A divergent movement exhibited by singularities of complexified classical trajectories plays an important role in multi-dimensional barrier tunneling. It induces a complex-domain heteroclinic entanglement between the stable manifold of a barrier-top unstable periodic orbit and incoming trajectories. Formation of such a complexified dynamical structure is generic in multi-dimensional barrier systems and enables multiple tunneling paths to contribute to the tunneling phenomena, which is observed as a “fringed” tunneling phenomenon, i.e., the tunneling probability is remarkably fringed due to the interference among multiple tunneling paths.
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