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Hopf bifurcation for neutral functional differential equations. (English) Zbl 1194.34137
Summary: We extend the computation of the properties of Hopf bifurcation, such as the direction of bifurcation and stability of bifurcating periodic solutions, of DDE to a kind of neutral functional differential equation (NFDE). As an example, a neutral delay logistic differential equation is considered, and the explicit formulas for determining the direction of bifurcation and the stability of bifurcating periodic solutions are derived. Finally, some numerical simulations are carried out to support the analytic results.

##### MSC:
 34K18 Bifurcation theory of functional differential equations 92D25 Population dynamics (general) 34K28 Numerical approximation of solutions of functional-differential equations 34K13 Periodic solutions of functional differential equations 34K20 Stability theory of functional-differential equations
##### Keywords:
NFDE; Hopf bifurcation; neutral logistic equation
Full Text:
##### References:
 [1] Brayton, R. K.: Bifurcation of periodic solutions in a nonlinear difference-differential equation of neutral type, Quart. appl. Math. 24, 215-224 (1966) · Zbl 0143.30701 [2] Gopalsamy, K.; Zhang, B.: On a neutral delay-logistic equation, Dyn. stab. Syst. 2, 183-195 (1988) · Zbl 0665.34066 · doi:10.1080/02681118808806037 [3] Györi, I.; Ladas, G.: Oscillation theory of delay differential equations with applications, (1991) · Zbl 0780.34048 [4] Hale, J.; Lunel, Y. H.: Introduction to functional differential equations, (1993) · Zbl 0787.34002 [5] Krawcewicz, W.; Ma, S.; Wu, J.: Multiple slowly oscillating periodic solutions in coupled lossless transmission lines, Nonlinear anal. 5, 309-354 (2004) · Zbl 1144.34365 · doi:10.1016/S1468-1218(03)00040-3 [6] Krawcewicz, W.; Wu, J.; Xia, H.: Global Hopf bifurcation theory for condensing fields and neutral equations with applications to lossless transmission problems, Can. appl. Math. Q. 1, 167-219 (1993) · Zbl 0801.34069 [7] Weedermann, M.: Normal forms for neutral functional differential equations, Fields inst. Commun. 29, 361-368 (2001) · Zbl 0989.34060 [8] Faria, T.; Magalhaes, L.: Normal forms for retarded functional differential equation and applications to bogdanov--Takens singularity, J. differential equations 122, 201-224 (1995) · Zbl 0836.34069 · doi:10.1006/jdeq.1995.1145 [9] Faria, T.; Magalhaes, L.: Normal forms for retarded functional differential equation with parameters and applications to Hopf bifurcation, J. differential equations 122, 181-200 (1995) · Zbl 0836.34068 · doi:10.1006/jdeq.1995.1144 [10] Weedermann, M.: Hopf bifurcation calculations for scalar delay differential equations, Nonlinearity 19, 2091-2102 (2006) · Zbl 1116.34057 · doi:10.1088/0951-7715/19/9/005 [11] Kazarinoff, N. D.; Den Driessche, P. Van; Wan, Y. H.: Hopf bifurcation and stability of periodic solutions of differential-difference and integro-differential equations, J. inst. Math. appl. 21, 461-477 (1978) · Zbl 0379.45021 · doi:10.1093/imamat/21.4.461 [12] Hassard, B.; Kazarinoff, N.; Wan, Y.: Theory and applications of Hopf bifurcation, (1981) · Zbl 0474.34002 [13] Wei, J.; Ruan, S.: Stability and global Hopf bifurcation for neutral differential equations, Acta math. Sinica 45, 94-104 (2002) · Zbl 1018.34068 [14] Carr, J.: Applications of centre manifold theory, (1981) · Zbl 0464.58001 [15] Chow, S. -N.; Lu, K.: Ck center unstable manifolds, Proc. roy. Soc. Edinburgh sect. A 108, 303-320 (1988) · Zbl 0707.34039 · doi:10.1017/S0308210500014682 [16] Hale, J.; Weedermann, M.: On perturbations of delay-differential equations with periodic orbits, J. differential equations 197, 219-246 (2004) · Zbl 1071.34074 · doi:10.1016/S0022-0396(02)00063-3 [17] Wiggins, S.: Introduction to applied nonlinear dynamical systems and chaos, (1990) · Zbl 0701.58001 [18] Wang, C.; Wei, J.: Normal forms for nfdes with parameters and application to the lossless transmission line, Nonlinear dynam. 52, 199-206 (2008) · Zbl 1187.34094 · doi:10.1007/s11071-007-9271-9