, 73:453 | Cite as

Chaotic systems in complex phase space

  • Carl M. BenderEmail author
  • Joshua Feinberg
  • Daniel W. Hook
  • David J. Weir


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.


PT symmetry approach to chaos kicked rotor standard map double pendulum 


05.45.-a 05.45.Pq 11.30.Er 02.30.Hq 


  1. [1]
    C M Bender, Contemp. Phys. 46, 277 (2005); Rep. Prog. Phys. 70, 947 (2007)CrossRefMathSciNetADSGoogle Scholar
  2. [2]
    P Dorey, C Dunning and R Tateo, J. Phys. A: Math. Gen. 40, R205 (2007)zbMATHCrossRefMathSciNetADSGoogle Scholar
  3. [3]
    C M Bender, S Boettcher and P N Meisinger, J. Math. Phys. 40, 2201 (1999)zbMATHCrossRefMathSciNetADSGoogle Scholar
  4. [4]
    A Nanayakkara, Czech. J. Phys. 54, 101 (2004); J. Phys. A: Math. Gen. 37, 4321 (2004)CrossRefMathSciNetADSGoogle Scholar
  5. [5]
    C M Bender, J-H Chen, D W Darg and K A Milton, J. Phys. A: Math. Gen. 39, 4219 (2006)zbMATHCrossRefMathSciNetADSGoogle Scholar
  6. [6]
    C M Bender and D W Darg, J. Math. Phys. 48, 042703 (2007)CrossRefMathSciNetADSGoogle Scholar
  7. [7]
    C M Bender, D D Holm and D W Hook, J. Phys. A: Math. Theor. 40, F81 (2007)zbMATHCrossRefMathSciNetADSGoogle Scholar
  8. [8]
    C M Bender, D D Holm and D W Hook, J. Phys. A: Math. Theor. 40, F793 (2007)CrossRefMathSciNetADSGoogle Scholar
  9. [9]
    C M Bender, D C Brody, J-H Chen and E Furlan, J. Phys. A: Math. Theor. 40, F153 (2007)zbMATHCrossRefMathSciNetADSGoogle Scholar
  10. [10]
    A Fring, J. Phys. A: Math. Theor. 40, 4215 (2007)zbMATHCrossRefMathSciNetADSGoogle Scholar
  11. [11]
    C M Bender and J Feinberg, J. Phys. A: Math. Theor. 41, 244004 (2008)CrossRefMathSciNetADSGoogle Scholar
  12. [12]
    C M Bender and D W Hook, J. Phys. A: Math. Theor. 41, 244005 (2008)CrossRefMathSciNetADSGoogle Scholar
  13. [13]
    C M Bender, D C Brody and D W Hook, J. Phys. A: Math. Theor. 41, 352003 (2008)CrossRefMathSciNetGoogle Scholar
  14. [13a]
    T Arpornthip and C M Bender, Pramana — J. Phys. 73, 259 (2009)Google Scholar
  15. [14]
    A V Smilga, J. Phys. A: Math. Theor. 41, 244026 (2008)CrossRefMathSciNetADSGoogle Scholar
  16. [15]
    A V Smilga, J. Phys. A: Math. Theor. 42, 095301 (2009)CrossRefMathSciNetADSGoogle Scholar
  17. [16]
    S Ghosh and S K Modak, Phys. Lett. A373, 1212 (2009)ADSGoogle Scholar
  18. [17]
    E Ott, Chaos in dynamical systems (Cambridge University Press, Cambridge, 2002), 2nd ed.zbMATHGoogle Scholar
  19. [18]
    M Tabor, Chaos and integrability in nonlinear dynamics: An introduction (Wiley-Interscience, New York, 1989)zbMATHGoogle Scholar
  20. [19]
    S Fishman, Quantum Localization in Quantum Chaos, Proc. of the International School of Physics “Enrico Fermi”, Varenna, July 1991 (North-Holland, New York, 1993)Google Scholar
  21. [19a]
    S R Jain, Phys. Rev. Lett. 70, 3553 (1993)zbMATHCrossRefMathSciNetADSGoogle Scholar
  22. [19b]
    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)Google Scholar
  23. [19c]
    S Fishman, D R Grempel and R E Prange, Phys. Rev. Lett. 49, 509 (1982)CrossRefMathSciNetADSGoogle Scholar
