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, $\langle$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: Xn+1 = f1(Xn, Yn; A, B) and Yn+1 = f2(Xn, Yn; 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$\rangle$ ~ |A – Ac|-γ, (λ+ – λc+) ~ |A – Ac|βA and (λ+ – λc+) ~ |B – Bc|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 = βB($\frac{1}{\beta_{A}}$ – 1), which is analogous to Widom's scaling law. We find strong agreement between the scaling relationship and numerical results