On the periodic solution of the van der Pol equation for small values of the damping parameter

International Journal of Circuit Theory and Applications

Volumn 26, Issue 1, 1998, Pages 39-52

  Buonomo, A ^a  

^a UNIVERSITY OF SALERNO   (Italy)

Author keywords

Non linear oscillation; Shohat expansion; Van der Pol's equation

Indexed keywords

CALCULATIONS; CIRCUIT THEORY; CONVERGENCE OF NUMERICAL METHODS; PERTURBATION TECHNIQUES;

NONLINEAR OSCILLATIONS; VAN DER POL EQUATION;

CIRCUIT OSCILLATIONS;

EID: 0031702596     PISSN: 00989886     EISSN: None     Source Type: Journal    
DOI: 10.1002/(SICI)1097-007X(199801/02)26:1<39::AID-CTA6>3.0.CO;2-X     Document Type: Article

References (18)

1. A. A. Andronov, A. A. Witt and S. E. Khaikin, Theory of Oscillations, Pergamon Press, Oxford, 1966.
2. A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations, Wiley, New York, 1979.
3. B. van der Pol, 'On relaxation-oscillations', Phil. Mag., 2(7), 978-992 (1926).
4. M. Urabe, 'Periodic solution of the van der Pol equation with damping coefficient λ = 0(0.2) 1.0', J. Sci. Hiroshima Univ., 21, Ser. A, 193-207 (1958).
5. M. Urabe, 'Periodic solutions of van der Pol's equation with damping coefficient λ = 0-10′, IRE Trans. Circuit Theory, CT-7, 382-386 (1960).
6. C. W. Clenshaw, 'The solution of van der Pol's equation in Chebychev series', in D. Greespan (ed.), Numerical Solution of Nonlinear Differential Equations, Wiley, New York, 1966, pp. 55-63.
7. J. A. Zonneveld, 'Periodic solutions of the van der Pol equation', Indag. Math., 28, 620-622 (1966).
8. E. A. Grebenikov and Yu. A. Ryabov, Constructive Methods in the Analysis of Nonlinear Systems, Mir Publ., Moscow, 1983, pp. 295-298.
9. A. Deprit and A. R. M. Rom, 'Asymptotic representations of the cycle of van der Pol's equation for small damping coefficients', Z. Agnew. Math. Phys., 18, 736-747 (1967).
10. A. A. Dorodnicyn, 'Asymptotic solution of van der Pol's equation', Amer. Mat. Soc. Trasl., 88 (1953).
11. J. Shohat, 'On Van der Pol's and related non-linear differential equations', J. Appl. Phys., 15, 568-574 (1944).
12. C. M. Andersen and J. F. Geer, 'Power series expansions for the frequency and period of the the limit cycle of the van der Pol equation', SIAM J. Appl. Math., 42, 678-693 (1982).
13. R. Bellman, Perturbation Techniques in Mathematics, Physics and Engineering, Holt, Rinehart and Winston, New York, 1964, pp. 61-64.
14. W. F. Ames, Nonlinear Ordinary Differential Equations in Transport Processes, Academic Press, New York, 1968, pp. 170-172.
15. F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Springer, Berlin, 1990.
16. Shui-Sheng Qiu and I. M. Filanovsky, 'Periodic solutions of the van der Pol equation with moderate values of damping coefficient', IEEE Trans. Circuits and Systems, CAS-34, 913-918 (1987).
17. E. Knuth, Fundamental Algorithms, vol. 50, Addison-Wesley, Reading, MA, 1968.
18. P. J. Ponzo and N. Wax, 'On the periodic solution of the van der Pol equation', IEEE Trans. Circuit Theory, CT-12, 135-136 (1965).