On the Dimensional Consistency Aware Fractional Domain Generalization of Simplest Chaotic Circuits

In this research, we generalize the simplest Chua’s chaotic circuit which is even more simpler than the four-element Chua’s circuit in terms of number of elements and the novel simplest chaotic circuit in the fractional domain by using the fractional circuit elements. Unlike the previous works, the time dimensional consistency aware generalization has been performed for the first time in this work. The dynamics of the generalized fractional nonlinear circuits have been analyzed by means of the fractional calculus based on the modified Riemann–Liouville fractional derivative where the Lyapunov exponents and dimensions have also been numerically calculated. We have found that including the dimensional consistency significantly alters the dynamic of the obtained fractional domain Chua’s circuit from that of the previous dimensional consistency ignored counterpart as different Lyapunov exponents and dimensions can be obtained. The conditions for both fractional domain circuits which cease to be chaotic have also been determined where such condition of Chua's circuit presented in this study is different from that of the previous work. This is because the time dimensionalconsistency has been included. The dynamical analyses of these circuits have also been performed where their conditions for being nonchaotic have been verified. Moreover, their emulators have also been realized.