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Triple-zero bifurcation in van der Pol’s oscillator with delayed feedback. (English) Zbl 1273.34079

Consider the scalar differential-delay equation

$\stackrel{¨}{x}+\epsilon \left({x}^{2}-1\right)\stackrel{˙}{x}+x=\epsilon g\left(x\left(t-\tau \right)\right)\phantom{\rule{2.em}{0ex}}\left(1\right)$

with $\epsilon >0$, $g\left(0\right)=0$, ${g}^{\text{'}}\left(0\right)=k$, ${g}^{\text{'}\text{'}}\left(0\right)\ne 0$, $\tau$$\epsilon$ and $k$ are bifurcation parameters. (1) exhibits codimension-one and codimension-two bifurcations. For $\tau =\epsilon =\sqrt{2}$, $k=\sqrt{2}/2$, a codimension-three bifurcation occurs, i.e., triple-zero bifurcation. Using center manifold theory, the authors derive the following normal form with universal unfolding

${\stackrel{˙}{z}}_{1}={z}_{2},\phantom{\rule{1.em}{0ex}}{\stackrel{˙}{z}}_{2}={z}_{3},$
${\stackrel{˙}{z}}_{3}={\lambda }_{1}{z}_{1}+{\lambda }_{2}{z}_{2}+{\lambda }_{3}{z}_{3}+{\eta }_{1}{z}_{1}^{2}+{\eta }_{2}{z}_{2}^{2}+{\eta }_{3}{z}_{1}{z}_{2}+{\eta }_{4}{z}_{1}{z}_{3}+\text{h.o.t.}$

They determine the bifurcations occurring in the truncated normal form, namely transcritical, Hopf, Takens-Bogdanov, zero-Hopf ones. Finally, numerical simulations are presented.

##### MSC:
 34K18 Bifurcation theory of functional differential equations
##### References:
 [1] Campbell, S. A.; Yuan, Y.: Zero singularities of codimension two and three in delay differential equations, Nonlinearity 21, 2671-2691 (2008) · Zbl 1149.92002 · doi:10.1088/0951-7715/21/11/010 [2] Eigoli, A. K.; Khodabakhsh, M.: A homotopy analysis method for limit cycle of the van der Pol oscillator with delayed amplitude limiting, Appl math comput (2011) [3] Faria, T.; Magalhaes, L. T.: Normal forms for retarded functional differential equations and applications to bogdanov – Takens singularity, J diff eqs 122, 01-224 (1995) · Zbl 0836.34069 · doi:10.1006/jdeq.1995.1145 [4] Faria, T.; Magalhaes, L. T.: Normal forms for retarded functional differential equations with parameters and applications to Hopf bifurcation, J diff eqs 122, 181-200 (1995) · Zbl 0836.34068 · doi:10.1006/jdeq.1995.1144 [5] Guckenheimer, J.; Holmes, P.: Nonlinear oscillations, dynamical systems and bifurcations of vector fields, (1983) [6] Hale JK. Oscillations in nonlinear systems. New York; 1963. · Zbl 0115.07401 [7] Ji, J. C.: Nonresonant Hopf bifurcations of a controlled van der Pol – Duffing oscillator, J sound vib 297, 183-199 (2006) [8] Jiang, W.; Wei, J.: Bifurcation analysis in van der Pol’s oscillator with delayed feedback, J comput appl math 213, 604-615 (2008) [9] Wei, J.; Jiang, W.: Stability and bifurcation analysis in van der Pol’s oscillator with delayed feedback, J sound vib 283, 801-819 (2005) [10] Kimiaeifar, A.; Saidi, A.; Sohouli, A.; Ganji, D.: Analysis of modified van der Pol’s oscillator using he’s parameter-expanding methods, Curr appl phys 10, 279-283 (2010) [11] Liao, X.: Hopf and resonant codimension two bifurcation in van der Pol equation with two time delays, Chaos solitons fractals 23, 857-871 (2005) · Zbl 1076.34087 · doi:10.1016/j.chaos.2004.05.048 [12] Lin, G.: Periodic solutions for van der Pol equation with time delay, Appl math comput 187, 1187-1198 (2007) · Zbl 1210.34096 · doi:10.1016/j.amc.2006.09.032 [13] Mickens, R. E.: Oscillations in planar dynamic systems, (1996) [14] Qiao, Z.; Liu, X.; Zhu, D.: Bifurcation in delay differential systems with triple-zero singularity, Chin annal math ser A 31, No. 1, 59-70 (2010) · Zbl 1224.34239 [15] Jiang, W.; Yuan, Y.: Bogdanov – Takens singularity in van der Pol’s oscillator with delayed feedback, Phys D nonlinear phenom 227, 149-161 (2007) · Zbl 1124.34048 · doi:10.1016/j.physd.2007.01.003 [16] Wang, H.; Jiang, W.: Hopf-pitchfork bifurcation in van der Pol’s oscillator with nonlinear delayed feedback, J math anal appl 368, 9-18 (2010) [17] Wu, X.; Wang, L.: Zero-Hopf bifurcation for van der Pol’s oscillator with delayed feedback, J comput appl math 235, 2586-2602 (2011) · Zbl 1213.34082 · doi:10.1016/j.cam.2010.11.011 [18] Xu, Y.; Huang, M.: Homoclinic orbits and Hopf bifurcations in delay differential systems with T – B singularity, J diff eqs 244, 582-598 (2008) · Zbl 1154.34039 · doi:10.1016/j.jde.2007.09.003 [19] Yu, W.; Cao, J.: Hopf bifurcation and stability of periodic solutions for van der Pol equation with time delay, Nonlinear anal 62, 141-165 (2005) · Zbl 1138.34348 · doi:10.1016/j.na.2005.03.017 [20] Zhang, J.; Gu, X.: Stability and bifurcation analysis in the delay-coupled van der Pol oscillators, Appl math model 34, 2291 (2010) · Zbl 1195.34109 · doi:10.1016/j.apm.2009.10.037