2 research outputs found
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 | 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 ( – 1), which is analogous to Widom’s scaling law. We find strong agreement between the scaling relationship and numerical results. Copyright EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2010
Interference of a Bose-Einstein condensate in a hard-wall trap: from nonlinear Talbot effect to formation of vorticity
We theoretically study the coherent expansion of a Bose-Einstein condensate in the presence of a confining impenetrable hard-wall potential. The nonlinear dynamics of the macroscopically coherent matter field results in rich and complex spatio-temporal self-interference patterns demonstrating a nonlinear Talbot effect, and the formation of vorticity and solitonlike structures