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, \(\langle\) 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 \(\rangle\) ~ |A – A c |-γ, (λ + – λ c +) ~ |A – A c |βA and (λ + – λ c +) ~ |B – B 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.
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Stynes, D., Hanan, W., Pouryahya, S. et al. Scaling relations and critical exponents for two dimensional two parameter maps. Eur. Phys. J. B 77, 469–478 (2010). https://doi.org/10.1140/epjb/e2010-00265-4
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DOI: https://doi.org/10.1140/epjb/e2010-00265-4