Periodic orbits time domain

This is inherently impossible in the time-independent approach because the wavefunction contains the entire history of the wavepacket. The real understanding, however, is provided by classical mechanics. Plotting individual trajectories easily shows the type of internal motion leading to the recurrences which subsequently cause the diffuse structures in the energy domain. The next obvious step, finding the underlying periodic orbits, is rather straightforward. 

A chaotic flow produces either transverse homocHnic or transverse heterocHnic intersections, and/or is able to stretch and fold material in such a way that it produces what is called a horseshoe map, and/or has positive Liapunov exponents. These definitions are not equivalent to each other, and their interrelations have been discussed by Doherty and Ottino [63]. The time-periodic perturbation of homoclinic and heteroclinic orbits can create chaotic flows. In bounded fluid flows, which are encountered in mixing tanks, the homoclinic and heteroclinic orbits are separate streamlines in an unperturbed system. These streamhnes prevent fluid flux from one region of the domain to the other, thereby severely limiting mixing. These separate streamlines generate stable and unstable manifolds upon perturbation, which in turn dictate the mass and energy transports in the system [64-66]. 

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