Robust parameter estimation of chaotic systems

Figure 1.  Example of possible measurements of the states $ (X, Y) $ done beyond the deterministic intervals for the Lorenz 63 chaotic attractor for 64 simulations with slightly randomized initial conditions

Figure 2.  The uncertainty quantification study of the first numerical experiment

Figure 3.  Lorenz 63: Analysis performed after the addition of the second feature vector $ (S, \dot S) $

Figure 4.  The solid line represents to the projection of the solution trajectory onto an $ (X, V) $ plane and the dashed line is the corresponding attractor ($ V = 0 $). The solution is already on the attractor for $ Y $ and $ Z $ variables (the initial values were taken to be $ X(0) = 2.51 $, $ Y(0) = 2.51 $, $ Z(0) = 19.92 $, $ V(0) = -20 $). The selected points on the trajectory projection corresponding to different instants of time are marked with the stars ($ t_1 = 1.0005 $, $ t_2 = 1.0015 $, $ t_3 = 1.0025 $, and $ t_4 = 1.0035 $). To make the results more visual, different scales were used for the $ X $ and $ V $ axes (otherwise the trajectory would have looked like a vertical line drawn to $ V $ axis representing the attractor); as a result, for the same reason of making the illustration more comprehensible, we have drawn the ellipses with main axes $ 1.3\times10^{-4} $ and $ 2 $ instead of circles of radius $ \tilde{R} = 2 $. The corresponding values of $ \frac{1}{K}|\hat{g}_4^0| $ for the chosen time points $ t_1 $, $ t_2 $, $ t_3 $, and $ t_4 $ are $ 13.67 $, $ 5.03 $, $ 1.85 $, and $ 0.68 $, respectively

Figure 5.  Parameter uncertainty quantification for the Wang system

Figure 6.  Chua 7 model: parameters uncertainty quantification