Poincaré maps design for the stabilization of limit cycles in non-autonomous nonlinear systems via time-piecewise-constant feedback controllers with application to the chaotic Duffing oscillator

• Published in: Chaos, solitons and fractals solitons and fractals, 2019-10, Vol.127, p.127-145
• Main Author: Gritli, Hassène
• Format: Article
• Language: English
• Subjects: Controlled Poincaré map; Duffing oscillator; Limit cycle; Non–autonomous nonlinear systems; Time–piecewise–constant controller
• ISSN: 0960-0779; 1873-2887
• DOI: 10.1016/j.chaos.2019.06.035

•A design of an explicit analytical expression of Poincaré maps for non-autonomous periodically forced nonlinear systems is achieved.•A linearization of the nonlinear dynamics around a desired period-m limit cycle is performed.•We design three different time-piecewise-constant control laws to stabilize the limit cycle.•An application to the chaotic Duffing oscillator is realized to stabilize period-1 and period-2 limit cycles. In this paper, a design of Poincaré maps and time–piecewise–constant state–feedback control laws for the stabilization of limit cycles in periodically–forced, non–autonomous, nonlinear dynamical systems is achieved. Our methodology is based mainly on the linearization of the nonlinear dynamics around a desired period–m unstable limit cycle. Thus, this strategy permits to construct an explicit mathematical expression of a controlled Poincaré map from the structure of several local maps. An expression of the generalized controlled Poincaré map is also developed. To make comparisons, we design three different time–piecewise–constant control laws: an mT–piecewise–constant control law, a T–piecewise–constant control law and a Tn–piecewise–constant control law. As an illustrative application, we adopt the chaotic Duffing oscillator. By applying the designed piecewise–constant control laws to the Duffing oscillator, the system is stabilized on its desired period–m limit cycle and hence the chaotic motion is controlled.