Some Comments on Nonlinear Dynamic Behavior and Control of a 3rd-order Duffing Oscillator with External Force

In this work, the nonlinear dynamic behavior of a 3rd-order Duffing oscillator with external force was numerically investigated. The oscillator has an additional nonlinear feedback equation of state coupled to the first equation and therefore has two parameters that determine coupling and control in 3rd-order Duffing oscillator with external force. With the nonlinear dynamic analyses, it was obtained using the maximum Lyapunov exponent sweeping the coupling parameters of the equations; in these analyses, we observed the emergence of shrimp patterns. Bifurcation diagrams, phase maps, and Poincaré maps were also used, which corroborated to determine and confirm the chaotic regions of the 3rd-order Duffing oscillator with external force. Once the regions were determined, a control design for chaos suppression was proposed that kept the 3rd-order Duffing oscillator with external force in periodic orbit, using the optimal linear feedback control due to its high computational efficiency.