Identification of parameters with different orders of magnitude in chaotic systems

The synchronization and parameter identification of six unknown parameters in a chaotic neuron model, which one parameter (about 0.006) is 3 orders of magnitude smaller than the others (about 1-5), is investigated by using Lyapunov stability theory and adaptive synchronization in detail. Two gain coefficients (delta(1), delta(2)) are introduced into the Lyapunov function to obtain certain optimized controllers and parameter observers. A selectable amplification factor k(0) is presented using scale conversion and it is used to improve the accuracy of parameter estimation with the smallest order. The parameter space for gain coefficient (delta) versus amplification factor k(0), and the parameter space delta(1) versus delta 2 at certain fixed amplification factor k(0) are calculated numerically. It is found that the selection values of optimized gain coefficients and amplification factor are critical to estimate the six unknown parameters, particularly for the smallest unknown parameters with an order 0.001. The extensive numerical results show that it is more effective to estimate the smallest unknown parameter r when the two gain coefficients delta(1) and delta(2) are given the same value and a higher amplification factor k(0) is used. It could be useful to estimate the unknown parameters with large deviation of order magnitude, such as a single chaotic Josephson junction coupled to a Resonant tank and other chaotic systems with potential application [Z.Y. Wang, H.Y. Liao, and S.P. Zhou, Study of the DC biased Josephson junction coupled to a Resonant tank, Acta. Phys. Sin. 50(10) (2001), pp. 1996-2000 (in Chinese)].