Stability control of fractional chaotic systems based on a simple Lyapunov function
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2002
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Authors
Tianzeng Li
- School of Mathematics and Statistics, Artificial Intelligence Key Laboratory of Sichuan Province, Sichuan university of Science and Engineering, Zigong 643000, China.
Yu Wang
- School of Mathematics and Statistics, Key Laboratory of Higher Education of Sichuan Province for Enterprise Informationalization and Internet of Things, Sichuan university of Science and Engineering, Zigong 643000, China.
Hongmei Li
- School of Economic and Management, Northwest University, Xi'an 710069, China.
Abstract
In this paper the stabilization of fractional-order chaotic systems and a new property of fractional derivatives are studied. Then we propose a new fractional-order extension of Lyapunov direct method and a control method based on a simple Lyapunov candidate function. The proposed control method can be applied to the stabilization of fractional-order chaotic and hyperchaotic systems. This control method is simple, universal, and theoretically rigorous. Numerical simulations are given for three fractional-order chaotic (or hyperchaotic) systems to verify the effectiveness and the universality of the proposed control method.
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ISRP Style
Tianzeng Li, Yu Wang, Hongmei Li, Stability control of fractional chaotic systems based on a simple Lyapunov function, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 9, 4876--4889
AMA Style
Li Tianzeng, Wang Yu, Li Hongmei, Stability control of fractional chaotic systems based on a simple Lyapunov function. J. Nonlinear Sci. Appl. (2017); 10(9):4876--4889
Chicago/Turabian Style
Li, Tianzeng, Wang, Yu, Li, Hongmei. "Stability control of fractional chaotic systems based on a simple Lyapunov function." Journal of Nonlinear Sciences and Applications, 10, no. 9 (2017): 4876--4889
Keywords
- Lyapunov function
- fractional-order
- stabilization
MSC
References
-
[1]
W. M. Ahmad, R. El-Khazali, Y. Al-Assaf, Stabilization of generalized fractional order chaotic systems using state feedback control , Chaos, Solitons & Fractals, 22 (2004), 141–150.
-
[2]
Z.-B. Bai , On solutions of some fractional m-point boundary value problems at resonance, Electron. J. Qual. Theory Differ. Equ., 2010 (2010), 15 pages.
-
[3]
Z.-B. Bai , Solvability for a class of fractional m-point boundary value problem at resonance, Comput. Math. Appl., 62 (2011), 1292–1302.
-
[4]
Z.-B. Bai, X.-Y. Dong, C. Yin , Existence results for impulsive nonlinear fractional differential equation with mixed boundary conditions, Bound. Value Probl., 2016 (2016), 11 pages.
-
[5]
Z.-B. Bai, T.-T. Qiu , Existence of positive solution for singular fractional differential equation, Appl. Math. Comput., 215 (2009), 2761–2767.
-
[6]
Z.-B. Bai, Y.-H. Zhang , The existence of solutions for a fractional multi-point boundary value problem, Comput. Math. Appl., 60 (2010), 2364–2372.
-
[7]
T. A. Burton , Fractional differential equations and Lyapunov functionals, Nonlinear Anal., 74 (2011), 5648–5662.
-
[8]
Y. Chen, J.-H. Lü, Z.-L. Lin, Consensus of discrete-time multi-agent systems with transmission nonlinearity, Automatica J., 49 (2013), 1768–1775.
-
[9]
K. Diethelm, The analysis of fractional differential equations, Springer, Berlin (2004)
-
[10]
Z. M. Ge, W. R. Jhuang , Chaos, control and synchronization of a fractional order rotational mechanical system with a centrifugal governor, Chaos, Solitons & Fractals, 33 (2007), 270–289.
-
[11]
S. Bhalekar, V. Daftardar-Gejji, Fractional ordered Liu system with time-delay, Commun. Nonlinear Sci. Numer. Simul., 15 (2010), 2178–2191.
-
[12]
R. Gorenflo, F. Mainardi, D. Moretti, P. Paradisi, Time fractional diffusion: A discrete random walk approach, Nonlinear Dynamic, 29 (2002), 129–143.
-
[13]
I. Grigorenko, E. Grigorenko, Chaotic dynamics of the fractional Lorenz system, Physical Review Letters, 2003 (2003), 4 pages.
-
[14]
L.-L. Huang, S.-J. He, Stability of fractional state space system and its application to fractional order chaotic system, Acta Physica Sinica, 60 (2011), 447031–447036.
-
[15]
V. Lakshmikantham, S. Leela, M. Sambandham, Lyapunov theory for fractional differential equations, Commun. Appl. Anal., 12 (2008), 365–376.
-
[16]
V. Lakshmikantham, S. Leela, S. J. Vasundara Devi , Theory of fractional dynamic systems, Cambridge Scientific Publishers, Cambridge (2009)
-
[17]
C.-G. Li, G.-R. Chen, Chaos in the fractional order Chen system and its control, Chaos, Solitons & Fractals, 22 (2004), 549–554.
-
[18]
Y. Li, Y.-Q. Chen, I. Podlubny , Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag Leffler stability, Comput. Math. Appl., 59 (2010), 1810–1821.
