Poincare 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

In this paper, a design of Poincare 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 Poincare map from the structure of several local maps. An expression of the generalized controlled Poincare 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 T/n-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. (C) 2019 Elsevier Ltd. All rights reserved.