Quasi-periodic Orbits in Siegel Disks/Balls and the Babylonian Problem

    2018, Volume 23, Number 6, pp.  735-750

    Author(s): Saiki Y., Yorke J. A.

    We investigate numerically complex dynamical systems where a fixed point is surrounded by a disk or ball of quasi-periodic orbits, where there is a change of variables (or conjugacy) that converts the system into a linear map. We compute this “linearization” (or conjugacy) from knowledge of a single quasi-periodic trajectory. In our computations of rotation rates of the almost periodic orbits and Fourier coefficients of the conjugacy, we only use knowledge of a trajectory, and we do not assume knowledge of the explicit form of a dynamical system. This problem is called the Babylonian problem: determining the characteristics of a quasi-periodic set from a trajectory. Our computation of rotation rates and Fourier coefficients depends on the very high speed of our computational method “the weighted Birkhoff average”.
    Keywords: quasi-periodic orbits, rotation rates, weighted Birkhoff averaging, Siegel disk, Siegel ball
    Citation: Saiki Y., Yorke J. A., Quasi-periodic Orbits in Siegel Disks/Balls and the Babylonian Problem, Regular and Chaotic Dynamics, 2018, Volume 23, Number 6, pp. 735-750

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