Hopf bifurcation in delayed van der Pol oscillators

Nonlinear Dynamics

Volumn 71, Issue 3, 2013, Pages 555-568

  Zhang, Ling ^a   Guo, Shangjiang ^a  

^a HUNAN UNIVERSITY   (China)

Author keywords

Center manifold; Delay; Hopf bifurcation; Non semisimple; Normal form; Stability

Indexed keywords

CIRCUIT OSCILLATIONS; CONVERGENCE OF NUMERICAL METHODS; EIGENVALUES AND EIGENFUNCTIONS;

BIFURCATING PERIODIC SOLUTIONS; CENTER MANIFOLD; CENTER MANIFOLD THEOREM; CHARACTERISTIC EQUATION; DELAY; GENERALIZED HOPF BIFURCATION; NON-SEMISIMPLE; NORMAL FORM;

HOPF BIFURCATION;

EID: 84890509387     PISSN: 0924090X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s11071-012-0681-y     Document Type: Article

References (16)

1. Van der Pol, B.: A theory of the amplitude of free and forced triode vibrations. Radio Rev. 1, 701-710, 754-762 (1920)
2. Atay, F.M.: Van der Pol's oscillator under delayed feedback. J. Sound Vib. 218(2), 333-339 (1998)
3. Wei, J., Jiang, W.: Stability and bifurcation analysis in van der Pol's oscillator with delayed feedback. J. Sound Vib. 283, 801-819 (2005)
4. Jiang, W., Wei, J.: Bifurcation analysis in van der Pol's oscillator with delayed feedback. J. Comput. Appl. Math. 213, 604-615 (2008)
5. van Gils, S.A., Krupa, M., Langford, W.F.: Hopf bifurcation with non-semisimple 1:1 resonance. Nonlinearity 3, 825-850 (1990)
6. Ruan, S., Wei, J.: Periodic solutions of planner systems with two delays. Proc. R. Soc. Edinb. 129A, 1017-1032 (1999)
7. Dieudonné, J.: Foundations of Modern Analysis. Academic Press, New York (1996)
8. Namachchivaya, N.S., Doyle, M.M., Langford, W.F., Evans, N.W.: Normal form for generalized Hopf bifurcation with non-semisimple 1:1 resonance. Z. Angew. Math. Phys. 45, 312-324 (1994)
9. Arnold, V.I.: On matrices depending on parameters. Russ. Math. Surv. 26, 29-43 (1971)
10. Golubitsky, M., Langford, W.F.: Classification and unfoldings of degenerate Hopf bifurcations. J. Differ. Equ. 41, 375-415 (1981)
11. Iooss, G., Langford, W.F.: Conjectures on the routes to turbulence via bifurcations. Nonlinear Dyn. 357, 489-505 (1980)
12. Liao, X., Wong, K., Wu, Z.: Hopf bifurcation and stability of periodic solutions for van der Pol equation with distributed delay. Nonlinear Dyn. 26, 23-44 (2001)
13. Song, Y.: Hopf bifurcation and spatio-temporal patterns in delay-coupled van der Pol oscillators. Nonlinear Dyn. 63, 223-237 (2011)
14. Algaba, A., Freire, E., Gamero, E., Rodriguez-Luis, A.J.: A tame degenerate Hopf-Pitchfork bifurcation in a modified van der Pol-Duffing oscillator. Nonlinear Dyn. 22, 249-269 (2000)
15. Algaba, A., Freire, E., Gamero, E., Rodriguez-Luis, A.J.: Analysis of Hopf and Takens-Bogdanov bifurcations in a modified van der Pol-Duffing oscillator. Nonlinear Dyn. 16, 369-404 (1998)
16. Wirkus, S., Rand, R.H.: The dynamics of two coupled van der Pol oscillators with delay coupling. Nonlinear Dyn. 30, 205-221 (2002)