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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
C M Bender, Contemp. Phys. 46, 277 (2005); Rep. Prog. Phys. 70, 947 (2007)
P Dorey, C Dunning and R Tateo, J. Phys. A: Math. Gen. 40, R205 (2007)
C M Bender, S Boettcher and P N Meisinger, J. Math. Phys. 40, 2201 (1999)
A Nanayakkara, Czech. J. Phys. 54, 101 (2004); J. Phys. A: Math. Gen. 37, 4321 (2004)
C M Bender, J-H Chen, D W Darg and K A Milton, J. Phys. A: Math. Gen. 39, 4219 (2006)
C M Bender and D W Darg, J. Math. Phys. 48, 042703 (2007)
C M Bender, D D Holm and D W Hook, J. Phys. A: Math. Theor. 40, F81 (2007)
C M Bender, D D Holm and D W Hook, J. Phys. A: Math. Theor. 40, F793 (2007)
C M Bender, D C Brody, J-H Chen and E Furlan, J. Phys. A: Math. Theor. 40, F153 (2007)
A Fring, J. Phys. A: Math. Theor. 40, 4215 (2007)
C M Bender and J Feinberg, J. Phys. A: Math. Theor. 41, 244004 (2008)
C M Bender and D W Hook, J. Phys. A: Math. Theor. 41, 244005 (2008)
C M Bender, D C Brody and D W Hook, J. Phys. A: Math. Theor. 41, 352003 (2008)
T Arpornthip and C M Bender, Pramana — J. Phys. 73, 259 (2009)
A V Smilga, J. Phys. A: Math. Theor. 41, 244026 (2008)
A V Smilga, J. Phys. A: Math. Theor. 42, 095301 (2009)
S Ghosh and S K Modak, Phys. Lett. A373, 1212 (2009)
E Ott, Chaos in dynamical systems (Cambridge University Press, Cambridge, 2002), 2nd ed.
M Tabor, Chaos and integrability in nonlinear dynamics: An introduction (Wiley-Interscience, New York, 1989)
S Fishman, Quantum Localization in Quantum Chaos, Proc. of the International School of Physics “Enrico Fermi”, Varenna, July 1991 (North-Holland, New York, 1993)
S R Jain, Phys. Rev. Lett. 70, 3553 (1993)
S Fishman, Quantum Localization in Quantum Dynamics of Simple Systems, Proc. of the 44th Scottish Universities Summer School in Physics, Stirling, August 1994, edited by G L Oppo, S M Barnett, E Riis and M Wilkinson (SUSSP Publications and Institute of Physics, Bristol, 1996)
S Fishman, D R Grempel and R E Prange, Phys. Rev. Lett. 49, 509 (1982)
D R Grempel, R E Prange and S Fishman, Phys. Rev. A29, 1639 (1984)
P H Richter and H-J Scholz, Chaos in classical mechanics: The double pendulum in stochastic phenomena and chaotic behaviour in complex systems edited by P Schuster (Springer-Verlag, Berlin, 1984)
J S Heyl, http://tabitha.phas.ubc.ca/wiki/index.php/Double pendulum (2007)
A J Lichtenberg and M A Lieberman, Regular and stochastic motion (Springer-Verlag, New York, 1983)
D Ben-Simon and L P Kadanoff, Physica D13, 82 (1984)
R S MacKay, J D Meiss and I C Percival, Phys. Rev. Lett. 52, 697 (1984); Physica D13, 55 (1984)
I Dana and S Fishman, Physica D17, 63 (1985)
B V Chirikov, Phys. Rep. 52, 263 (1979)
D L Shepelyansky, Physica D8, 208 (1983)
J M Greene, J. Math. Phys. 20, 1183 (1981)
The idea to study chaotic systems in complex phase space was introduced in A Tanaka and A Shudo, J. Phys. A: Math. Theor. 40, F397 (2007) ref. [27]. The motivation in these papers was to study the effects of classical chaos on semiclassical tunnelling. In the instanton calculus one must deal with a complex configuration space. Additional complex studies are found in refs [28–30]
A Ishikawa, A Tanaka and A Shudo, J. Phys. A: Math. Theor. 40, F397 (2007)
T Onishi, A Shudo, K S Ikeda and K Takahashi, Phys. Rev. E68, 056211 (2003)
A Shudo, Y Ishii and K S Ikeda, J. Phys. A: Math. Gen. 35, L31 (2002)
T Onishi, A Shudo, K S Ikeda and K Takahashi, Phys. Rev. E64, 025201 (2001)
J M Greene and I C Percival, Physica D3, 540 (1982)
I C Percival, Physica D6, 67 (1982)
A Berretti and L Chierchaia, Nonlinearity 3, 39 (1990)
A Berretti and S Marmi, Phys. Rev. Lett. 68, 1443 (1992)
S Marmi, J. Phys. A23, 3447 (1990)
V F Lazutkin and C Simo, Int. J. Bifurcation Chaos Appl. Sci. Eng. 2, 253 (1997)
R I McLachlan and P Atela, Nonlinearity 5, 541 (1992)
B Leimkuhler and S Reich, Simulating Hamiltonian dynamics (Cambridge University Press, Cambridge, 2004)
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
C M Bender and S A Orszag, Advanced mathematical methods for scientists and engineers (McGraw Hill, New York, 1978)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12043-009-0099-3