Scaling relations and critical exponents for two dimensional two parameter maps

. 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 = 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: 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.