Chaotic systems in complex phase space

CARL M BENDER^1,*, JOSHUA FEINBERG², DANIEL W HOOK³

and DAVID J WEIR³

¹Department of Physics, Washington University, St. Louis, MO 63130, USA

²Department of Physics, University of Haifa at Oranim, Tivon 36006,

Israel and Department of Physics, Technion, Haifa 32000, Israel

³Blackett Laboratory, Imperial College London, London SW7 2AZ, UK

^*Corresponding author

E-mail: cmb@wustl.edu; joshua@technion.ac.il; d.hook@imperial.ac.uk;

david.weir03@imperial.ac.uk

Abstract. This paper examines numerically the complex classical

trajectories of the kicked rotor and the double pendulum.\ Both of

these systems exhibit a transition to chaos, and this feature is

studied in complex phase space. Additionally, it is shown that the

short-time and long-time behaviours of these two $\cP\cT$-symmetric

dynamical models in complex phase space exhibit strong qualitative

similarities.

Keywords. ${\cal PT}$ symmetry; approach to chaos; kicked rotor;

standard map; double pendulum.

PACS Nos 05.45.-a; 05.45.Pq; 11.30.Er; 02.30.Hq