To deal with the unique characteristics of route-to-chaos, it is necessary to capture the features of time and frequency simultaneously. The reason that conventional control theories fail to control nonlinear system is because they assume static system dynamics, and they don’t control the simultaneous deterioration on time and frequency domain, a signature of route-to-chaos. Based on the investigations, a novel nonlinear control theory is formulated to address and retain the fundamental characteristic inherent of all nonlinear systems undergoing route-to-chaos. One requires no linearization or closed form solution so that it doesn’t have drawbacks of all previous methods and preserves the genuine underlying features of the system. It controls in the time and frequency domain simultaneously without distorting or misinterpreting the true dynamics. The proposed control theory has huge impact on a broad range of applications including precision manufacturing and national security. It restrains the decaying instability of high-speed micro-milling process which often fails the current nonlinear controller, and it is proofed to be a universal decipher in communication.
The following is an example to synchronize two nonautonomous chaotic circuits with different initial conditions and driving frequencies. The synchronization procedure is corrupted with high frequency noise. It is shown that even with the noise, the time domain errors in x, y, and z remain adequately constrained within limited range called practical synchronization. And the frequencies of the response circuit are restored to be of the same bandwidth of the driving signal, though not of exactly the same characteristics.