A method for distinguishing chaotic from quasi-periodic motions in orbit tracking programs

In the design stage of an electron storage ring, its dynamic aperture (non-linear acceptance) is evaluated by means of orbit tracking programs. The closed orbit corresponds to an elliptic fixed point of the complete one-turn transfer map. Moving outwards from it, one generally observes a transition from regular stable motion to either unstable chaotic motion or quasi-periodic motion. For finite tracking times a band of sufficiently stable but chaotic motion is usually included in the notional dynamic aperture. In order to distinguish quasi-periodic from chaotic motions one can calculate fractal dimensions of the computed power spectra of the motion. A method for provisionally assigning a dimension d to a sample of power spectra of an orbit is described and illustrated. A value of d sufficiently different from unity is the hallmark of chaos