Controllability of fractional-order Chua’s circuit

The complex derivative D α±jβ , with α, β ∈ R+ is a generalization of the concept of integer derivative, where α = 1, β = 0. Fractional-order electric elements and circuits are becoming more and more attractive. In this paper, the complexorder electric elements concept is proposed for the first time, and the complex-order elements are modeled and analyzed. Some interesting phenomena are found that the real part of the order affects the phase of output signal, and the imaginary part affects the amplitude for both the complex-order capacitor and complex-order memristor. More interesting is that the complex-order capacitor can do well at the time of fitting electrochemistry impedance spectra. The complex-order memristor is also analyzed. The area inside the hysteresis loops increases with the increasing of the imaginary part of the order and decreases with the increasing of the real part. Some complex case of complex-order memristors hysteresis loops are analyzed at last, whose loop has touching points beyond the origin of the coordinate system.

“…1(b). system, [25,26] and filter circuit. [27,28] In this paper, a complexorder derivative is introduced to circuit elements, and many interesting phenomena are found.…”

Section: Introductionmentioning

confidence: 99%

Attempt to generalize fractional-order electric elements to complex-order ones

Diao

Zhu

et al. 2017

“…Being a counterpart of the integer-order systems, the fractional-order systems are rich in complex dynamics, such as bifurcations, chaos, hyperchaos, etc. [18][19][20][21] The control and synchronization issues of these systems have also been studied. [18][19][20]22] To deal with the associated FDEs, different versions of fractional derivatives, such as Grunwald-Letnikov fractional derivative, Reimann-Liouville fractional derivative, and Caputo fractional derivative, [23] are possible.…”

Section: Introductionmentioning

confidence: 99%

“…[18][19][20][21] The control and synchronization issues of these systems have also been studied. [18][19][20]22] To deal with the associated FDEs, different versions of fractional derivatives, such as Grunwald-Letnikov fractional derivative, Reimann-Liouville fractional derivative, and Caputo fractional derivative, [23] are possible. Since analytical solutions of nonlinear FDEs are usually not obtainable, the use of the numerical integration method such as the Adams-Basforth-Moulton predictor-corrector scheme (ABM) [24] is common.…”

Section: Introductionmentioning

confidence: 99%

Parrondo’s paradox for chaos control and anticontrol of fractional-order systems

Danca

Tang

2016

We present the generalized forms of Parrondo's paradox existing in fractional-order nonlinear systems. The generalization is implemented by applying a parameter switching (PS) algorithm to the corresponding initial value problems associated with the fractional-order nonlinear systems. The PS algorithm switches a system parameter within a specific set of N ≥ 2 values when solving the system with some numerical integration method. It is proven that any attractor of the concerned system can be approximated numerically. By replacing the words "winning" and "loosing" in the classical Parrondo's paradox with "order" and "chaos", respectively, the PS algorithm leads to the generalized Parrondo's paradox: chaos 1 + chaos 2 + • • • + chaos N = order and order 1 + order 2 + • • • + order N = chaos. Finally, the concept is well demonstrated with the results based on the fractional-order Chen system.

“…[9] Intensive studies confirmed theoretically and physically that the chaotic and hyperchaotic behaviors exist the nonlinear fractional-order systems. [3,[8][9][10][11][12][13][14][15] Meanwhile, synchronization of fractional-order chaotic systems also has attracted increasing attention due to its potential applications in secure communication [16,17] and digital cryptography. [18][19][20] A variety of control schemes have been proposed to control and synchronize fractional-order chaos in infinite or finite time, [21,22] such as active control, [23,24] sliding mode control, [25][26][27] adaptive control, [28] passive control, [29] and impulsive control.…”

Section: Introductionmentioning

confidence: 99%

A novel adaptive-impulsive synchronization of fractional-order chaotic systems

Andrew

Chu

et al. 2015

A novel adaptive-impulsive scheme is proposed for synchronizing fractional-order chaotic systems without the necessity of knowing the attractors' bounds in priori. The nonlinear functions in these systems are supposed to satisfy local Lipschitz conditions but which are estimated with adaptive laws. The novelty is that the combination of adaptive control and impulsive control offers a control strategy gathering the advantages of both. In order to guarantee the convergence is no less than an expected exponential rate, a combined feedback strength design is created such that the symmetric axis can shift freely according to the updated transient feedback strength. All of the unknown Lipschitz constants are also updated exponentially in the meantime of achieving synchronization. Two different fractional-order chaotic systems are employed to demonstrate the effectiveness of the novel adaptive-impulsive control scheme.