It is of interest to determine whether two scalar time series can belong to the same chaotic dynamical system. In geophysical and biological systems, where the observables relevant to modeling may need to be uncovered, and also in extended physical systems such as turbulent ones, the question of dynamical connectedness is important. Different parts of an extended turbulent system can possess different values of the correlation dimension D2 , and one easily finds different systems with nearly equal D2. Therefore, equality of D2 is neither a necessary nor sufficient condition for dynamical connectedness. Also, the cross-correlation function can be misleading. In Fig. 1 we give the plots of the cross-correlation functions for (x1,x2) of the Roessler and (x1,x3) of the Lorenz attractors . The Lorenz case shows a cross-correlation amplitude of only a few percent. Surprisingly, the amplitude is slightly increased when a multiplicative noise signal of up to 0.1 is injected. The Roessler case, with little correlation at nearly equal times, is more typical of strongly correlated systems. Clearly, a small cross-correlation function is an indecisive test of whether two different observables are dynamically connected. Consequently, for example, the interpretation of the disappearance of the cross-correlation function at two different points in a Rayleigh-Bénard cell at the onset of soft turbulence  is ambiguous.
Chaotic System Correlation Dimension Time Shift Strange Attractor Chaotic Dynamical System
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