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Research Papers

Chaotic Behavior and Its Control in a Fractional-Order Energy Demand–Supply System

[+] Author and Article Information
Dongqin Chen

College of Mathematics and Statistics,
Nanjing University of Information
Science and Technology,
Nanjing 210044, China
e-mail: cdqlinsey@163.com

Wenjun Liu

College of Mathematics and Statistics,
Nanjing University of Information
Science and Technology,
Nanjing 210044, China
e-mail: wjliu@nuist.edu.cn

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received March 15, 2016; final manuscript received June 20, 2016; published online July 22, 2016. Assoc. Editor: Dumitru Baleanu.

J. Comput. Nonlinear Dynam 11(6), 061010 (Jul 22, 2016) (7 pages) Paper No: CND-16-1138; doi: 10.1115/1.4034048 History: Received March 15, 2016; Revised June 20, 2016

In this paper, we first propose a fractional-order energy demand–supply system, with the background of the energy resources demand in the eastern regions of China and the energy resources supply in the western regions of China. Then, we confirm the energy resource attractor with a necessary condition about the existence of chaotic behaviors. By employing an improved version of Adams Bashforth Moulton algorithm, we use three cases with different fractional values to verify the necessary condition. Finally, chaos control of fractional-order energy demand–supply system is investigated by two different control strategies: a linear feedback control and an adaptive switching control strategy via a single control input. Numerical simulations show that the energy demand and import in Eastern China and energy supply in Western China are self-feedback controlled around the system’s equilibrium point.

