"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."
Real chaos and complex time. Bernold Fiedler, Institut fur Mathematik, Freie Universitat Berlin
https://arxiv.org/abs/2310.08136