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Hopf-pitchfork bifurcation in van der Pol’s oscillator with nonlinear delayed feedback. (English) Zbl 1364.34100
Summary: First, we identify the critical values for Hopf-pitchfork bifurcation. Second, we derive the normal forms up to third order and their unfolding with original parameters in the system near the bifurcation point, by the normal form method and center manifold theory. Then we give a complete bifurcation diagram for original parameters of the system and obtain complete classifications of dynamics for the system. Furthermore, we find some interesting phenomena, such as the coexistence of two asymptotically stable states, two stable periodic orbits, and two attractive quasi-periodic motions, which are verified both theoretically and numerically.

MSC:
34K18 Bifurcation theory of functional-differential equations
34K17 Transformation and reduction of functional-differential equations and systems, normal forms
34K19 Invariant manifolds of functional-differential equations
34K14 Almost and pseudo-almost periodic solutions to functional-differential equations
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[1] Algaba, A.; Fernández-Sánchez, F.; Freire, E.; Gamero, E.; Rodríguez-Luis, A.J., Oscillation-sliding in a modified Van der pol – duffing electronic oscillator, J. sound vibration, 249, 5, 899-907, (2002) · Zbl 1237.34049
[2] Ashkenazi, M.; Chow, S.N., Normal forms near critical points for differential equations and maps, IEEE trans. circuits syst., 35, 850-862, (1988) · Zbl 0702.34033
[3] Atay, F.M., Van der Pol’s oscillator under delayed feedback, J. sound vibration, 218, 22, 333-339, (1998) · Zbl 1235.70142
[4] Carr, J., Applications of centre manifold theory, (1981), Springer-Verlag New York · Zbl 0464.58001
[5] Chow, S.N.; Li, C.; Wang, D., Normal forms and bifurcation of planar vector fields, (1994), Cambridge University Press New York
[6] Chua, L.O.; Kokubu, H., Normal forms for nonlinear vector fields – part I: theory and algorithm, IEEE trans. circuits syst., 35, 863-880, (1988) · Zbl 0683.58021
[7] Chua, L.O.; Kokubu, H., Normal forms for nonlinear vector fields – part II: applications, IEEE trans. circuits syst., 36, 51-70, (1988) · Zbl 0702.58047
[8] Erneux, T.; Grasman, J., Limit-cycle oscillators subject to a delayed feedback, Phys. rev. E, 78, 2, 026209, (2008)
[9] Faria, T., On a planar system modelling a neuron network with memory, J. differential equations, 168, 129-149, (2000) · Zbl 0961.92002
[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] 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
[12] Guckenheimer, J.; Holmes, P., Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, (1983), Springer-Verlag New York · Zbl 0515.34001
[13] Jiang, W.; Wei, J., Bifurcation analysis in Van der Pol’s oscillator with delayed feedback, J. comput. appl. math., 213, 2, 604-615, (2008) · Zbl 1354.34125
[14] Jiang, W.; Yuan, Y., Bogdanov – takens singularity in Van der Pol’s oscillator with delayed feedback, Phys. D, 227, 2, 149-161, (2007) · Zbl 1124.34048
[15] Hale, J., Theory of functional differential equations, (1977), Springer-Verlag New York
[16] Ma, S.; Lu, Q.; Feng, Z., Double Hopf bifurcation for Van der pol – duffing oscillator with parametric delay feedback control, J. math. anal. appl., 338, 993-1007, (2008) · Zbl 1141.34044
[17] Olien, L.; Blair, J., Bifurcations, stability, and monotonicity properties of a delayed neural network model, Phys. D, 102, 349-363, (1997) · Zbl 0887.34069
[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, Phys. D, 166, 131-146, (2002) · Zbl 1012.34069
[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.; Han, M., Dynamics in a ratio-dependent predator-prey model with predator harvesting, J. math. anal. appl., 324, 14-29, (2006) · Zbl 1122.34035
[22] Xiao, D.; Ruan, S., Multiple bifurcations in a delayed predator-prey system with non-monotonic functional response, J. differential equations, 176, 494-510, (2001) · Zbl 1003.34064
[23] Xu, J.; Chung, K.W., Effects of time delayed position feedback on a Van der pol – duffing oscillator, Phys. D, 180, 17-39, (2003) · Zbl 1024.37028
[24] Xu, X.; Hu, H.; Wang, H., Stability, bifurcation and chaos of a delayed oscillator with negative damping and delayed feedback control, Nonlinear dynam., 49, 117-129, (2007) · Zbl 1181.70020
[25] Yan, X., Bifurcation analysis in a simplified tri-neuron BAM network model with multiple delays, Nonlinear anal. real world appl., 9, 3, 963-976, (2008) · Zbl 1152.34051
[26] Yuan, Y.; Wei, J., Multiple bifurcation analysis in a neural network model with delays, Internat. J. bifur. chaos, 16, 2903-2913, (2006) · Zbl 1185.37136
[27] Yuan, Y.; Wei, J., Singularity analysis on a planar system with multiple delays, J. dynam. differential equations, 19, 2, 437-456, (2007) · Zbl 1133.34039
[28] Yuan, Y.; Yu, P., Computation of simplest normal forms of differential equations associated with a double-zero eigenvalues, Internat. J. bifur. chaos, 11, 5, 1307-1330, (2001) · Zbl 1090.37539
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