# Fractional-Order Analysis of Modified Chua’s Circuit System with the Smooth Degree of 3 and Its Microcontroller-Based Implementation with Analog Circuit Design

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## Abstract

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## 1. Introduction

## 2. The Modified Chua’s Circuit System with the Smooth Degree of 3

## 3. Fractional-Order Model of Modified Chua’s Circuit

## 4. Dynamical Properties of the FOABS System

#### 4.1. Lyapunov Exponents

#### 4.2. Route to Chaos

#### 4.3. Multistability and Coexisting Attractors

## 5. Microcontroller-Based Implementation of FOABS System

## 6. Electronic Circuit Realization of Modified Chua’s Circuit (1)

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**When $\alpha =9.267,\beta =14,a=-1/6,b=1/16$, chaos can be obtained in system (1) with initial values (−1.01,−0.01,−0.01).

**Figure 2.**Phase portraits of the fractional-order abs system (FOABS) (8) for (

**a**) $\beta =12.8$; (

**b**) $\beta =14$.

**Figure 5.**(

**a**) Bifurcation of the FOABS with respect to q (the blue plot is forward continuation and the red plot is backward continuation); (

**b**) Bifurcation of the FOABS with respect to $\beta $ (the red plot is forward continuation and the blue plot is backward continuation).

**Figure 6.**The coexisting attractors shown by the FOABS system (8).

**Figure 10.**2D phase portraits of the FOABS system obtained from the microcontroller, $\beta =12.8$ (

**a**) $x-y$; (

**b**) $y-z$.

**Figure 11.**The circuit schematic of the electronic design for system (1).

**Figure 12.**Time series of the system (1) state variables.

**Figure 13.**The all phase portraits of electronic circuit design in ORCAD-PSpice for parameters $\alpha =9.267,\beta =14,a=-1/6$ and $b=1/16$ in system (1).

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**MDPI and ACS Style**

Wang, J.; Xiao, L.; Rajagopal, K.; Akgul, A.; Cicek, S.; Aricioglu, B.
Fractional-Order Analysis of Modified Chua’s Circuit System with the Smooth Degree of 3 and Its Microcontroller-Based Implementation with Analog Circuit Design. *Symmetry* **2021**, *13*, 340.
https://doi.org/10.3390/sym13020340

**AMA Style**

Wang J, Xiao L, Rajagopal K, Akgul A, Cicek S, Aricioglu B.
Fractional-Order Analysis of Modified Chua’s Circuit System with the Smooth Degree of 3 and Its Microcontroller-Based Implementation with Analog Circuit Design. *Symmetry*. 2021; 13(2):340.
https://doi.org/10.3390/sym13020340

**Chicago/Turabian Style**

Wang, Junxia, Li Xiao, Karthikeyan Rajagopal, Akif Akgul, Serdar Cicek, and Burak Aricioglu.
2021. "Fractional-Order Analysis of Modified Chua’s Circuit System with the Smooth Degree of 3 and Its Microcontroller-Based Implementation with Analog Circuit Design" *Symmetry* 13, no. 2: 340.
https://doi.org/10.3390/sym13020340