2023/10/12 by Fiedler, Bernold
#34C27 #34C29 #34C37 #34M05 #65L20 #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.2310.08136
Real vector fields z = f(z) in ℝN extend to ℂN, for complex entire f. One known consequence are exponentially small upper bounds Cηexp(-η/ε) on homoclinic splittings under discretizations of step size ε>0, or under rapid forcings of that period. Here the complex time extension of Γ(t) is assumed to be analytic in the complex horizontal strip |Im t|≤ η. The phenomenon relates to adiabatic elimination, infinite order averaging, invisible chaos, and backward error analysis. However, what if Γ(t) itself were complex entire? Then η could be chosen arbitrarily large. We consider connecting orbits Γ(t) between limiting hyperbolic equilibria f(v_±)=0, for real t→±∞. For the linearizations f'(v_±), we assume real eigenvalues which are nonresonant, separately at v_±. We then show the existence of singularities of Γ(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 η 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 Γ(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.