Chaotic dynamics reconstruction from noisy data: Phenomenon of predictability worsening for incomplete set of observables

The phenomenon of predictability worsening is studied, which is characteristic for chaotic dynamics, reconstructed from incomplete set of observational data. It is pointed out that in conditions of data deficiency, when there are fewer observables than independent variables, reconstruction procedure inevitably has to deal with additional differentiations of noisy observables, which is the main reason for the phenomenon of predictability worsening to take place, especially in the presence of short-correlated noise.The phenomenon of predictability worsening is illustrated with a theoretical analysis and numerical simulations for the third order system (Rössler attractor), which can be reconstructed using the least squares method on the basis of three, two and one noisy observables. For all the three cases the admissible noise intensity is estimated, which provides acceptable quality of prediction. It is shown that a deficiency of every observable is responsible for a significant (up to 20 ÷ 100 times) decrease of the admissible noise. Numerical results are in a satisfactory agreement with theoretical expectations.