By Henry Abarbanel
When I encountered the belief of chaotic habit in deterministic dynami cal structures, it gave me either nice pause and nice reduction. The beginning of the nice reduction was once paintings I had performed past on renormalization staff houses of homogeneous, isotropic fluid turbulence. on the time I labored on that, it used to be prevalent to ascribe the it seems that stochastic nature of turbulent flows to a couple form of stochastic using of the fluid at huge scales. It used to be easily no longer imagined that with simply deterministic using the fluid might be turbulent from its personal chaotic movement. I bear in mind a colleague remarking that there has been whatever essentially unsettling approximately requiring a fluid to be pushed stochastically to have even the appearance of complicated movement within the speed and strain fields. I definitely agreed with him, yet neither people have been capable of offer the other average advice for the saw, it seems that stochastic motions of the turbulent fluid. So it was once with reduction that chaos in nonlinear structures, specifically, complicated evolution, indistinguish capable from stochastic motions utilizing common instruments resembling Fourier research, seemed in my bag of physics notions. It enabled me to have a physi cally moderate conceptual framework during which to count on deterministic, but stochastic having a look, motions. the nice pause got here from now not understanding what to make of chaos in non linear systems.
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Extra resources for Analysis of Observed Chaotic Data
Since the selection of the first minimum of J(T) is a prescription, this means that for values of T near this minimum the coordinate system produced by time delay vectors is essentially as good as that for the T which is the actual first minimum of J(T). The visual picture of the reconstructed attractor for systems such as the Lorenz attractor or other low dimensional systems changes gracefully for variations around the minimum of J(T) [FS86]. Choosing the actual first minimum appears a useful and workable prescription.
Since none of us will ever acquire an infinite amount of infinitely accurate data and still have time to compute anything else, this means one must turn to some principle other than the same geometric construction used in the unfolding of the attractor to establish a workable value for the time delay T. So that absent another theorem-and such a theorem is so far absent-we must find some prescription for choosing T. To put it another way, we have no hope of finding the very best or optimal time delay without giving some additional rationale for that optimality.
Thus we will have confidence that our choice is founded in some property of the system we wish to describe. Stable linear systems generate zero information, so selecting information generation as the critical aspect of nonlinear systems assures us from the outset that the quantity we focus on is a property of nonlinear dynamics not shared by linear evolution. 2 Chaos as an Information Source In a system undergoing chaotic motion two nearby points in the state space move exponentially rapidly apart in time.
Analysis of Observed Chaotic Data by Henry Abarbanel