Prog. Theor. Phys. Vol. 108 No. 6 (2002) pp. 987-997

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Non-Symmetric Periodic Points in Reversible Maps

— Examples from the Standard Map —

Kiyotaka Tanikawa¹ and Yoshihiro Yamaguchi²

¹National Astronomical Observatory, Mitaka 181-8588, Japan
²Teikyo Heisei University, Ichihara 290-0193, Japan

(Received July 15, 2002)

Abstract:

The existence of non-symmetric non-Birkhoff periodic points and non-symmetric periodic points of the accelerator mode in standard-like maps is proved. Positions of these points are approximately determined for large parameter values. The number of such points is shown to diverge as the parameter value goes to infinity. We have found two routes for the appearance of non-symmetric periodic points. One is the equi-period bifurcation of a symmetric periodic point and the other is simultaneous saddle-node bifurcations. These two bifurcations seem to disprove the necessity of `hidden symmetry' introduced by Murakami et al.(2001). We do not know whether or not these are the only routes for the appearance of these points. Numerical examples are considered for the standard map.

URL : http://ptp.ipap.jp/link?PTP/108/987/
DOI : 10.1143/PTP.108.987

References:

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Citing Article(s) :

Progress of Theoretical Physics Vol. 109 No. 2 (2003) pp. 187-202 :
Non-Birkhoff Orbits with 2n Turning Points in the Standard Map

Kiyotaka Tanikawa and Yoshihiro Yamaguchi
Progress of Theoretical Physics Vol. 120 No. 1 (2008) pp. 175-180 :
Birkhoff Periodic Orbits in the Standard-Like Maps

Yoshihiro Yamaguchi and Kiyotaka Tanikawa