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    Scaling relations and critical exponents for two dimensional two parameter maps


    Stynes, D. and Hanan, G.W. and Pouryahya, S. and Heffernan, Daniel (2010) Scaling relations and critical exponents for two dimensional two parameter maps. European Phyical Journal B, 77. pp. 469-478. ISSN 1434-6028

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    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: 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: τ ∼ |A−Ac|−γ, (λ+ −λ+c ) ∼| A−Ac |βA and (λ+ −λ+c ) ∼| 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( 1 βA − 1), which is analogous to Widom’s scaling law. We find strong agreement between the scaling relationship and numerical results.

    Item Type: Article
    Additional Information: The definitive version of this article is available at DOI: 10.1140/epjb/e2010-00265-4
    Keywords: Scaling relations; critical exponents; two dimensional two parameter maps;
    Academic Unit: Faculty of Science and Engineering > Mathematical Physics
    Item ID: 4450
    Depositing User: Prof. Daniel Heffernan
    Date Deposited: 03 Sep 2013 13:51
    Journal or Publication Title: European Phyical Journal B
    Publisher: EDP Sciences
    Refereed: Yes
    URI:

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