  24. [19d]
    D R Grempel, R E Prange and S Fishman, Phys. Rev. A29, 1639 (1984)ADSGoogle Scholar
  25. [20]
    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)Google Scholar
  26. [21]
  27. [22]
    A J Lichtenberg and M A Lieberman, Regular and stochastic motion (Springer-Verlag, New York, 1983)zbMATHGoogle Scholar
  28. [23]
    D Ben-Simon and L P Kadanoff, Physica D13, 82 (1984)MathSciNetADSGoogle Scholar
  29. [23a]
    R S MacKay, J D Meiss and I C Percival, Phys. Rev. Lett. 52, 697 (1984); Physica D13, 55 (1984)CrossRefMathSciNetADSGoogle Scholar
  30. [23b]
    I Dana and S Fishman, Physica D17, 63 (1985)MathSciNetADSGoogle Scholar
  31. [24]
    B V Chirikov, Phys. Rep. 52, 263 (1979)CrossRefMathSciNetADSGoogle Scholar
  32. [25]
    D L Shepelyansky, Physica D8, 208 (1983)ADSGoogle Scholar
  33. [26]
    J M Greene, J. Math. Phys. 20, 1183 (1981)CrossRefADSGoogle Scholar
  34. [26a]
    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]Google Scholar
  35. [27]
    A Ishikawa, A Tanaka and A Shudo, J. Phys. A: Math. Theor. 40, F397 (2007)zbMATHCrossRefMathSciNetADSGoogle Scholar
  36. [27a]
    T Onishi, A Shudo, K S Ikeda and K Takahashi, Phys. Rev. E68, 056211 (2003)ADSGoogle Scholar
  37. [27b]
    A Shudo, Y Ishii and K S Ikeda, J. Phys. A: Math. Gen. 35, L31 (2002)CrossRefMathSciNetGoogle Scholar
  38. [27c]
    T Onishi, A Shudo, K S Ikeda and K Takahashi, Phys. Rev. E64, 025201 (2001)ADSGoogle Scholar
  39. [28]
    J M Greene and I C Percival, Physica D3, 540 (1982)MathSciNetGoogle Scholar
  40. [28a]
    I C Percival, Physica D6, 67 (1982)MathSciNetADSGoogle Scholar
  41. [29]
    A Berretti and L Chierchaia, Nonlinearity 3, 39 (1990)zbMATHCrossRefMathSciNetADSGoogle Scholar
  42. [29a]
    A Berretti and S Marmi, Phys. Rev. Lett. 68, 1443 (1992)CrossRefADSGoogle Scholar
  43. [29b]
    S Marmi, J. Phys. A23, 3447 (1990)MathSciNetADSGoogle Scholar
  44. [30]
    V F Lazutkin and C Simo, Int. J. Bifurcation Chaos Appl. Sci. Eng. 2, 253 (1997)MathSciNetGoogle Scholar
  45. [31]
    R I McLachlan and P Atela, Nonlinearity 5, 541 (1992)zbMATHCrossRefMathSciNetADSGoogle Scholar
  46. [32]
    B Leimkuhler and S Reich, Simulating Hamiltonian dynamics (Cambridge University Press, Cambridge, 2004)zbMATHGoogle Scholar
  47. [33]
    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.Google Scholar
  48. [33a]
    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) symmetryGoogle Scholar
  49. [34]
    C M Bender and S A Orszag, Advanced mathematical methods for scientists and engineers (McGraw Hill, New York, 1978)zbMATHGoogle Scholar

Copyright information

© Indian Academy of Sciences 2009

Authors and Affiliations

  • Carl M. Bender
    • 1
    Email author
  • Joshua Feinberg
    • 2
    • 3
  • Daniel W. Hook
    • 4
  • David J. Weir
    • 4
  1. 1.Department of PhysicsWashington UniversitySt. LouisUSA
  2. 2.Department of PhysicsUniversity of Haifa at OranimTivonIsrael
  3. 3.Department of Physics, TechnionHaifaIsrael
  4. 4.Blackett LaboratoryImperial College LondonLondonUK

Personalised recommendations