Abstract
In this paper, independent, symmetric, periodic motions in a van der Pol oscillator are predicted through a semi-analytical method. This semi-analytic method is based on the discretization of the corresponding continuous nonlinear system for an implicit mapping. Through the implicit mapping structures, stable and unstable periodic motions are obtained analytically. A sequence of periodic motions to chaos via 1(S) ◁ 3(S) ◁ ··· ◁ (2m − 1)(S) ◁ ··· is discovered. The stability and bifurcations of periodic motions are determined through eigenvalue analysis. The frequency–amplitude characteristics of periodic motions are discussed. Numerical simulations of the periodic motions are carried out for comparison of numerical and analytical results. Such a periodic motion sequence is for a better understanding of dynamics of the van der Pol oscillator.
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References
van der Pol B (1920) A theory of the amplitude of free and forced triode vibrations. Radio Rev 1:701–710, 754–762
van der Pol B, van der Mark J (1927) Frequency demultiplication. Nature 120:363–364
Van der Pol B (1934) The nonlinear theory of electric oscillations. Proc Inst Radio Eng 22(9):1051–1086
Cartwright ML, Littlewood JE (1945) On nonlinear differential equations of the second order I. The equation \( \ddot{y} - k(1 - y^{2} )\dot{y} + y = b\lambda k\cos (\lambda t + \alpha ) \), k large. J Lond Math Soc 20:180–189
Littlewood JE (1947) On non-linear differential equations of the second order: III. The equation \( \ddot{y} - k(1 - y^{2} )\dot{y} + y = b\mu k\cos (\mu t + \alpha ) \) for large k, and its generalizations. Acta Math 97(1–4): 267–308
Levinson N (1948) A simple second order differential equation with singular motions. Proc Natl Acad Sci USA 34(1):13–15
Levinson N (1949) A second order differential equation with singular solutions. Ann Math II Ser 50(1):127–153
Smale S (1967) Differentiable dynamical systems. Bull Am Math Soc 73:747–817
Buonomo A (1998) The periodic solution of van der Pol’s equation. SIAM J Appl Math 59(1):156–171
Buonomo A (1998) On the periodic solution of the van der Pol equation for small values of the damping parameter. Int J Circuit Theory Appl 26(1):39–52
Mickens RE (2001) Analytical and numerical study of a nonstandard finite difference scheme for the unplugged van der Pol equation. J Sound Vib 245:757–761
Mickens RE (2002) Step-size dependence of the period for a forward-Euler scheme of the van der Pol equation. J Sound Vib 258:199–202
Waluya SB, van Horssen WT (2003) On the periodic solutions of a generalized non-linear van der Pol oscillator. J Sound Vib 268:209–215
Andrianov IV, van Horssen WT (2006) Analytical approximations of the period of a generalized nonlinear van der Pol oscillator. J Sound Vib 295:10991104
Luo ACJ (2012) Continuous dynamical systems. HEP/L&H Scientific, Beijing/Glen Carbon
Luo ACJ, Lakeh AB (2013) Analytical solutions for period-m motions in a periodically forced van der Pol oscillator. Int J Dyn Control 1(2):99–115
Luo ACJ, Lakeh AB (2014) An approximate solution for period-1 motions in a periodically forced van der Pol oscillator. J Comput Nonlinear Dyn 9(3):031001
Luo ACJ, Lakeh AB (2014) Period-m motions and bifurcation trees in a periodically forced, van der Pol–Duffing oscillator. Int J Dyn Control 2(4):474–493
Luo ACJ (2015) Periodic flows to chaos based on discrete implicit mappings of continuous nonlinear systems. Int J Bifurc Chaos 25(03):1550044
Luo ACJ (2015) Discretization and implicit mapping dynamics. HEP/Springer, Beijing/Berlin
Luo ACJ, Guo Y (2015) A semi-analytical prediction of periodic motions in Duffing oscillator through mapping structures. Discontin Nonlinear Complex 4(2):121–150
Guo Y, Luo ACJ (2015) On complex periodic motions and bifurcations in a periodically forced, damped, hardening Duffing oscillator. Chaos Solitons Fractals 81:378–399
Guo Y, Luo ACJ (2017) Periodic motions in a double-well Duffing oscillator under periodic excitation through discrete implicit mappings. Int J Dyn Control 5(2):223–238
Guo Y, Luo ACJ (2017) Routes of periodic motions to chaos in a periodically forced pendulum. Int J Dyn Control 5(3):551–569
Luo ACJ, Xing S (2017) On frequency responses of period-1 motions to chaos in a periodically forced, time-delayed quadratic nonlinear system. Int J Dyn Control 5(3):466–476
Luo ACJ, Xing S (2016) Multiple bifurcation trees of period-1 motions to chaos in a periodically forced, time-delayed, hardening Duffing oscillator. Chaos Solitons Fractals 89:405–434
Xing S, Luo ACJ (2018) On possible infinite bifurcation trees of period-3 motions to chaos in a time-delayed, twin-well Duffing oscillator. Int J Dyn Control. https://doi.org/10.1007/s40435-018-0418-y
Xu Y, Luo ACJ (2018) A series of symmetric period-1 motion to chaos in a two degree of freedom van der Pol duffing oscillator. J Vib Test Syst Dyn 2(2):119–153
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Xu, Y., Luo, A.C.J. Sequent period-(2m − 1) motions to chaos in the van der Pol oscillator. Int. J. Dynam. Control 7, 795–807 (2019). https://doi.org/10.1007/s40435-018-0468-1
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DOI: https://doi.org/10.1007/s40435-018-0468-1