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No-chattering sliding mode control chaos in Hindmarsh-Rose neurons with uncertain parameters. (English) Zbl 1222.37106
Summary: A Hindmarsh-Rose (HR) model was constructed from voltage clamp data to provide a simple description of the patterned activity seen in molluscan neurons. Its complex dynamics characters are presented, including the phase trajectory, the Lyapunov exponents and the Poincaré map. Furthermore, a no-chattering sliding mode control method for the Hindmarsh-Rose (HR) model with uncertain parameters and bounded external disturbances is proposed, and it can control the system to any point and any periodic orbit. Both the theoretical analysis and the simulation results are presented to confirm the validity of the control method.

MSC:
37N35 Dynamical systems in control
92C20 Neural biology
34C28 Complex behavior and chaotic systems of ordinary differential equations
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
93B12 Variable structure systems
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[1] Khadra, A.; Liu, X.Z.; Shen, X., Impulsively synchronizing chaotic systems with delay and applications to secure communication, Automatica, 41, 1491-1502, (2005) · Zbl 1086.93051
[2] Liu, Y.J.; Yang, Q.G., Dynamics of a new Lorenz-like chaotic system, Nonlinear analysis. real world applications, 11, 2563-2572, (2010) · Zbl 1202.34083
[3] Harb, A.M.; Nabil, A.J., Controlling Hopf bifurcation and chaos in a small power system, Chaos, solitons and fractals, 18, 1055-1063, (2003) · Zbl 1074.93522
[4] Ditto, W.L., Applications of chaos in biology and medicine, Chaos and the changing nature of science and medicine, 376, 175-202, (1996)
[5] Ma, J.; Wang, C.N.; Tang, J., Suppression of the spiral wave and turbulence in the excitability-modulated media, International journal of theoretical physics, 48, 150-157, (2009)
[6] Zhang, J.; Sun, J.; Luo, X.; Zhang, K.; Nakamura, T.; Small, M., Characterizing topology of pseudoperiodic time series via complex network approach, Physica D, 237, 2856-2865, (2008) · Zbl 1153.37447
[7] Wei, Z.C.; Yang, Q.G., Controlling the diffusionless Lorenz equations with periodic parametric perturbation, Computers & mathematics with applications, 58, 1979-1987, (2009) · Zbl 1189.34118
[8] Fang, X.L.; Yu, H.J.; Jiang, Z.L., Chaotic synchronization of nearest-neighbor diffusive coupling hindmarsh – rose neural networks in noisy environments, Chaos, solitons and fractals, 39, 2426-2441, (2009)
[9] Hindmarsh, J.L.; Rose, R.M., A model of neuronal bursting using three coupled first order differential equations, Proceedings of the royal society of London, series B, 22, 87-102, (1984)
[10] Rosa, M.L.; Rabinovich, M.I.; Huerta, R., Slow regularization through chaotic oscillation transfer in an unidirectional chain of hindmarsh – rose models, Physics letters A, 266, 88-93, (2000)
[11] He, D.H.; Hu, G.; Zhan, M.; Lu, H.P., Periodic states with functional phase relation in weakly coupled chaotic hindmarsh – rose neurons, Physica D, 156, 314-324, (2001) · Zbl 1026.92008
[12] Wu, Y.; Xu, J.X.; He, D.H., Generalized synchronization induced by noise and parameter mismatching in hindmarsh – rose neurons, Chaos, solitons and fractals, 23, 1605-1611, (2005) · Zbl 1066.92015
[13] Yu, H.J.; Peng, J.H., Chaotic synchronization and control in nonlinear-coupled hindmarsh – rose neural systems, Chaos, solitons and fractals, 29, 342-348, (2006) · Zbl 1095.92020
[14] Arena, P.; Fortuna, L.; Frasca, M.; Rosa, M.L., Locally active hindmarsh – rose neurons, Chaos, solitons and fractals, 27, 405-412, (2006) · Zbl 1101.37320
[15] Wei, D.Q.; Luo, X.S.; Qin, Y.H., Random long-range connections induce activity of complex hindmarsh – rose neural networks, Physica A, 387, 2155-2160, (2008)
[16] Wang, Z.L.; Shi, X.R., Chaotic bursting lag synchronization of hindmarsh – rose system via a single controller, Applied mathematics and computation, 215, 1091-1097, (2009) · Zbl 1205.37025
[17] Wu, Q.J.; Zhou, J.; Xiang, L.; Liu, Z.R., Impulsive control and synchronization of chaotic hindmarsh – rose models for neuronal activity, Chaos, solitons and fractals, 41, 2706-2715, (2009) · Zbl 1198.37136
[18] Che, Y.Q.; Wang, J.; Tsang, K.M.; Chan, W.L., Unidirectional synchronization for hindmarsh – rose neurons via robust adaptive sliding mode control, Nonlinear analysis: real world applications, 11, 1096-1104, (2010) · Zbl 1183.37149
[19] Ott, E.; Grebogi, G.; Yorke, J.A., Controlling chaos, Physical review letters, 64, 1196-1199, (1990) · Zbl 0964.37501
[20] Nakajima, H., On analytical properties of delayed feedback control of chaos, Physics letters A, 232, 207-210, (1997) · Zbl 1053.93509
[21] Yang, S.K.; Chien, C.L.; Yau, H.T., Control of chaos in Lorenz system, Chaos, solitons and fractals, 13, 767-780, (2002) · Zbl 1031.34042
[22] Ahamad, W.M.; Harb, A.M., On nonlinear control design for autonomous chaotic systems of integer and fractional orders, Chaos, solitons and fractals, 18, 693-701, (2003) · Zbl 1073.93027
[23] Hovel, P.; Scholl, E., Control of unstable steady states by time-delayed feedback methods, Physical review E, 72, 046203, (2005)
[24] Tavazoei, M.S.; Haeri, M., Chaos control via a simple fractional-order controller, Physics letters A, 372, 798-807, (2008) · Zbl 1217.70022
[25] Wei, Z.C.; Yang, Q.G., Controlling the diffusionless Lorenz equations with periodic parametric perturbation, Computers & mathematics with applications, 58, 1979-1987, (2009) · Zbl 1189.34118
[26] Yu, J.P.; Chen, B.; Yu, H.S.; Gao, J.W., Adaptive fuzzy tracking control for the chaotic permanent magnet synchronous motor drive system via backstepping, Nonlinear analysis. real world applications, 12, 671-681, (2011) · Zbl 1203.93112
[27] Konishi, K.; Kokame, H.; Hara, N., Delayed feedback control based on the act-and-wait concept, Nonlinear dynamics, 63, 513-519, (2011)
[28] Fuh, C.C.; Wang, M.C., A combined input-state feedback linearization scheme and independent component analysis filter for the control of chaotic systems with significant measurement noise, Journal of vibration and control, 17, 215-221, (2011) · Zbl 1271.70060
[29] Hindmarsh, J.L.; Rose, R.M., A model of neuronal bursting using three coupled first order differential equations, Proceedings of the royal society of London, series B, 22, 87-102, (1984)
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