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

Abstract

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.

The Fractional Order Generalization of HP Memristor Based Chaotic Circuit with Dimensional Consistency

“…where {i} = {2, 3, 4} as the representative of λ 2 , λ 3 and λ 4 because the effect of m i to the location of λ i on the complex plane is equivalent to that of Re[λ i ] if λ i is an eigenvalue for the integer-order circuit (Banchuin, 2020), (Abdelouahab et al, 2012). Noted also that we do not need to formulate m 1 as λ 1 = 0; thus, its location cannot be altered by any parameter changing.…”

“…A fractional derivative with nonlocal-nonsingular kernel (Atangana & Baleanu, 2016) has been applied by Alkahtani (Alkahtani, 2016). Recently, both simplest Chua's chaotic circuit and the simplest current mode chaotic circuit proposed by Jin et al (Jin et al, 2017) have been generalized by Banchuin (Banchuin, 2020). In all of these works but Banchuin, (2020), the dimensional consistency has been ignored.…”

“…On the other hand, the conformable fractional derivative (Khalil et al, 2014) has been applied by Martinez et al (Martinez et al, 2018). For nonlinear circuit, on the other hand, the dimensional consistency has been mentioned only in Banchuin, (2020) which stated that such consistency significantly affects the dynamics of the circuit generalized chaotic circuit. However, the memristors assumed in this previous work are merely the hypothetical ones unlike the HP memristor which is a physically realized device.…”

“…However, the memristors assumed in this previous work are merely the hypothetical ones unlike the HP memristor which is a physically realized device. In addition, the physical meaning of fractional time component has never been assigned in Banchuin, (2020) despite that such meaning is necessary for obtaining a physically meaningful fractional-order generalization because it can be uniquely assigned depending on the circuit under consideration. As an example, it can be the fraction of time constant for an RC circuit (Gómez-Aguilar et al, 2014), (Martinez et al, 2018).…”

“…Therefore, a novel fractional-order generalization of a memristor-based chaotic circuit with dimensional consistency awareness has been performed in this work. Unlike (Banchuin, 2020), the HP memristor has been assumed and the physical meaning of fractional time component assigned. The simplest HP memristor-based chaotic circuit with correct memristor assumption has been chosen as our candidate circuit.…”