A Technique for Studying a Class of Fractional-Order Nonlinear Dynamical Systems

Abstract

In this work, we propose a technique to study nonlinear dynamical systems with fractional-order. The main idea of this technique is to transform the fractional-order dynamical system to the integer one based on Jumarie’s modified Riemann–Liouville sense. Many systems in the interdisciplinary fields could be described by fractional-order nonlinear dynamical systems, such as viscoelastic systems, dielectric polarization, electrode-electrolyte polarization, heat conduction, resistance-capacitance-inductance (RLC) interconnect and electromagnetic waves. To deal with integer order dynamical system it would be much easier in contrast with fractional-order system. Two systems are considered as examples to illustrate the validity and advantages of this technique. We have calculated the Lyapunov exponents of these examples before and after the transformation and obtained the same conclusions. We used the integer version of our example to compute numerically the values of the fractional-order and the system parameters at which chaotic and hyperchaotic behaviors exist.

Frequency Splitting and Transmission Characteristics of MCR-WPT System Considering Non-Linearities of Compensation Capacitors

“…u0 is the excitation source of the system, M represents the coupling inductor ( = ), and k is the coupling coefficient which reflects a transmission distance of the MCR-WPT system. Considering the non-linearities of a capacitor [28,29], the coulomb-volt characteristics of capacitors can be expressed by = ⁄ + + (n = 1,2), where C0 is a linear capacitance, and κ1 and κ2 are non-linear charge coefficients of capacitors. The non-linear coupled dynamics equations can be rewritten following form:…”

“…Then, the following fourth-order differential equation can be obtained: Considering the non-linearities of a capacitor [28,29], the coulomb-volt characteristics of capacitors can be expressed by…”

Fractional‐order–dependent global stability analysis and observer‐based synthesis for a class of nonlinear fractional‐order systems

“…19 It should be noted that similar conclusion is obtained when we compare our method in Theorem 2 with -independent methods in other works. [20][21][22][23][24][25][26][27] The similar procedure for stability test of system (47) for 1.5 < < 2 is performed and the result is depicted in Figure 4. It is noted that by -independent methods, [19][20][21][22][23][24][25][26][27]…”

“…According to this method, the stability of the integer-order system guarantees the stability of the main FOS. This method was extended for stability (check the works of Mahmoud et al 25 and Shao and Zuo 26 ) and observer-based control of the class of No-FOSs, 27 respectively. Similar to the stability analysis approaches based on Lyapunov's direct methods in other works, [19][20][21][22][23][24] the stability conditions of Mahmoud et al, 25 Shao and Zuo, 26 and Qiu and Ji 27 are only true for 0 < < 1 and they are independent of .…”

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

“…[30], which can be thought of as the parallel structured counterpart of Chua's simplest chaotic circuit, has also been considered. e analyses of the generalized circuits have been performed based on Jumarie's modified Riemann-Liouville fractional derivative [31] and nonlinear transformation [32] where the Lyapunov exponents and dimensions have also been calculated. Such derivative has been chosen despite the fact that there exist recent fractional derivatives, e.g., Caputo and Fabrizio [33] and Atangana and Baleanu fractional derivatives [34], because these new derivatives have been found to be controversial.…”