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Keywords:
a three-dimensional energy demand-supply system; stability; equilibrium point; delayed feedback control; Hopf bifurcation
a three-dimensional energy demand-supply system; stability; equilibrium point; delayed feedback control; Hopf bifurcation
Summary:
This paper considers a three-dimensional energy demand-supply system which typically demonstrates the relationship between the amount of energy supply and that of energy demand for the two regions in China. A delayed feedback controller is proposed to stabilize the system which was originally unstable even under some other controllers. The stability properties of the equilibrium points are subsequently analyzed and it is found that the Hopf bifurcation appears under some conditions. By using the center manifold theorem and normal form method, we obtain the explicit formulae revealing the properties of the periodic solutions of Hopf bifurcation to show stabilizing effects of the delayed feedback controller. Numerical simulations illustrate effectiveness of our results.
References:
[1] Chen, L., Han, Z. Z.: A survey on time-delayed feedback control for chaotic systems. Control Decision 19 (2004), 1-6. MR 2034040
[2] Huang, C. F., Cheng, K. H., Yan, J. J.: Robust chaos synchronization of four-dimensional energy resource systems subject to unmatched uncertainties. Commun. Nonlinear Sci. Numer. Simul. 14 (2009), 2784-2792. DOI 10.1016/j.cnsns.2008.09.017
[3] Lei, A. Z., Ji, L., Xu, W. G.: Delayed feedback control of a chemical chaotic model. Applied Math. Modelling 33 (2009), 677-682. DOI 10.1016/j.apm.2007.12.001
[4] Ott, E., Grebogi, C., Yorke, Y. A.: Controlling chaos. Physical Rev. Lett. 64 (1990), 1196-1199. DOI 10.1103/physrevlett.64.1196 | MR 1041523 | Zbl 0964.37502
[5] Ruan, S. G., Wei, J. J.: On the zeros of transcendental functions with applications to stability of delay differential equations with two delays. Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis 10 (2003), 863-874. MR 2008751 | Zbl 1068.34072
[6] Sun, M., Tian, L. X.: An energy resources demand-supply system and its dynamical analysis. Chaos, Solitons and Fractals 32 (2007), 168-180. DOI 10.1016/j.chaos.2005.10.085 | MR 2271110 | Zbl 1133.91524
[7] Sun, M., Tian, L. X., Fu, Y., Wei, Q.: Dynamics and adaptive synchronization of the energy resource system. Chaos, Solitons and Fractals 31 (2007), 879-888. DOI 10.1016/j.chaos.2005.10.035 | MR 2262183 | Zbl 1149.34032
[8] Sun, M., Tian, L. X., Jia, Q.: Adaptive control and synchronization of a four-dimensional energy resources system with unknown parameters. Chaos, Solitons and Fractals 39 (2009), 1943-1949. DOI 10.1016/j.chaos.2007.06.117 | Zbl 1197.93100
[9] Sun, M., Tian, L. X., Yin, J.: Hopf bifurcation analysis of the energy resource chaotic system. Int. J. Nonlinear Sci. 1 (2006), 49-53. MR 2299989
[10] Sun, M., Wang, X. F., Chen, Y., Tian, L. X.: Energy resources demand-supply system analysis and empirical research based on non-linear approach. Energy 36 (2011), 5460-5465. DOI 10.1016/j.energy.2011.07.036
[11] Sun, M., Tian, L. X., Xu, J.: Feedback control and adaptive control of the energy resource chaotic system. Chaos, Solitons and Fractals 32 (2007), 1725-1734. DOI 10.1016/j.chaos.2005.12.008 | MR 2299087 | Zbl 1129.93403
[12] Sun, M., Tian, L. X., Zeng, C. Y.: The energy resources system with parametric perturbations and its hyperchaos control. Nonlinear Analysis: Real World Appl. 10 (2009), 2620-2626. DOI 10.1016/j.nonrwa.2008.04.019 | MR 2508472 | Zbl 1163.34308
[13] Sun, Z. K., Xu, W., Yang, X. L., Fang, T.: Inducing or suppressing chaos in a double-well Duffing oscillator by time delay feedback. Chaos, Solitons and Fractals 27 (2006), 705-714. DOI 10.1016/j.chaos.2005.04.041 | Zbl 1091.93008
[14] Tian, Y. Q., Zhu, J. D., Chen, G. R.: A survey on delayed feedback control of chaos. J. Control Theory Appl. 3 (2005), 311-319. DOI 10.1007/s11768-005-0018-1 | MR 2229964 | Zbl 1284.93113
[15] Wang, Z. L.: Chaos synchronization of an energy resource system based on linear control. Nonlinear Analysis: Real World Appl. 11 (2010), 3336-3343. DOI 10.1016/j.nonrwa.2009.11.026 | MR 2683793 | Zbl 1216.34061
[16] Wang, L. J., Chen, X. M., Sun, M., Tian, L X.: Stability of the energy supply-demand stochastic system. Mathematics in Practice and Theory 42 (2012), 105-111.
[17] Wang, M. G., Tian, L. X.: A new four-dimensional energy-saving andemission-reduction system and its linear feedback control. J. Systems Sci. Math. Sci. 32 (2012), 811-820. MR 3056150
[18] Wang, Z., Hu, H. Y.: Stability switches of time-delayed dynamic systems with unknown parameters. J. Sound Vibration 233 (2000), 215-233. DOI 10.1006/jsvi.1999.2817 | MR 1762567 | Zbl 1237.93159
[19] Wang, X., Zhang, F. Q., Zhang, Y. J.: Hopf bifurcation of three species system with time delays. J. Systems Sci. Math. Sci. 30 (2010), 530-540. MR 2771905
[20] Wei, J. J., Wang, H. B., Jiang, W. H.: Theory and Application of Delay Differential Equations. Sciences Press, Beijing 2012.
[21] Xin, B. G., Chen, T., Liu, Y. Q.: Projective synchronization of chaotic fractional-order energy resources demand-supply systems via linear control. Comm. Nonlinear Sci. Numer. Simul. 16 (2011), 4479-4486. DOI 10.1016/j.cnsns.2011.01.021 | MR 2806760 | Zbl 1222.93108
[22] Xu, C. J., Liao, M. X., He, X. F.: Stability and Hopf bifurcation analysis for a Lotka-Volterra predator-prey model with two delays. Int. J. Appl. Math. Computer Sci. 21 (2011), 97-107. DOI 10.2478/v10006-011-0007-0 | MR 2814465 | Zbl 1231.34151
[23] Ye, Z. Y., Yang, G., Deng, C. B.: Time-delay feedback control in a delayed dynamical chaos system and its applications. Chinese Physics B 20 (2011), 1-5. DOI 10.1088/1674-1056/20/1/010207
[24] Zou, E., Li, X. F., Chen, J. G.: Chaos Control and Optimization Applications. National University of Defence Technology Press, Changsha 2002.
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