Tang, Jiashi; Han, Feng; Xiao, Han; Wu, Xiao Amplitude control of a limit cycle in a coupled van der Pol system. (English) Zbl 1175.34038 Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 71, No. 7-8, 2491-2496 (2009). Summary: In order to control the amplitude of a limit cycle of a coupled van der Pol system, feedback controllers are designed. The control equations of the weakly nonlinear systems are obtained by using an approximate method, and the relationship between the amplitude of the limit cycle and the control parameter is derived. Hence, the amplitude of the limit cycle can be controlled effectively. The method may also be applied to other coupled van der Pol systems. Cited in 7 Documents MSC: 34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations 34H05 Control problems involving ordinary differential equations 93B52 Feedback control 34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations Keywords:coupled van der Pol system; limit cycle; amplitude; feedback controller PDFBibTeX XMLCite \textit{J. Tang} et al., Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 71, No. 7--8, 2491--2496 (2009; Zbl 1175.34038) Full Text: DOI References: [1] Chen, G.; Moiola, J. L.; Wang, H. O., Bifurcation control: Theories, methods, and applications, International Journal of Bifurcation Chaos, 10, 3, 511-548 (2000) · Zbl 1090.37552 [2] Nayfeh, A. H.; Harb, A. M.; Chin, C. M., Bifurcations in a power system model, International Journal of Bifurcation Chaos, 6, 3, 497-512 (1996) · Zbl 0874.34036 [3] J.L. Moiola, D.W. Berns, G. Chen, Feedback control of limit cycle amplitudes, in: Proc. IEEE Conf. Decis. Contr., San Diego, CA, 1997, pp. 1479-1485; J.L. Moiola, D.W. Berns, G. Chen, Feedback control of limit cycle amplitudes, in: Proc. IEEE Conf. Decis. Contr., San Diego, CA, 1997, pp. 1479-1485 [4] Maccari, A., Vibration control for the primary resonance of the van der Pol oscillator by a time delay state feedback, International Journal of Non-Linear Mechanics, 38, 123-131 (2003) · Zbl 1346.74086 [5] Mickens, R. E., Fractional van der Pol equations, Journal of Sound and Vibration, 259, 2, 457-460 (2003) · Zbl 1237.34066 [6] Tang, J. S.; Fu, W. B.; Li, K. A., Bifurcations of a parametrically excited oscillator with strong nonlinearity, Chinese Physics, 11, 10, 1004-1007 (2002) [7] Lopez-Ruiz, Ricardo, Symmetry induced oscillations in four-dimensional models deriving from the van der Pol equation, Chaos, Solitons & Fractals, 21, 1, 55-61 (2004) · Zbl 1060.37073 [8] Mickens, R. E.; Gumel, A. B., Numerical study of a non-standard finite-difference scheme for the van der Pol equation, Journal of Sound and Vibration, 250, 5, 955-963 (2002) · Zbl 1237.65084 [9] Tang, J. S.; Chen, Z. L., Amplitude control of limit cycle in van der Pol system, International Journal Bifurcation and Chaos, 16, 2, 487-495 (2006) · Zbl 1105.34323 [10] Tang, J. S.; Qian, C. Z., The asymptotic solution of the strongly nonlinear Klein-Gordon equation, Journal of Sound and Vibration, 268, 5, 1036-1040 (2003) · Zbl 1236.35013 [11] Qian, C. Z.; Tang, J. S., Asymptotic solution for a kind of boundary layer problem, Nonlinear Dynamics, 45, 1-2, 15-24 (2006) · Zbl 1138.35400 [12] Tang, J. S.; Qin, J. Q.; Xiao, H., Bifurcations of a generalized van der Pol oscillator with strong nonlinearity, Journal of Sound and Vibration, 306, 3-5, 890-896 (2007) [13] Nayfeh, A. H.; Mook, D. T., Nonlinear Oscillations (1978), Wiley: Wiley New York This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.