DYNAMICAL ANALYSIS OF NONAUTONOMOUS RLC CIRCUIT WITH THE ABSENCE AND PRESENCE OF ATANGANA-BALEANU FRACTIONAL DERIVATIVE

The Fractional-order derivative (FOD) has a rich history in mathematical science. In comparison, the half derivative has not been extensively used in applied science and engineering. On the other hand, it is widely applicable in many branches of engineering modeling, especially control fractional-order proportional integral derivative(PID), semi-infinite lossy transmission, and many more. Thus, this work investigates the classical RLC circuit with the sense of the Atangana-Baleanu FOD. Concurrently, the Lyapunov spectral analysis is applied to determine whether or not stability and instability appear in the RLC circuit. In the first look, the RLC likes a straightforward dynamical system when the external driving force is zero. But, the dynamical system of the RLC evolves more complicated, and the wandering phase spaces do not illustrate the stability or instability when the time-dependent driving force is not zero. As a result, the Lyapunov exponents play a significant role to analyze the trait of the trajectories of the phase state. It is found from the bifurcation plots and Lyapunov exponent's spectral analysis that the nonautonomous dynamical systems lead to instability and the chaotic state with the manifestation of the local dynamical system, which switches to a stable spiral node with the presence Atangana-Baleanu FOD.

Automated summary

Fractional-order derivatives have a long history in mathematical science, while half derivatives are less commonly used in applied science and engineering. This study investigates the relationship between the classic RLC circuit and the Atangana-Baleanu fractional-order derivative, using Lyapunov spectral analysis to determine the circuit's stability and instability. Bifurcation plots and Lyapunov exponents analysis reveal that nonautonomous dynamical systems lead to instability and chaotic states, which can be transformed into stable spiral nodes with the presence of the Atangana-Baleanu fractional-order derivative.