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Bifurcation Cascades and Self-Similarity of Periodic Orbits with Analytical Scaling Constants in Hénon–Heiles Type Potentials

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Abstract

We investigate the isochronous bifurcations of the straight-line librating orbit in the Hénon–Heiles and related potentials. With increasing scaled energy e, they form a cascade of pitchfork bifurcations that cumulate at the critical saddle-point energy e=1. The stable and unstable orbits created at these bifurcations appear in two sequences whose self-similar properties possess an analytical scaling behavior. Different from the standard Feigenbaum scenario in area preserving two-dimensional maps, here the scaling constants α and β corresponding to the two spatial directions are identical and equal to the root of the scaling constant δ that describes the geometric progression of bifurcation energies en in the limit n→∞. The value of δ is given analytically in terms of the potential parameters.

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  1. Institute for Theoretical Physics, University of Regensburg, D-9304, Regensburg, Germany

    Matthias Brack

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Brack, M. Bifurcation Cascades and Self-Similarity of Periodic Orbits with Analytical Scaling Constants in Hénon–Heiles Type Potentials. Foundations of Physics 31, 209–232 (2001). https://doi.org/10.1023/A:1017582218587

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