The European Physical Journal B

, Volume 77, Issue 4, pp 469–478 | Cite as

Scaling relations and critical exponents for two dimensional two parameter maps

  • D. Stynes
  • W. G. Hanan
  • S. Pouryahya
  • D. M. Heffernan


In this paper we calculate the critical scaling exponents describing the variation of both the positive Lyapunov exponent, λ +, and the mean residence time, \(\langle\) τ \(\rangle\), near the second order phase transition critical point for dynamical systems experiencing crisis-induced intermittency. We study in detail 2-dimensional 2-parameter nonlinear quadratic mappings of the form: X n+1 = f 1(X n , Y n ; A, B) and Y n+1 = f 2(X n , Y n ; A, B) which contain in their parameter space (A, B) a region where there is crisis-induced intermittent behaviour. Specifically, the Henon, the Mira 1, and Mira 2 maps are investigated in the vicinity of the crises. We show that near a critical point the following scaling relations hold: \(\langle\) τ \(\rangle\) ~ |AA c |-γ , (λ +λ c +) ~ |AA c |βA and (λ +λ c +) ~ |BB c |βB. The subscript c on a quantity denotes its value at the critical point. All these maps exhibit a chaos to chaos second order phase transition across the critical point. We find these scaling exponents satisfy the scaling relation γ = β B (\(\frac{1}{\beta_{A}}\) – 1), which is analogous to Widom’s scaling law. We find strong agreement between the scaling relationship and numerical results.


Critical Exponent Order Phase Transition Scaling Relation Spontaneous Magnetization Positive Lyapunov Exponent 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    E. Ott, C. Grebogi, J.A. Yorke, Phys. Rev. A 36, 5365 (1987) CrossRefMathSciNetADSGoogle Scholar
  2. 2.
    H.L. Yang, Z.Q. Huang, E.J. Ding, Phys. Rev. E 55, 6598 (1997) CrossRefADSGoogle Scholar
  3. 3.
    J.C. Sommerer, C. Grebogi, Int. J. Bifur. Chaos 2, 383 (1992) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    C. Grebogi, E. Ott, Physica D 7, 181 (1983) CrossRefMathSciNetADSGoogle Scholar
  5. 5.
    H.J.H. Clercx, C.J.F. van Heijst, D. Molenaar, Z. Yin. Phys. Rev. E 75, 036309 (2007) CrossRefMathSciNetADSGoogle Scholar
  6. 6.
    E.L. Rempe, F.A. Borott, T. Hada, A.C.L. Chain, W.M. Santana, Y. Kamide, Nonlinear Processes in Geophysics 14, 17 (2007) CrossRefADSGoogle Scholar
  7. 7.
    M.A.F. Sanjuan, G. Tanaka, K. Aihara, Phys. Rev. E 71, 016219 (2005) CrossRefADSGoogle Scholar
  8. 8.
    A. Barugola, J.C. Cathala, C. Mira, L. Gardini, Chaotic dynamics in two-dimensional noninvertible maps (World Scientific, 1996), vol. 20 of A Google Scholar
  9. 9.
    K. Huang, Statistical Mechanics (Wiley, New York, 1987) Google Scholar
  10. 10.
    V. Mehra, R. Ramaswamy, Phys. Rev. E 53, 3420 (1996) CrossRefADSGoogle Scholar
  11. 11.
    J.A. Yorke, C. Grebogi, E. Ott, Phys. Rev. Lett. 57, 1284 (1986) CrossRefMathSciNetADSGoogle Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • D. Stynes
    • 1
  • W. G. Hanan
    • 1
  • S. Pouryahya
    • 1
  • D. M. Heffernan
    • 1
    • 2
  1. 1.Department of Mathematical PhysicsNational University of Ireland MaynoothCo. KildareIreland
  2. 2.School of Theoretical Physics, Dublin Institute for Advanced StudiesDublin 4Ireland

Personalised recommendations