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March  2018, 23(2): 667-700. doi: 10.3934/dcdsb.2018038

## Global Hopf bifurcations of neutral functional differential equations with state-dependent delay

 1 College of Mathematics, Taiyuan University of Technology, Taiyuan 030024, China 2 School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China 3 School of Science, Tianjin Polytechnic University, Tianjin 300387, China

* Corresponding author: Xiuli Sun

The second author is supported by NSFC grant No.11371058 and the Fundament Research Funds for the Central University. The third author is supported by NSFC grant No.11501409

Received  July 2016 Revised  September 2017 Published  December 2017

A global Hopf bifurcation theory for a system of neutral functional differential equations (NFDEs) with state-dependent delay is investigated by applying the $S^{1}$-equivariant degree theory. We use the information about the characteristic equation of the formal linearization with frozen delay to detect the local Hopf bifurcation and to describe the global continuation of periodic solutions for such a system. The results are important in studying bifurcations of NFDEs with state-dependent delay.

Citation: Xiuli Sun, Rong Yuan, Yunfei Lv. Global Hopf bifurcations of neutral functional differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 667-700. doi: 10.3934/dcdsb.2018038
##### References:
 [1] W. G. Aiello, H. I. Freedman and J. Wu, Analysis of a model representing stage-structured population growth with state-dependent time delay, SIAM J. Appl. Math., 52 (1992), 855-869. doi: 10.1137/0152048. Google Scholar [2] J. F. M. Al-Omari and S. A. Gourley, Dynamics of a stage-structured population model incorporating a state-dependent maturation delay, Nonlinear Anal. Real World Appl., 6 (2005), 13-33. doi: 10.1016/j.nonrwa.2004.04.002. Google Scholar [3] O. Arino, M. L. Hbid and R. Bravo de la Parra, A mathematical model of growth of population of fish in the larval stage: density-dependence effects, Math. Biosci., 150 (1998), 1-20. doi: 10.1016/S0025-5564(98)00008-X. Google Scholar [4] O. Arino and E. S$\acute{a}$nchez, Delays included in population dynamics, in: Mathematical Modeling of Population Dynamics, Banach Center Publ., 63 (2004), 9–46. Google Scholar [5] O. Arino, E. S$\acute{a}$nchez and A. Fathallah, State-dependent delay differential equations in population dynamics: Modeling and analysis, Fields Inst. Commun., 29 (2001), 19-36. Google Scholar [6] Z. Balanov, Q. W. Hu and W. Krawcewicz, Global Hopf bifurcation of differential equations with threshold type state-dependent delay, J. Differential Equations, 257 (2014), 2622-2670. doi: 10.1016/j.jde.2014.05.053. Google Scholar [7] R. K. Brayton, Bifurcations of periodic solutions in a nonlinear difference-differential equation of neutral type, Quarterly Appl. Math., 24 (1966), 215-224. doi: 10.1090/qam/204800. Google Scholar [8] R. K. Brayton, Nonlinear oscillations in a distributed network, Quaterly Appl. Math., 24 (1967), 289-301. doi: 10.1090/qam/99914. Google Scholar [9] Y. L. Cao, J. P. Fan and T. C. Gard, The effect of state-dependent delay on a stage-structured population growth model, Nonlinear Anal.-Theor., 19 (1992), 95-105. doi: 10.1016/0362-546X(92)90113-S. Google Scholar [10] S. N. Chow and J. Mallet-Paret, The Fuller index and global Hopf bifurcation, J. Differential Equations, 29 (1978), 66-85. doi: 10.1016/0022-0396(78)90041-4. Google Scholar [11] K. Gopalsamy and B. G. Zhang, On a neutral delay logistic equation, Dynam. Stabil. Syst., 2 (1987), 183-195. Google Scholar [12] S. J. Guo and J. S. W. Lamb, Equivariant Hopf bifurcation for neutral functional differential equations, Proc. Amer. Math. Soc., 136 (2008), 2031-2041. doi: 10.1090/S0002-9939-08-09280-0. Google Scholar [13] J. K. Hale, Theory of Functional Differential Equations, 2 $^{nd}$, Springer-Verlag, New York-Heidelberg, 1977. Google Scholar [14] Q. W. Hu and J. H. Wu, Global Hopf bifurcation for differential equations with state-dependent delay, J. Differential Equations, 248 (2010), 2801-2840. doi: 10.1016/j.jde.2010.03.020. Google Scholar [15] Q. W. Hu and J. H. Wu, Global continua of rapidly oscillating periodic solutions of state-dependent delay differential equations, J. Dynam. Differential Equations, 22 (2010), 253-284. doi: 10.1007/s10884-010-9162-5. Google Scholar [16] G. S. Jones, On the nonlinear differential-difference equation $f'(x) = -α f(x-1)[1+f(x)]$, J Math. Anal. Appl., 4 (1962), 440-469. doi: 10.1016/0022-247X(62)90041-0. Google Scholar [17] G. S. Jones, The existence of periodic solutions of $f'(x) = -α f(x-1)[1+f(x)]$, J. Math. Anal. Appl., 5 (1962), 435-450. Google Scholar [18] G. S. Jones, Periodic motions in Banach space and applications to functional-differential equations, Contrib. Diff. Eqns., 3 (1964), 75-106. Google Scholar [19] W. Krawcewicz and J. H. Wu, Theory of Degrees with Applications to Bifurcations and Differential Equations, Wiley-Interscience, John Wiley & Sons, Inc., New York, 1997. Google Scholar [20] Y. Kuang, On neutral-delay two-species Lotka-Volterra competitive systems, J. Austral. Math. Soc. Ser. B, 32 (1991), 311-326. doi: 10.1017/S0334270000006895. Google Scholar [21] S. Lang, Real and Functional Analysis, 3 $^{nd}$, edition Springer-Verlag, New York, 1993. Google Scholar [22] O. Lopes, Forced oscillations in nonlinear neutral differential equations, SIAM J. Appl. Math., 29 (1975), 196-207. doi: 10.1137/0129017. Google Scholar [23] J. Mallet-Paret, R. Nussbaum and P. Paraskevopoulos, Periodic solutions for functional-differential equations with multiple state-dependent time lags, Topol. Methods Nonlinear Anal., 3 (1994), 101-162. doi: 10.12775/TMNA.1994.006. Google Scholar [24] R. D. Nussbaum, A global bifurcation theorem with application to functional differential equations, J. Funct. Anal., 19 (1975), 319-338. doi: 10.1016/0022-1236(75)90061-0. Google Scholar [25] R. D. Nussbaum, Global bifurcation of periodic solutions of some autonomous functional differential equations, J. Math. Anal. Appl., 55 (1976), 699-725. doi: 10.1016/0022-247X(76)90076-7. Google Scholar [26] R. D. Nussbaum, A Hopf global bifurcation theorem for retarded functional differential equations, Trans. Amer. Math. Sot., 238 (1978), 139-164. doi: 10.1090/S0002-9947-1978-0482913-0. Google Scholar [27] E. C. Pielou, Mathematical Ecology, 2$^{nd}$, edition, Wiley Interscience, New York, 1977. Google Scholar [28] S. Rai and R. L. Robertson, Analysis of a two-stage population model with space limitations and state-dependent delay, Canad. Appl. Math. Quart., 8 (2000), 263-279. Google Scholar [29] S. Rai and R. L. Robertson, A stage-structured population model with state-dependent delay, Int. J. Differ. Equ. Appl., 6 (2002), 77-91. Google Scholar [30] F. E. Smith, Population dynamics in Daphnia magna and a new model for population growth, Ecology, 44 (1963), 651-663. doi: 10.2307/1933011. Google Scholar [31] H. Smith, Hopf bifurcation in a system of functional equations modelling the spread of infectious disease, SIAM J. Appl. Math., 43 (1983), 370-385. doi: 10.1137/0143025. Google Scholar [32] G. Vidossich, On the structure of periodic solutions of differential equations, J. Differential Equations, 21 (1976), 263-278. doi: 10.1016/0022-0396(76)90122-4. Google Scholar [33] J. H. Wu, Global continua of periodic solutions to some difference-differential equations of neutral type, Tohoku Math. J., 45 (1993), 67-88. doi: 10.2748/tmj/1178225955. Google Scholar