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References

Sun, M. , Tian, L. , and Fu, Y. , 2007, “ An Energy Resources Demand-Supply System and Its Dynamical Analysis,” Chaos Solitons Fractals, 32(1), pp. 168–180. [CrossRef]
Grigorenko, I. , and Grigorenko, E. , 2003, “ Chaotic Dynamics of the Fractional Lorenz System,” Phys. Rev. Lett., 91(3), p. 034101. [CrossRef] [PubMed]
Hu, G. , Li, X. , and Wang, Y. , 2015, “ Pattern Formation and Spatiotemporal Chaos in a Reaction-Diffusion Predator-Prey System,” Nonlinear Dyn., 81(1–2), pp. 265–275. [CrossRef]
Li, C. , and Chen, G. , 2004, “ Chaos in the Fractional Order Chen System and Its Control,” Chaos Solitons Fractals, 22(3), pp. 549–554. [CrossRef]
Daftardar-Gejji, V. , and Bhalekar, S. , 2010, “ Chaos in Fractional Ordered Liu System,” Comput. Math. Appl., 59(3), pp. 1117–1127. [CrossRef]
Sun, M. , Tian, L. , and Yin, J. , 2006, “ Hopf Bifurcation Analysis of the Energy Resource Chaotic System,” Int. J. Nonlinear Sci., 1(1), pp. 49–53.
Podlubny, I. , 1999, Fractional Differential Equations (Mathematics in Science and Engineering), Vol. 198, Academic Press, San Diego, CA.
Zubair, M. , Mughal, M. J. , and Naqvi, Q. A. , 2011, “ On Electromagnetic Wave Propagation in Fractional Space,” Nonlinear Anal. Real World Appl., 12(5), pp. 2844–2850. [CrossRef]
Tarasov, V. E. , and Trujillo, J. J. , 2013, “ Fractional Power-Law Spatial Dispersion in Electrodynamics,” Ann. Phys., 334, pp. 1–23. [CrossRef]
Wu, B. , and Wu, S. , 2014, “ Existence and Uniqueness of an Inverse Source Problem for a Fractional Integrodifferential Equation,” Comput. Math. Appl., 68(10), pp. 1123–1136. [CrossRef]
Luo, Y. , Chen, Y. , and Pi, Y. , 2011, “ Experimental Study of Fractional Order Proportional Derivative Controller Synthesis for Fractional Order Systems,” Mechatronics, 21(1), pp. 204–214. [CrossRef]
Shen, Y. J. , Wei, P. , and Yang, S. P. , 2014, “ Primary Resonance of Fractional-Order Vander Pol Oscillator,” Nonlinear Dyn., 77(4), pp. 1629–1642. [CrossRef]
Baleanu, D. , Golmankhaneh, A. K. , Nigmatullin, R. , and Golmankhaneh, A. K. , 2010, “ Fractional Newtonian Mechanics,” Open Phys., 8(1), pp. 120–125. [CrossRef]
Golmankhaneh, A. K. , Arefi, R. , and Baleanu, D. , 2015, “ Synchronization in a Nonidentical Fractional Order of a Proposed Modified System,” J. Vib. Control, 21(6), pp. 1154–1161. [CrossRef]
Liu, W. J. , and Chen, K. W. , 2015, “ Chaotic Behavior in a New Fractional-Order Love Triangle System With Competition,” J. Appl. Anal. Comput., 5(1), pp. 103–113.
Chen, K. W. , Liu, W. J. , and Park, J. K. , 2016, “ Modified Models for Love and Their Dynamical Properties,” Miskolc Math. Notes, 17(1), pp. 119–132.
Wang, Z. , Huang, X. , and Shi, G. , 2011, “ Analysis of Nonlinear Dynamics and Chaos in a Fractional Order Financial System With Time Delay,” Comput. Math. Appl., 62(3), pp. 1531–1539. [CrossRef]
Wu, G. C. , and Baleanu, D. , 2014, “ Chaos Synchronization of the Discrete Fractional Logistic Map,” Signal Process., 102(9), pp. 96–99. [CrossRef]
Wu, G. C. , Baleanu, D. , and Zeng, S. D. , 2014, “ Discrete Chaos in Fractional Sine and Standard Maps,” Phys. Lett. A, 378(5–6), pp. 484–487. [CrossRef]
Wu, G. C. , and Baleanu, D. , 2014, “ Discrete Fractional Logistic Map and Its Chaos,” Nonlinear Dyn., 75(1–2), pp. 283–287. [CrossRef]
Zhang, R. , and Yang, S. , 2011, “ Response to the Comments on “Adaptive Synchronization of Fractional-Order Chaotic Systems Via a Single Driving Variable,”” Nonlinear Dyn., 66(4), pp. 843–844. [CrossRef]
Li, S.-J. , and Respondek, W. , 2015, “ Orbital Feedback Linearization for Multi-Input Control Systems,” Int. J. Robust Nonlinear Control, 25(9), pp. 1352–1378. [CrossRef]
Xue, Y.-M. , Zheng, B.-C. , and Ye, D. , 2015, “ Quantized Feedback Control Design of Nonlinear Large-Scale Systems Via Decentralized Adaptive Integral Sliding Mode Control,” Math. Probl. Eng., 2015, p. 718924.
Yin, C. , Dadras, S. , and Zhong, S. , 2012, “ Design an Adaptive Sliding Mode Controller for Drive-Response Synchronization of Two Different Uncertain Fractional-Order Chaotic Systems With Fully Unknown Parameters,” J. Franklin Inst., 349(10), pp. 3078–3101. [CrossRef]
Yin, C. , Dadras, S. , Zhong, S. , and Chen, Y.-Q. , 2013, “ Control of a Novel Class of Fractional-Order Chaotic Systems Via Adaptive Sliding Mode Control Approach,” Appl. Math. Model., 37(4), pp. 2469–2483. [CrossRef]
Jia, H. Y. , Chen, Z. Q. , and Qi, G. Y. , 2013, “ Topological Horseshoe Analysis and Circuit Realization for a Fractional-Order Lü System,” Nonlinear Dyn., 74(1–2), pp. 203–212. [CrossRef]
Deng, W. , Li, C. , and Lü, J. , 2007, “ Stability Analysis of Linear Fractional Differential System With Multiple Time Delays,” Nonlinear Dyn., 48(4), pp. 409–416. [CrossRef]
Tavazoei, M. S. , and Haeri, M. , 2008, “ Chaotic Attractors in Incommensurate Fractional Order Systems,” Physica D, 237(20), pp. 2628–2637. [CrossRef]
Diethelm, K. , Ford, N. J. , and Freed, A. D. , 2002, “ A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations,” Nonlinear Dyn., 29(1–4), pp. 3–22. [CrossRef]
Diethelm, K. , Ford, N. J. , and Freed, A. D. , 2004, “ Detailed Error Analysis for a Fractional Adams Method,” Numer. Algorithms, 36(1), pp. 31–52. [CrossRef]
He, J. H. , 1997, “ A New Approach to Nonlinear Partial Differential Equations,” Commun. Nonlinear Sci. Numer. Simul., 2(4), pp. 230–235. [CrossRef]
Matouk, A. E. , 2011, “ Chaos, Feedback Control and Synchronization of a Fractional-Order Modified Autonomous Van der Pol-Duffing Circuit,” Commun. Nonlinear Sci. Numer. Simul., 16(2), pp. 975–986. [CrossRef]
Wang, X. Y. , and Wang, M. J. , 2007, “ Dynamic Analysis of the Fractional-Order Liu System and Its Synchronization,” Chaos, 17(3), p. 033106. [CrossRef] [PubMed]
Haghighi, A. R. , 2014, “ Robust Stabilization of a Class of Three-Dimensional Uncertain Fractional-Order Non-Autonomous Systems,” Int. J. Ind. Math., 6(2), pp. 133–139.
Aghababa, M. P. , Haghighi, A. R. , and Roohi, M. , 2015, “ Stabilisation of Unknown Fractional-Order Chaotic Systems: An Adaptive Switching Control Strategy With Application to Power Systems,” IET Gener. Transm. Distrib., 9(14), p. 1883. [CrossRef]
Weitzner, H. , and Zaslavsky, G. M. , 2003, “ Some Applications of Fractional Equations,” Commun. Nonlinear Sci. Numer. Simul., 8(3–4), pp. 273–281. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

The numerical simulation result for system (4) when (q1,q2,q3)=(0.98,0.97,0.96)

Grahic Jump Location
Fig. 2

The numerical simulation result for system (4) when (q1,q2,q3)=(0.86,0.86,0.84)

Grahic Jump Location
Fig. 3

The numerical simulation result for system (4) when (q1,q2,q3)=(0.44,0.84,0.75)

Grahic Jump Location
Fig. 4

The control of chaotic energy demand–supply system (15) with (k1,k2,k3)=(−0.12,−0.02,−0.042) is stable in the equilibrium point (0.8,0.1255,0.1412)

Grahic Jump Location
Fig. 5

The control processes of energy demand–supply system’s variables

Grahic Jump Location
Fig. 6

State trajectories of the fractional-order chaotic energy demand–supply system

Grahic Jump Location
Fig. 7

Time history of the single control input in control of chaotic system

Grahic Jump Location
Fig. 8

Time response of the adaptive parameter in control of chaotic system

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