zbMATH — the first resource for mathematics

Bogdanov-Takens singularity in Van der Pol’s oscillator with delayed feedback. (English) Zbl 1124.34048
The paper studies the local bifurcations of the classical Van der Pol equation with nonlinear delayed feedback \[ \ddot x+\varepsilon (x^2-1)\dot x+x =g(x(t-\tau)), \] where \(g(0)=0\). The main bifurcation parameters are \(k:=g'(0)\) and \(\tau\). For \(k=1/\varepsilon\) and \(\tau=\varepsilon\) the authors discover a Takens-Bogdanov bifurcation (plus additional structure because the origin is always a fixed point). Then the authors reduce the system to its normal form using the method of T. Faria and L. Magalhaes [J. Differ. Equations 122, No. 2, 201–224 (1995; Zbl 0836.34069)] for two cases: the generic case \(g''(0)\neq 0\), and the odd case \(g(x)=-g(-x)\) (with \(g'''(0)\neq0\)). The bifurcation diagrams are only sketched in the normal form parameters.

34K18 Bifurcation theory of functional-differential equations
34K17 Transformation and reduction of functional-differential equations and systems, normal forms
Full Text: DOI
[1] Ashkenazi, M.; Chow, S.N., Normal forms near critical points for differential equations and maps, IEEE trans. circuits syst., 35, 850-862, (1998) · Zbl 0702.34033
[2] Atay, F.M., Van der pol’s oscillator under delayed feedback, J. sound vibration, 218, 333-339, (1998) · Zbl 1235.70142
[3] Bold, K.; Edwards, C.; Guckenheimer, J.; Guharay, S.; Hoffman, K.; Hubbard, J.; Oliva, R.; Weckesser, W., The forced van der Pol equation II: canards in the reduced system, SIAM J. appl. dyn. syst., 2, 4, 570-608, (2003) · Zbl 1089.37013
[4] Buonomo, A., The periodic solution of Van der pol’s equation, SIAM J. appl. math., 59, 1, 156-171, (1998) · Zbl 0920.34013
[5] Carr, J., Applications of centre manifold theory, (1981), Spring-Verlag New York · Zbl 0464.58001
[6] Chow, S.N.; Li, C.; Wang, D., Normal forms and bifurcation of planar vector fields, (1994), Cambridge University Press New York
[7] Chua, L.O.; Kokubu, H., Normal forms for nonlinear vector fields — part I: theory and algorithm, IEEE trans. circuits syst., 35, 863-880, (1998) · Zbl 0683.58021
[8] Chua, L.O.; Kokubu, H., Normal forms for nonlinear vector fields — part II: applications, IEEE trans. circuits syst., 35, 51-70, (1998) · Zbl 0702.58047
[9] Faria, T.; Magalhães, L.T., Normal forms for retarded functional differential equations with parameters and applications to Hopf bifurcation, J. differential equations, 122, 181-200, (1995) · Zbl 0836.34068
[10] Faria, T.; Magalhães, L.T., Normal forms for retarded functional differential equations and applications to bogdanov – takens singularity, J. differential equations, 122, 201-224, (1995) · Zbl 0836.34069
[11] Faria, T., On the study of singularities for a plane system with two delays, J. dyn. contin. discrete impuls. syst. ser. A, 10, 357-371, (2003) · Zbl 1036.34083
[12] Giannkopoulos, F.; Zapp, A., Bifurcation in a planar system of differential delay equations modelling neural activity, Physica D, 159, 215-232, (2001)
[13] Guckenheimer, J.; Holmes, P., Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, (1983), Springer-Verlag New York · Zbl 0515.34001
[14] Guckenheimer, J.; Hoffman, K.; Weckesser, W., The forced van der Pol equation I: the slow flow and its bifurcation, SIAM J. appl. dyn. syst., 2, 1, 1-35, (2003) · Zbl 1088.37504
[15] Hale, J., Theory of functional differential equations, (1977), Springer-Verlag New York
[16] E. Horozov, Versal deformations of equivariant vector fields for the cases of symmetry of order 2 and 3, in: Proceedings of Petrovskii Seminar, 5, Moscow State University, 1979, pp. 163-192 (in Russian)
[17] Murakami, K., Bifurcated periodic solutions of for delayed van der Pol equation, Neural parallel sci. comput., 7, (1999) · Zbl 0933.34081
[18] Oliveira, J.C.F., Oscillations in a van der Pol equation with delayed argument, J. math. anal. appl., 275, 789-803, (2002) · Zbl 1022.34067
[19] Redmond, B.F.; LeBlanc, V.G.; Longtin, A., Bifurcation analysis of a class of first-order nonlinear delay-differential equations with reflectional symmetry, Physica D, 166, 121-145, (2002)
[20] Wei, J.; Jiang, W., Stability and bifurcation analysis in Van der pol’s oscillator with delayed feedback, J. sound vibration, 283, 801-819, (2005) · Zbl 1237.70091
[21] Xiao, D.; Ruan, S., Multiple bifurcations in a delayed predator – prey system with non-monotonic function response, J. differential equations, 176, 494-510, (2001) · Zbl 1003.34064
[22] Yuan, Y.; Wei, J., Multiple bifurcation analysis in a neural network model with delays, Internat. J. bifur. chaos, 10, 2903-2913, (2006) · Zbl 1185.37136
[23] Yuan, Y.; Yu, P., Computation of simplest normal forms of differential equations associated with a double-zero eigenvalues, Internat. J. bifur. chaos, 12, 5, 1307-1330, (2001) · Zbl 1090.37539
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.