zbMATH — the first resource for mathematics

Chaos control of chaotic limit cycles of real and complex van der Pol oscillators. (English) Zbl 1046.70014
From the summary: We examine the chaos control of chaotic unstable limit cycles of real and complex (or coupled) nonlinear van der Pol oscillators. The presence of chaotic limit cycles is verified by calculating largest Lyapunov exponent and the power spectrum. The problem of chaos control of these limit cycles is studied using a feedback control method, which is based on the construction of a special form of a time-continuous perturbation. We investigate both real and complex (or coupled) van der Pol oscillators.

70K55 Transition to stochasticity (chaotic behavior) for nonlinear problems in mechanics
70K05 Phase plane analysis, limit cycles for nonlinear problems in mechanics
70Q05 Control of mechanical systems
93C10 Nonlinear systems in control theory
Full Text: DOI
[1] Hajnovicova, J.; Halasova, D., Fundamentals of dynamical systems and bifurcation theory, (1992), Adam Hilgar Bristol, Philadelphia, New York
[2] Mahmoud, G.M., Approximate solutions of a class of complex nonlinear dynamical systems, Physica A, 253, 211-222, (1998)
[3] Mahmoud, G.M.; Rauh, A.; Mohamed, A.A., On modulated complex nonlinear dynamical systems, Il nuovo cimento B, 114, 31-47, (1999)
[4] Maccari, A., Periodic and quasi-periodic motion for complex nonlinear systems, Int J non-linear mech, 38, 575-584, (2003) · Zbl 1346.34034
[5] Cveticanin, L., Analytical approach for the solution of the complex-valued strongly nonlinear differential equation of Duffing type, Physica A, 297, 348-360, (2001) · Zbl 0969.34501
[6] Mahmoud, G.M.; Aly, S.A., On periodic solutions of parametrically excited complex nonlinear dynamical systems, Physica A, 278, 390-404, (2000)
[7] Mahmoud, G.M.; Aly, S.A., On periodic attractors of complex damped nonlinear systems, Int J non-linear mech, 35, 309-323, (2000) · Zbl 1068.70525
[8] Mahmoud, G.M.; Farghaly, A.A.M., Stabilization of unstable periodic attractors of complex damped nonlinear dynamical systems, Chaos, soliton & fractals, 17, 105-112, (2003) · Zbl 1098.70527
[9] Qi Bi, Dynamical analysis of two coupled parametrically excited van der Pol oscillators, Int J non-linear mech, 39, 33-54, (2004) · Zbl 1225.34049
[10] Leung, H.K., Synchronization dynamics of coupled van der Pol systmes, Physica A, 321, 248-255, (2003) · Zbl 1020.37014
[11] Maccari, A., Modulated motion and infinite-period bifurcation for two non-linearly coupled and parametrically excited van der Pol oscillators, Int J non-linear mech, 36, 335-347, (2001) · Zbl 1345.70039
[12] Chung, K.W.; Chan, C.L.; Xu, Z.; Mahmoud, G.M., A perturbation-incremental method for strongly nonlinear autonomous oscillators with many degrees of freedom, Nonlinear dyn, 28, 3, 243-259, (2000) · Zbl 1015.70017
[13] Rand, R.H.; Holmes, P.J., Bifurcation of periodic motions into weakly coupled van der Pol oscillators, Int J non-linear mech, 15, 387-399, (1980) · Zbl 0447.70028
[14] Storti, D.W.; Rand, R.H., Dynamics of two strongly coupled van der Pol oscillators, Int J non-linear mech, 17, 143-152, (1982) · Zbl 0498.70037
[15] Dieci, L.; Lrenz, R.; Russell, R.D., Numerical calculations of invariant tori for two weakly coupled van der Pol oscillators, SIAM J sci statist comput, 12, 607-647, (1990)
[16] Minorsky, N., Nonlinear oscillations, (1962), Van Nostrand New York · Zbl 0123.06101
[17] Hayashi, C.; Kuramitsu, M., Self-excited oscillations in a system with two degrees of freedom, Mem fac engng Kyoto univ, 36, 87-104, (1974)
[18] van der Pol, B.; van der Mrak, J., Frequency demultiplication, Nature, 120, 363-364, (1927)
[19] Pyragas, K., Continuous control of chaos by self controlling feedback, Phys lett A, 170, 421-428, (1992)
[20] Ott, E.; Grebogi, C.; Yorke, J.A., Controlling chaos, Phys rev lett, 64, 1196-1199, (1990) · Zbl 0964.37501
[21] Filipe, J.; Grebogi, C.; Ott, E.; Dayawansa, W.P., Controlling chaotic dynamical systems, Physica D, 58, 165-192, (1992) · Zbl 1194.37140
[22] Osipov, G.V.; Kozlov, A.K.; Shalfeev, V.P., Impulse control of chaos in continuous systems, Phys lett A, 247, 119-128, (1998)
[23] Breeden, J.L., Open-loop control of nonlinear systems, Phys lett A., 190, 264-272, (1994)
[24] Yagasaki, K.; Uozumi, T., A new approach for controlling chaotic dynamical systems, Phys lett A, 238, 349-357, (1998) · Zbl 0946.37023
[25] Hegazi, A.; Agiza, H.N.; El-Dessoky, M.M., Controlling chaotic behavior for spin generator and rossler dynamical systems with feedback control, Chaos, solitons & fractals, 12, 631, (2001) · Zbl 1016.37050
[26] Mahmoud, G.M.; Rauh, A.; Farghaly, A.A.M., Applying chaos control to a modulated complex nonlinear dynamical system, Il nuovo cimento B, 116, 10, 1113-1126, (2001)
[27] Mahmoud, G.M.; Mohamed, A.A.; Aly, S.A., Strange attractors and chaos control in periodically forced complex duffing’s oscillators, Physica A, 292, 193-206, (2001) · Zbl 0972.37054
[28] Mahmoud GM. In: Chaotic behavior of nonlnear oscillators, International Nonlinear Sciences Conference, University of Vienna, Vienna, Austria, February 7-9, 2003
[29] Wolf, A.; Swift, J.; Swinney, H.; Vastano, J., Determining Lyapunov exponents from a time series, Physica D, 16, 285-317, (1985) · Zbl 0585.58037
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.