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Dynamical Systems: Chaotic Attractors and Synchronization using Time-Averaged Partial Observations of the Phase Space
Mathematics and Statistics
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We study the synchronization of chaotic systems when the coupling between themcontains both time averages and stochastic noise. Our model dynamics are givenby the Lorenz equations which are a system of three ordinary differential equationsin the variables X, Y and Z. Our theoretical results show that coupling two copiesof the Lorenz equations using a feedback control which consists of time averages ofthe X variable leads to exact synchronization provided the time-averaging windowis known and sufficiently small. In the presence of noise the convergence is towithin a factor of the variance of the noise. The novelty of our investigationhinges on the analysis of the time averages. We also consider the case whenthe time-averaging window is not known and show that it is possible to tunethe feedback control to recover the size of the time-averaging window. Furthernumerical computations show that synchronization is more accurate and occursunder much less stringent conditions than our theory requires.