The Lyapunov exponents of the Van der Pol oscillator

Mathematical Methods in the Applied Sciences

Volumn 28, Issue 10, 2005, Pages 1131-1139

  Grasman, Johan ^a   Verhulst, Ferdinand ^b   Shih, Shagi Di ^c  

^a WAGENINGEN UNIVERSITY   (Netherlands)

^b UTRECHT UNIVERSITY   (Netherlands)

^c UNIVERSITY OF WYOMING   (United States)

Author keywords

Lyapunov exponents; Relaxation oscillation; Van der Pol oscillator; Weakly oscillation

Indexed keywords

APPROXIMATION THEORY; ASYMPTOTIC STABILITY; COMPUTER SOFTWARE; DIFFERENTIAL EQUATIONS; OSCILLATIONS; RELAXATION OSCILLATORS;

ASYMPTOTIC APPROXIMATION; LYAPUNOV EXPONENTS; RELAXATION OSCILLATION; VAN DER POL OSCILLATOR; WEAKLY OSCILLATION;

LYAPUNOV METHODS;

EID: 20544448513     PISSN: 01704214     EISSN: None     Source Type: Journal    
DOI: 10.1002/mma.606     Document Type: Article

References (13)

1. Van der Pol B. On relaxation oscillations. Philosophical Magazine 1926; 2:978-992.
2. Zonneveld JA. Periodic solutions of the Van der Pol equation. Nederlandse Akademie van Wetenschappen Proceedings A69 = Indagationes Mathematicae 1966; 28:620-622.
3. Andersen CM, Geer JF. Power series expansions for the frequency and period of the limit cycle of the Van der Pol equation. SIAM Journal on Applied Mathematics 1982; 42:678-693.
4. Dadfar MB, Geer J, Andersen CM. Perturbation analysis of the limit cycle of the free Van der Pol equation. SIAM Journal on Applied Mathematics 1984; 44:881-895.
5. Buonomo A. The periodic solution of Van der Pol's equation. SIAM Journal of Applied Mathematics 1999; 59:156-171.
6. Grasman J. Asymptotic Methods for Relaxation Oscillations and Applications. Springer: New York, 1987.
7. Tondl A, Ruijgrok T, Verhulst F, Nabergoj R. Autoparametric Resonance in Mechanical Systems. Cambridge University Press: New York, 2000.
8. Verhulst F, Abadi. Autoparametric resonance of relaxation oscillations. Zeitschrift für Angewandte Mathematik and Mechanik, in press.
9. Wolf A, Swift JB, Swinney HL, Vastano JA. Determining Lyapunov exponents from a time series. Physica D 1985; 16:285-317.
10. Verhulst F. Nonlinear Differential Equations and Dynamical Systems. Springer: New York, 1996.
11. Carrier GF, Lewis JA. The relaxation oscillations of the Van der Pol oscillator. Advances in Applied Mechanics 1953; 3:12-16.
12. Dorodnicyn AA. Asymptotic solution of Van der Pol's equation. Akad. Nauk SSSR Prikl. Mat. Mech. 1947; 11:313-328 (Russian)
13. American Mathematical Society Translation 1953; 88:24.