Simultaneous time–frequency control of bifurcation and chaos

Control scheme facilitated either in the time- or frequency-domain alone is insufficient in controlling route-to-chaos, where the corresponding response deteriorates in the time and frequency domains simultaneously. A novel chaos control scheme is formulated by addressing the fundamental characteristics inherent of chaotic response. The proposed control scheme has its philosophical basis established in simultaneous time–frequency control, on-line system identification, and adaptive control. Physical features that embody the concept include multiresolution analysis, adaptive Finite Impulse Response (FIR) filter, and Filtered-x Least Mean Square (FXLMS) algorithm. A non-stationary Duffing oscillator is investigated to demonstrate the effectiveness of the control methodology. Results presented herein indicate that for the control of dynamic instability including chaos to be deemed viable, mitigation has to be adaptive and engaged in the time and frequency domains at the same time.

► Control theory either designed in time-domain or frequency-domain is insufficient to resolve route-to-chaos.
► A novel chaos control scheme is formulated from addressing the fundamental characteristics inherent of chaotic response.
► Result implies that a chaos controller must be adaptive and developed in joint time–frequency domain to be viable.