-
[19]
X.-P. Li, X.-Y. Lin, Y.-Q. Lin, Lyapunov-Type Conditions and Stochastic Differential Equations Driven By G-Brownian Motion, J. Math. Anal. Appl., 439 (2016), 235–255.
-
[20]
Y.-X. Li, W. K. Tang, G.-R. Chen, Generating hyperchaos via state feedback control, International Journal of Bifurcation and Chaos, 15 (2005), 3367–3371.
-
[21]
T.-Z. Li, Y. Wang, M.-K. Luo , Control of chaotic and hyperchaotic systems based on a fractional order controller, Chinese Physics B, 2014 (2014), 12 pages.
-
[22]
F. Li, C.-N. Wang, J. Ma , Reliability of linear coupling synchronization of hyperchaotic systems with unknown parameters, Chinese Phys. B, 2013 (2013), 9 pages.
-
[23]
T.-Z. Li, Y. Wang, Y. Yong, Designing synchronization schemes for fractional-order chaoticsystem via a single state fractional-order controller, Optik, 125 (2014), 6700–6705.
-
[24]
C.-X. Liu, L. Liu, A novel four-dimensional autonomous hyperchaotic system, Chinese Phys. B, 18 (2009), 2188–2193.
-
[25]
C.-X. Liu, L. Liu, T. Liu, A novel three-dimensional autonomous chaos system , Chaos, Solitons & Fractals, 39 (2009), 1950–1958.
-
[26]
J.-H. Lü, G.-R. Chen, A new chaotic attractor coined, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 12 (2002), 659–661.
-
[27]
J.-H. Lü, G.-R. Chen, Generating multiscroll chaotic attractors: theories, methods and applications, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 16 (2006), 775–858.
-
[28]
D. Matignon , Representations en variables detat de modeles de guides dondes avec drivation fractionnaire , [Ph.D. thesis], Universit Paris (1994)
-
[29]
A. Oustaloup, J. Sabatier, P. Lanusse, From fractal robustness to the CRONE control, Fract. Calc. Appl. Anal., 2 (1999), 1–30.
-
[30]
I. Petras, Fractional-order nonlinear systems: modeling, analysis and simulation, Higher Education Press, Beijing (2011)
-
[31]
I. Podlubny, Fractional Differential Equations, Academic Press, New York (1999)
-
[32]
O. E. Rössler, An equation for hyperchaos, Phys. Lett. A, 71 (1979), 155–157.
-
[33]
J. J. Slotine, W. Li, Applied nonlinear control, Prentice Hall, London (1991)
-
[34]
Z. Wang, A Numerical Method For Delayed Fractional-Order Differential Equations, J. Appl. Math., 2013 (2013), 7 pages.
-
[35]
S.-D. Wang, Y. Chen, Q.-Y. Wang, E. Li, Y.-S. Su, D. Meng, Analysis For Gene Networks Based On Logic Relationships, J. Syst. Sci. Complex., 23 (2010), 999–1011.
-
[36]
Z. Wang, X. Huang, Y.-X. Li, X.-N. Song, A new image encryption algorithm based on fractional-order hyperchaotic Lorenz system, Chinese Phys. B, 2013 (2013), 8 pages.
-
[37]
Z. Wang, X. Huang, G.-D. Shi , Analysis of Nonlinear Dynamics and Chaos In A Fractional Order Financial System With Time Delay, Comput. Math. Appl., 62 (2011), 1531–1539.
-
[38]
Z. Wang, X. Huang, J.-P. Zhou, A Numerical Method For Delayed Fractional-Order Differential Equations: Based On G-L Definition, Appl. Math. Inf. Sci., 7 (2013), 525–529.
-
[39]
Y. Wang, T.-Z. Li, Stability analysis of fractional-order nonlinear systems with delay, Math. Probl. Eng., 2014 (2014), 8 pages.
-
[40]
Y. Wang, T.-Z. Li , Synchronization of fractional order complex dynamical networks, Phys. A, 428 (2015), 1–12.
-
[41]
X.-J. Wang, J. Li, G.-R. Chen, Chaos in the fractional order unified system and its synchronization, J. Franklin Inst., 345 (2008), 392–401.
-
[42]
X.-Y. Wang, M.-J. Wang, A hyperchaos generated from Lorenz system, Phys. A, 387 (2008), 3751–3758.
-
[43]
N.-N. Yang, C.-X. Liu, C.-J.Wu, A hyperchaotic system stabilization via inverse optimal control and experimental research, Chinese Physics B, 2010 (2010), 10 pages.
-
[44]
S. Yuste, J. Murillo, On three explicit difference schemes for fractional diffusion and diffusionwave equations, Physicas Scripta, 2009 (2009), 7 pages.
-
[45]
Y.-L. Zhang, K.-B. Lv, S.-D. Wang, J.-L. Su, D.-Z. Meng, Modeling Gene Networks In Saccharomyces Cerevisiae Based On Gene Expression Profiles, Comput. Math. Methods Med., 2015 (2015), 10 pages.
-
[46]
R.-X. Zhang, S.-P. Yang , Chaos in the fractional-order conjugate Chen system and its circuit emulation, Acta Physica Sinica, 58 (2009), 29571–29576.
-
[47]
Q.-S. Zhong, Impulsive Control for Fractional-Order Chaotic Systems, Chinese Phys. Lett., 25 (2008), 2812–2815.