Eur. Phys. J. B 77, 469-478 (2010)
https://doi.org/10.1140/epjb/e2010-00265-4

Scaling relations and critical exponents for two dimensional two parameter maps

¹ Department of Mathematical Physics, National University of Ireland Maynooth, Maynooth, Co. Kildare, Ireland
² School of Theoretical Physics, Dublin Institute for Advanced Studies, Dublin 4, Ireland

Corresponding author: ^a dmh@thphys.nuim.ie

Received: 3 November 2009
Revised: 15 July 2010
Published online: 16 September 2010

Abstract

In this paper we calculate the critical scaling exponents describing the variation of both the positive Lyapunov exponent, λ⁺, and the mean residence time, τ, 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 = f₁(X_n, Y_n; A, B) and Y_n+1 = f₂(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: τ ~ |A – A_c|^-γ, (λ⁺ – λ_c⁺) ~ |A – A_c|^β_A and (λ⁺ – λ_c⁺) ~ |B – B_c|^β_B. 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( – 1), which is analogous to Widom's scaling law. We find strong agreement between the scaling relationship and numerical results.