Symbolic dynamics and Periodic orbits

Symbolic dynamics for Helium atom

Note, that an orbit passing the (r1 = r2) diagonal must collide with the nucleus (r1 = 0 or r2 = 0 lines) before coming back to the diagonal. Empirically, there exists a one-to-one correspondence between binary symbol sequences and collinear trajectories. The symbols 0 and 1 are defined as

    1   if a trajectory crosses the diagonal between two collisions with nucleus (r1 = 0 or r2 = 0);
    0   if the trajectory does not cross the diagonal between two collisions.

Self-retracing periodic orbits

The time reversal symmetric (self-retracing) unstable periodic orbit with symbolic dynamics 1 is shown below

These are the 01 orbit

the 001 orbit

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[1] P.Cvitanovic et al. Classical and Quantum Chaos Chap.24: Helium atom

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updated 3 Apr 2003