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:	http://epjb.epj.org/
Use Licence:	This item is available under a Creative Commons Attribution Non Commercial Share Alike Licence (CC BY-NC-SA). Details of this licence are available here

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