For citation:
Mchedlova E. S., Krasichkov L. V. The piecewise-linear model of the Van der Pol oscillator under external periodic force: complex dynamics, peculiarities of behaviour at the parameter plane. Izvestiya VUZ. Applied Nonlinear Dynamics, 2002, vol. 10, iss. 6, pp. 54-61. DOI: 10.18500/0869-6632-2002-10-6-54-61
The piecewise-linear model of the Van der Pol oscillator under external periodic force: complex dynamics, peculiarities of behaviour at the parameter plane
The electronic model of Van der Pol oscillator with piecewise-linear currentvoltage characteristics of nonlinear element is proposed. Experimental investigations of behaviour of the Van der Pol oscillator under external sinusoidal force were carried out. The parameter plane (external force amplitude - frequency) for the dynamical regimes of the oscillator was experimentally plotted. Numerical simulations of peculiarities of the oscillator model were carried out. In natural and numerical experiments it was shown that transition to chaos in the oscillator takes place through the period doubling cascade. It was revealed that the internal structure of the synchronization tongues looks typically as for the crossroad area.
1. Van der Роl B, Van der Mark J. Frequency demultiplication. Nature. 1927;120(3019):363–364. DOI; 10.1038/120363a0.
2. Andronov AA, Vitt AA, Khaikin SE. Theory of Oscillations. Pergamon Press; 1966. 815 p. DOI: 10.1016/C2013-0-06631-5.
3. Rabinovich MI, Trubetskov DI. Introduction to the Theory of Oscillations and Waves. Moscow: Nauka; 1984. 560 p. (in Russian).
4. Kennedy MP, Chua LO. Van der Pol and chaos. IЕЕЕ Transactions оn Circuits and Systems. 1986;33(10):974–980. DOI: 10.1109/TCS.1986.1085855.
5. Ueda Y, Akamatsu N. Chaotically transitional phenomena in the forced negative-resistance oscillator. IЕЕЕ Transactions оn Circuits and Systems. 1981;28(3):217–223. DOI: 10.1109/TCS.1981.1084975.
6. Qin GR, Gong DC, Wen XD. Rich bifurcational behaviour оf the driven Van der Pol oscillator. Phys. Lett. А. 1989;141(8–9):412–416. DOI: 10.1016/0375-9601(89)90859-1.
7. Parlitz U, Lauterborn W. Period doubling cascades and devil’s staircases оf the driven Van der Pol oscillator. Phys. Rev. А. 1987;36(3):1428–1434. DOI: 10.1103/physreva.36.1428.
8. Cartwright ML, Litlewood JE. On non-linear differential equations of the second order: The equation y+k(1-y2)y+y=bλkcos(λt+a), k large. J. London Math. Soc. 1945;20:180–189.
9. Levinson N. A second order differential equation with singular solutions. Ann. Math. 1949;50(1):127–153.
10. Matsumoto T, Chua LO, Komuro M. The double scroll. IЕЕЕ Transactions оn Circuits and Systems. 1985;32(8):798–818. DOI: 10.1109/TCS.1985.1085791.
11. Mchedlova ES. Chaotic oscillations in the piece-wise linear approximation оf the driven Van der Pol equation. In: the Proceedings оf thе 5th International Specialist Workshop оn Nonlinear Dynamics оf Electronic Systems (NDES-97) (Moscow, Russia, June 26-27, 1997). Moscow; 1997. P. 449–452.
12. Tietze U, Schenk C. Halbleiter-Schaltungstechnik. 8th edition. Berlin: Springer; 1986. 860 p. DOI: 10.1007/978-3-662-07641-5.
13. Chua LO, Desoer C, Kuh Е. Linear and Nonlinear Circuits. McGrow-Hill Book Company; 1987. 839 p.
14. Kuznetsov AP, Kuznetsov SP, Sataev IR. Critical dynamics of one-dimensional mappings. Part II. Two-parameter transition to chaos. Izvestiya VUZ. Applied Nonlinear Dynamics. 1993;1(3–4):17–35 (in Russian).
- 429 reads