Complex Pendulum Oscillations at Complex Times

The last few years have seen a rebirth of the importance of complex trajectories. In addition to the classic uses (saddle point approximation of path integrals and complex paths for -symmetric Hamiltonians), the transitions on the Riemann surface sheets as a possible (deterministic) explanation of classical chaos are also popular. The pendulum is the perfect example to study complex trajectories. This shows trajectories of a pendulum in the complex plane for complex initial conditions and complex time running along a ray through the origin of the complex time plane.