Dependence of extensive chaos on the spatial correlation length (substantial revision).pdfVIP

a r X i v : c h a o - d y n / 9 3 0 7 0 1 0 v 2 1 1 N o v 1 9 9 3 Dependence of extensive chaos on the spatial correlation length David A. Egolf? and Henry S. Greenside?? Department of Physics, Duke University, Durham, North Carolina, 27708-0305 USA ? Also the Department of Computer Science, Duke University ? Also the Center for Nonlinear and Complex Studies, Duke University September 30, 1993 E-mail: dae@phy.duke.edu and hsg@cs.duke.edu We consider spatiotemporal chaotic systems1–5 for which spatial correlation func- tions decay substantially over a length scale ξ (the spatial correlation length) that is small compared to the system size L. Numerical simulations6–8 suggest that such systems generally will be extensive, with the fractal dimension D growing in proportion to the system volume for sufficiently large systems (L ? ξ). Intuitively, extensive chaos arises because of spatial disorder. Subsystems that are sufficiently separated in space should be uncorrelated and so contribute to the fractal dimension in propor- tion to their number. We report here the first numerical calculation that examines quantitatively how one important characterization of extensive chaos—the Lyapunov dimension density—depends on spatial disorder, as measured by the spatial correla- 1 tion length ξ. Surprisingly, we find that a representative extensively chaotic system does not act dynamically as many weakly interacting regions of size ξ. More specifically, researchers have conjectured5 that the fractal dimension D of a sufficiently- large homogeneous spatiotemporal chaotic system should obey the following simple scaling relation: D = C(L/ξ)d for L ? ξ. (1) Here C is a constant and d is the spatial dimensionality of the system, e.g., d = 2 for a large- aspect-ratio convection experiment.1 Eq. (1) follows from an assumption of spatial disorder, that parts of a large homogeneous system are uncorrelated and hence dynamically independent when separated by distances larger than the len