separately at $v_\pm$. We then show the existence of singularities
of $\Gamma(t)$ in complex time $t$. In that sense, real connecting orbits are
accompanied by finite time blow-up, in imaginary time. Moreover, the
singularities bound admissible $\eta$ in exponential estimates \eqref{*}.
  The cases of complex or resonant eigenvalues are completely open. We
therefore offer a 1,000 Euro reward to any mathematician, up to and including
non-permanent PostDoc level, who first comes up with a complex entire
homoclinic orbit $\Gamma(t)$, in the above setting. Such an example would
exhibit ultra-exponentially small separatrix splittings, and ultra-invisible
chaos, under discretization.
  We also provide a time-reversible example of an entire periodic orbit with
ultra-sharp Arnold tongues, alias ultra-invisible phaselocking, under
discretization.
