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 PT-symmetric dynamical models in complex phase space exhibit strong qualitative similarities.
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These qualitative changes in behaviour were mentioned briefly in talks given by C M Bender and D W Hook at the Workshop on Pseudo-Hermitian Hamiltonians in Quantum Physics VI, held in London, July 2007.
When K and g become imaginary the system becomes invariant under combined PT reflection. However, now P is the spatial reflection, P: θ → θ + π, so that both cos θ and sin θ, and thus the Cartesian coordinates, change sign. The sign of the angular momentum now remains unchanged under parity reflection. This explains the symmetry of the plots when K and g are pure imaginary (see figure 9 and the lower-right plot in figure 11 respectively). This change of symmetry of the system as its couplings vary in complex parameter space is not unusual. For example, at a generic point in coupling space for the three-dimensional anisotropic harmonic oscillator, the only symmetry is parity. However, when any two couplings coincide and are different from the third, the reflection symmetry is enhanced and becomes a continuous symmetry, namely, an O(2) symmetry around the third axis. (There remains parity-time reflection symmetry in the third direction.) When all three couplings coincide, the symmetry is enhanced further and becomes a full O(3) symmetry
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Bender, C.M., Feinberg, J., Hook, D.W. et al. Chaotic systems in complex phase space. Pramana - J Phys 73, 453–470 (2009). https://doi.org/10.1007/s12043-009-0099-3
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DOI: https://doi.org/10.1007/s12043-009-0099-3