Research Fields — Foundamentals of Chaos

Statitical shadowing Computing or measuring averages is a common practice in science. A key question is whether statistical averages in chaotic systems can be computed or measured reliably under the influence of noise. Recently, a paper Published in Phys. Rev. Lett (89, 184101(2002)), coauthored by Arizona State University team and NUS team (leading by Prof. Lai), has made a breakthrough in this direction. The situations are identified where the invariance of such averages breaks down as the noise amplitude increases through a critical level and, an algebraic scaling law is obtained which relates the change of the averages to the noise variation. This breakdown of shadow ability of statistical averages, as characterized by the algebraic scaling law, can be expected in both low- and high-dimensional chaotic systems. The principal result of this study is that there are situations in chaotic dynamical systems where, if the noise amplitude D exceeds a critical value Dc, the statistical average G can change with noise and scales with D in the following algebraic manner: where α> 0 is a scaling exponent that depends on the details of the system such as the dimensionality, and ΔG(D) = 0 for D<Dc.  The scaling behavior holds in both low and high dimensions, and it is expected to be observable because it occurs in parameter regions of positive Lebesgue measure. An example when calculating the statistical average from a chaotic system: Lorenz system is shown below.   Figure. For the Lorenz system, (a) Statistical average of the function G(x) = z2(t) versus the noise amplitude, where the average is constant for D<Dc, increases for D > Dc, and Dc = 10-2.61. (b) Algebraic scaling between G(D) and D - Dc, where the dashed line represents the theoretical slope Related publications: Y. C. Lai, Zonghua Liu, Guo-Wei Wei, and C. H. Lai, Phys. Rev. Lett. 89, 184101(2002).