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Chaos synchronization of an energy resource system based on linear control. (English) Zbl 1216.34061
The author considers a system of ordinary differential equations of third order
$x'(t)=f(x(t)),\quad x\in\mathbb{R}^{3}$
proposed by M. Sun, L. Tian and Y. Fu [Chaos Solitons Fractals 32, No. 1, 168–180 (2007; Zbl 1133.91524)]. For this energy resource demand-supply system, the following problem is studied: find a function (controller) $$u(x)$$ such that the following “controlled” system
$y'=f(y)+u(x-y),\quad x'=f(x),\quad x,y\in\mathbb{R}^{3}$
is synchronized, i.e., $$\|x(t)-y(t)\|\to 0$$ as $$t\to \infty$$ for all initial conditions.
For the specific system, the authors prove that the functions $$u_{1}(x)=\mathrm{diag}\{g,g,g\}x$$, $$u_{1}(x)=\mathrm{diag}\{g,0,g\}x$$, and $$u_{1}(x)=\mathrm{diag}\{g,0,0\}x$$ can be used as controllers if $$g$$ is sufficiently large.

##### MSC:
 34H10 Chaos control for problems involving ordinary differential equations 34H05 Control problems involving ordinary differential equations 34D06 Synchronization of solutions to ordinary differential equations
##### Keywords:
linear control; energy resource system
Full Text:
##### References:
 [1] Pecora, L.M.; Carroll, T.L., Synchronization in chaotic systems, Phys. rev. lett., 64, 821-824, (1990) · Zbl 0938.37019 [2] Lu, J.; Cao, J., Adaptive complete synchronization of two identical or different chaotic (hyperchaotic) dynamical systems with fully unknown parameters, Chaos, 15, 1-10, (2005), (043901) · Zbl 1144.37378 [3] Ma, J.; Ying, H.P.; Pu, Z.S., An anti-control scheme for spiral under Lorenz chaotic signal, Chin. phys. lett., 22, 1065-1068, (2005) [4] Ghosh, D.; Banerjee, S., Adaptive scheme for synchronization-based multiparameter estimation from a single chaotic time series and its applications, Phys. rev. E, 78, 056211, (2008) [5] Ma, J.; Wang, Q.Y.; Jin, W.Y.; Xia, Y.F., Control chaos in the hindmarsh – rose neuron by using intermittent feedback with one variable, Chin. phys. lett., 25, 10, 3582-3585, (2008) [6] Wang, Z.L., Anti-synchronization in two non-identical hyperchaotic systems with known or unknown parameters, Commun. nonlinear sci. numer. simul., 14, 2366-2372, (2009) [7] Ma, J.; Jin, W.Y.; Li, Y.L., Chaotic signal-induced dynamics of degenerate optical parametric oscillator, Chaos solitons fractals, 36, 494-499, (2008) [8] Park, J.H., Synchronization of Genesio chaotic system via backstepping approach, Chaos solitons fractals, 27, 1369-1375, (2006) · Zbl 1091.93028 [9] Li, Y.N.; Chen, L.; Cai, Z.S.; Zhao, X.Z., Experimental study of chaos synchronization in the belousov – zhabotinsky chemical system, Chaos solitons fractals, 22, 767-771, (2004) · Zbl 1067.92069 [10] Corron, N.J.; Hahs, D.W., A new approach to communications using chaotic signals, IEEE trans. circuits syst., 44, 373-382, (1997) · Zbl 0902.94003 [11] Ims, R.A.; Andreassen, H.P., Spatial synchronization of vole population dynamics by predatory birds, Nature, 408, 194-196, (2000) [12] Cao, J.; Li, H.X.; Ho, Daniel W.C., Synchronization criteria of lur’e systems with time-delay feedback control, Chaos solitons fractals, 23, 1285-1298, (2005) · Zbl 1086.93050 [13] Liang, J.; Cao, J.; Lam, J., Convergence of discrete-time recurrent neural networks with variable delay, Int. J. bifurcation chaos, 15, 581-595, (2005) · Zbl 1098.68107 [14] Tian, L.X.; Xu, J.; Sun, M., Chaos synchronization of the energy resource chaotic system with active control, Int. J. nonlinear sci., 3, 3, 228-234, (2006) · Zbl 1394.34132 [15] Sun, M.; Tian, L.X.; Jiang, S.; Xu, J., Feedback control and adaptive control of the energy resource chaotic system, Chaos solitons fractals, 32, 1725-1734, (2007) · Zbl 1129.93403 [16] Sun, M.; Tian, L.X.; Fu, Y.; Qian, W., Dynamics and adaptive synchronization of the energy resource system, Chaos solitons fractals, 31, 879-888, (2007) · Zbl 1149.34032 [17] Ghosh, D.; Chowdhury, A.R.; Saha, P., On the various kinds of synchronization in delayed duffing – van der Pol system, Commun. nonlinear sci. numer. simul., 13, 4, 790-803, (2008) · Zbl 1221.34196 [18] Sun, M.; Tian, L.; Xu, J., Time-delayed feedback control of the energy resource chaotic system, Int. J. nonlinear sci., 3, 3, 172-177, (2006) · Zbl 1394.93291 [19] Sun, S.M.; Tian, L.X.; Fu, Y., An energy resources demand – supply system and its dynamical analysis, Chaos solitons fractals, 32, 168-180, (2007) · Zbl 1133.91524 [20] Jiang, G.P.; Zheng, W.X., An LMI criterion for linear-state-feedback based chaos synchronization of a class of chaotic systems, Chaos solitons fractals, 26, 437-443, (2005) · Zbl 1153.93390 [21] Liu, F.; Ren, Y.; Shan, X.M.; Qiu, Z.L., A linear feedback synchronization theorem for a class of chaotic systems, Chaos solitons fractals, 13, 723-730, (2002) · Zbl 1032.34045 [22] Burton, T.A., Stability and periodic solutions of ordinary and functional differential equations, (1985), Academic Press New York, pp. 22-84 [23] Mei, S.W.; Shen, T.L.; Liu, K.Z., Modern robust control theory and application, (2003), Qinghua University Press Beijing, p. 59 (in Chinese)
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