Numerical Analysis of a Liénard Oscillator with Fractional Order Circuit Elements

The Liénard oscillator is a nonlinear dynamical system governed by a differential equation that incorporates both damping and restoring terms. It is widely used in physics and engineering to model self-sustained oscillations, including electrical circuits and biological rhythms. A special case of this oscillator is the Van der Pol oscillator, which exhibits limit cycles and nonlinear behavior. The Liénard oscillator plays a crucial role in understanding complex oscillatory phenomena in both classical and quantum mechanics. In the last half century, fractional-order circuit elements have become very popular. They can be used to model electrical components, circuits, systems, etc. A Liénard oscillator can also be modified to have a fractional-order capacitor and a fractional-order inductor instead of the LTI ones. In this case, the oscillator is still nonlinear, and a numerical method should be used to solve it. Fractional-order derivatives introduce more design parameters that allow designing a more flexible Liénard oscillator with more complex dynamical behavior than one with classical integer-order derivatives or the LTI circuit elements. This article models the use of a fractional-order capacitor and a fractional-order inductor in a Liénard oscillator and examines its dynamical behavior parametrically using the Grünwald-Letnikov Fractional derivative. The performance of fractional- order circuit elements in the Liénard oscillator is evaluated considering their parameters. The results show that usage of the fractional order circuit elements results in a more complex behavior of Liénard oscillators due to having extra parameters.