Real chaos and complex time

Abstract

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.