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Control of chaos in a piecewise smooth nonlinear system. (English) Zbl 1102.37303

Summary: This paper shows the stabilization of the unstable periodic orbit of any given piecewise smooth system with linear and/or nonlinear characteristics. By utilizing the periodicity of the switching action, we construct a Poincaré mapping including all information of the original system. This mapping offers a first step toward extending a novel technique for controlling chaos based on the appropriate state feedback in piecewise smooth nonlinear systems. We also apply this approach to Rayleigh-type oscillator described by piecewise smooth nonlinear systems.

MSC:

37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
93C10 Nonlinear systems in control theory
37N05 Dynamical systems in classical and celestial mechanics
37C27 Periodic orbits of vector fields and flows
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[1] Saito, T.; Mitsubori, K., Control of chaos from a piecewise linear hysteresis circuit, IEEE Trans Circ Syst I: Fundam Theory Appl, 42, 168-172 (1995) · Zbl 0846.93076
[2] Banerjee, S.; Yorke, J.; Grebogi, C., Robust Chaos Phys Rev Lett, 80, 3049-3052 (1998) · Zbl 1122.37308
[3] (Banerjee, S.; Verghese, G. C., Nonlinear phenomena in power electronics: attractors bifurcations chaos and non-linear control (2001), IEEE Press: IEEE Press New York)
[4] Bernardo, M.di; Tse, C. K., Chaos in power electronics: an overview, chaos in circuits and systems (2002), World Scientific, [Chapter 16]
[5] Robert, B.; Robert, C., Border collision bifurcations in a one-dimension piecewise smooth map for a PWM current-programmed H-bridge inverter, Int J Cont, 11, 1356-1367 (2002) · Zbl 1078.93529
[6] Ma, Y.; Kawakami, H.; Tse, C. K., Bifurcation analysis of switched dynamical systems with periodically moving borders, IEEE Trans Circ Syst I: Fundam Theory Appl, 51, 1184-1193 (2004) · Zbl 1374.37068
[7] Chakrabarty, K.; Banerjee, S., Control of chaos in piecewise linear systems with switching nonlinearity, Phys Lett A, 200, 115-120 (1995)
[8] Poddar, G.; Chakrabarty, K.; Banerjee, S., Control of chaos in DC-DC converters, IEEE Trans Circ Syst I: Fundam Theory Appl, 45, 672-676 (1998)
[9] Bernardo, M.di; Chen, G., Controlling bifurcations in nonsmooth dynamical systems, (Chen, G., Controlling chaos and bifurcations in engineering systems (1999), CRC Press), 391-415 · Zbl 0938.93539
[10] Batlle, C.; Fossas, E.; Olivar, G., Stabilization of periodic orbits of the buck converter by time-delayed feedback, Int J Circ Theory Appl, 11, 617-631 (1999) · Zbl 0965.93090
[11] Iu, H. H.C.; Robert, B., Control of chaos in a PWM current-mode H-bridge inverter using time-delayed feedback, IEEE Trans Circ Syst I: Fundam. Theory Appl, 50, 1125-1129 (2003)
[12] Kousaka, T.; Yasuhara, Y.; Ueta, T.; Ma, Y.; Kawakami, H., Experimental realization of controlling chaos in the periodically switched nonlinear circuit, Int J Bifurcat Chaos, 14, 3655-3660 (2004) · Zbl 1067.94599
[13] Kousaka, T.; Ueta, T.; Kawakami, H., Chaos in a simple hybrid system and its control, Electron Lett, 1, 1-2 (2001)
[14] Kousaka T, Ueta T, Ma Y, Kawakami H. Bifurcation analysis of a piecewise smooth system with nonlinear characteristics. Int J Circuit Theory Appl, in press.; Kousaka T, Ueta T, Ma Y, Kawakami H. Bifurcation analysis of a piecewise smooth system with nonlinear characteristics. Int J Circuit Theory Appl, in press. · Zbl 1093.37019
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