Bifurcation Analysis of a Class Fractional-Oder Nonlinear Chua’s Circuit System
Zhe Zhang*, Toshimitsu Ushio**, Jing Zhang*, Can Ding*, and Feng Liu***
*College of Electrical and Information Engineering, Hunan University
Lushan Road, Yuelu District, Changsha, Hunan 410082, China
**Graduate School of Engineering Science, Osaka University
1-3 Machikaneyama, Toyonaka, Osaka 560-8531, Japan
***School of Automation, China University of Geosciences (Wuhan)
388 Lumo Road, Hongshan District, Wuhan 430074, China
In recent years, with the rapid development of science and technology, dynamic characterization and control of the research circuit system has become not only theoretical but also practical consideration in academic research and practical engineering applications. Therefore, the complex behavior of a research circuit system has become a hot spot in the theoretical field. This thesis is aimed toward the stability criterion and bifurcation of the fractional-order Chua’s circuit system. Despite numerous studies relating to the Chua’s system, most of them focus on its sum of delays. Different from traditional bifurcation analysis of Chua’s circuit system, the parameters are chosen as the bifurcation parameters in this paper such that the stability and bifurcation of the fractional-order Chua’s system is analyzed from a new angle. Then, the conditions of the existence for Hopf bifurcations are achieved by analyzing its characteristic equation. Finally, the validity and rationality of the theory are verified by numerical simulation.
-  A. Rezaei, J. B. Burl, B. Zhou, and M. Rezaei, “A New Real-Time Optimal Energy Management Strategy for Parallel Hybrid Electric Vehicles,” IEEE Trans. on Control Systems Technology, Vol.27, No.2, pp. 830-837, 2017.
-  B. Xiao and S. Yin, “A New Disturbance Attenuation Control Scheme for Quadrotor Unmanned Aerial Vehicles,” IEEE Trans. on Industrial Informatics, Vol.13, No.6, pp. 2922-2932, 2017.
-  M. Yousefikhoshbakht, F. Didehvar, and F. Rahmati, “An Effective Rank Based Ant System Algorithm for Solving the Balanced Vehicle Routing Problem,” Int. J. of Industrial Engineering: Theory, Applications and Practice, Vol.23, No.1, 2016.
-  M. Messadi and A. Mellit, “Control of chaos in an induction motor system with LMI predictive control and experimental circuit validation,” Chaos, Solitons and Fractals, Vol.97, pp. 51-58, 2017.
-  H. Liu, S. Li, G. Li, and H. Wang, “Robust adaptive control for fractional-order financial chaotic systems with system uncertainties and external disturbances,” Information Technology and Control, Vol.46, No.2, pp. 246-259, 2017.
-  R. Almeida, “A Caputo fractional derivative of a function with respect to another function,” Communications in Nonlinear Science and Numerical Simulation, Vol.44, pp. 460-481, 2017.
-  D. Baleanu, G. Wu, and S. Zeng, “Chaos analysis and asymptotic stability of generalized Caputo fractional differential equations,” Chaos Solitons & Fractals, 2017.
-  B. Bao, N. Wang, M. Chen, Q. Xu, and J. Wang, “Inductor-free simplified Chua’s circuit only using two-op-amp-based realization,” Nonlinear Dynamics, Vol.84, No.2, pp. 511-525, 2016.
-  R. Rocha, J. Ruthiramoorthy, and T. Kathamuthu, “Memristive oscillator based on Chua’s circuit: stability analysis and hidden dynamics,” Nonlinear Dynamics, Vol.88, No.4, pp. 2577-2587, 2017.
-  J. Kengne, “On the Dynamics of Chua’s oscillator with a smooth cubic nonlinearity: occurrence of multiple attractors,” Nonlinear Dynamics, Vol.87, pp. 363-375, 2017.
-  Z. Zhang, J. Zhang, and Z. Ai, “A novel stability criterion of the time-lag fractional-order gene regulatory network system for stability analysis,” Communications in Nonlinear Science and Numerical Simulation, Vol.66, pp. 96-108, 2019.
-  O. Guner, “Exp-Function Method and Fractional Complex Transform for Space-Time Fractional KP-BBM Equation,” Communications in Theoretical Physics, Vol.68, No.8, pp. 149-154, 2017.
-  W. S. Chung, S. Zare, and H. Hassanabadi, “Investigation of Conformable Fractional Schrödinger Equation in Presence of Killingbeck and Hyperbolic Potentials,” Communications in Theoretical Physics, Vol.67, No.3, pp. 250-254, 2017.
-  S. M. A. Pahnehkolaei, A. Alfi, and J. A. T. Machado, “Dynamic stability analysis of fractional order leaky integrator echo state neural networks,” Communications in Nonlinear Science and Numerical Simulation, Vol.47, pp. 328-337, 2017.
-  G. Fernandez-Anaya, G. Nava-Antonio, J. Jamous-Galante, R. Muñoz-Vega, and E. G. Hernández-Martínez, “Lyapunov functions for a class of nonlinear systems using Caputo derivative,” Communications in Nonlinear Science and Numerical Simulation, Vol.43, pp. 91-99, 2017.
-  Y. Yang, Y. He, Y. Wang, and M. Wu, “Stability analysis for impulsive fractional hybrid systems via variational Lyapunov method,” Communications in Nonlinear Science and Numerical Simulation, Vol.45, pp. 140-157, 2017.
-  J. T. Machado, V. Kiryakova, and F. Mainardi, “Recent history of fractional calculus,” Communications in Nonlinear Science and Numerical Simulation, Vol.16, No.3, pp. 1140-1153, 2011.
-  J. Sabatier, O. P. Agrawal, and J. A. Tenreiro Machado (Eds.), “Advances in fractional calculus: Theoretical developments and applications in physics and engineering,” Springer, 2008.
-  D. Baleanu, G.-C. Wu, and S.-D. Zeng, “Chaos analysis and asymptotic stability of generalized Caputo fractional differential equations,” Chaos, Solitons and Fractals, Vol.102, pp. 99-105, 2017.
This article is published under a Creative Commons Attribution-NoDerivatives 4.0 Internationa License.