Triple-zero bifurcation in van der Pol’s oscillator with delayed feedback

► We discuss triple-zero singularity of codimension three in van der Pol’s equation with delayed feedback. ► We give the versal unfolding of the norm forms at the triple-zero bifurcation point. ► We numerically find the complicated dynamics near the triple-zero point. In this paper, we study a class...

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Bibliographic Details
Published in:Communications in nonlinear science & numerical simulation 2012-12, Vol.17 (12), p.5229-5239
Main Authors: He, Xing, Li, Chuandong, Shu, Yonglu
Format: Article
Language:English
Subjects:
Bifurcations
Bogdanov–Takens bifurcation
Computer simulation
Differential equations
Feedback
Mathematical analysis
Mathematical models
Norms
Oscillators
Triple-zero bifurcation
Zero-Hopf bifurcation
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Summary:► We discuss triple-zero singularity of codimension three in van der Pol’s equation with delayed feedback. ► We give the versal unfolding of the norm forms at the triple-zero bifurcation point. ► We numerically find the complicated dynamics near the triple-zero point. In this paper, we study a classical van der Pol’s equation with delayed feedback. Triple-zero bifurcation is investigated by using center manifold reduction and the normal form method for retarded functional differential equation. We get the versal unfolding of the norm forms at the triple-zero bifurcation and show that the model can exhibit transcritical bifurcation, Hopf bifurcation, Bogdanov–Takens bifurcation and zero-Hopf bifurcation. Some numerical simulations are given to support the analytic results.
ISSN:1007-5704
1878-7274
DOI:10.1016/j.cnsns.2012.05.001