show all references

##### References:
 [1] W. G. Aiello, H. I. Freedman and J. Wu, Analysis of a model representing stage-structured population growth with state-dependent time delay, SIAM J. Appl. Math., 52 (1992), 855-869. doi: 10.1137/0152048. Google Scholar [2] J. F. M. Al-Omari and S. A. Gourley, Dynamics of a stage-structured population model incorporating a state-dependent maturation delay, Nonlinear Anal. Real World Appl., 6 (2005), 13-33. doi: 10.1016/j.nonrwa.2004.04.002. Google Scholar [3] O. Arino, M. L. Hbid and R. Bravo de la Parra, A mathematical model of growth of population of fish in the larval stage: density-dependence effects, Math. Biosci., 150 (1998), 1-20. doi: 10.1016/S0025-5564(98)00008-X. Google Scholar [4] O. Arino and E. S$\acute{a}$nchez, Delays included in population dynamics, in: Mathematical Modeling of Population Dynamics, Banach Center Publ., 63 (2004), 9–46. Google Scholar [5] O. Arino, E. S$\acute{a}$nchez and A. Fathallah, State-dependent delay differential equations in population dynamics: Modeling and analysis, Fields Inst. Commun., 29 (2001), 19-36. Google Scholar [6] Z. Balanov, Q. W. Hu and W. Krawcewicz, Global Hopf bifurcation of differential equations with threshold type state-dependent delay, J. Differential Equations, 257 (2014), 2622-2670. doi: 10.1016/j.jde.2014.05.053. Google Scholar [7] R. K. Brayton, Bifurcations of periodic solutions in a nonlinear difference-differential equation of neutral type, Quarterly Appl. Math., 24 (1966), 215-224. doi: 10.1090/qam/204800. Google Scholar [8] R. K. Brayton, Nonlinear oscillations in a distributed network, Quaterly Appl. Math., 24 (1967), 289-301. doi: 10.1090/qam/99914. Google Scholar [9] Y. L. Cao, J. P. Fan and T. C. Gard, The effect of state-dependent delay on a stage-structured population growth model, Nonlinear Anal.-Theor., 19 (1992), 95-105. doi: 10.1016/0362-546X(92)90113-S. Google Scholar [10] S. N. Chow and J. Mallet-Paret, The Fuller index and global Hopf bifurcation, J. Differential Equations, 29 (1978), 66-85. doi: 10.1016/0022-0396(78)90041-4. Google Scholar [11] K. Gopalsamy and B. G. Zhang, On a neutral delay logistic equation, Dynam. Stabil. Syst., 2 (1987), 183-195. Google Scholar [12] S. J. Guo and J. S. W. Lamb, Equivariant Hopf bifurcation for neutral functional differential equations, Proc. Amer. Math. Soc., 136 (2008), 2031-2041. doi: 10.1090/S0002-9939-08-09280-0. Google Scholar [13] J. K. Hale, Theory of Functional Differential Equations, 2 $^{nd}$, Springer-Verlag, New York-Heidelberg, 1977. Google Scholar [14] Q. W. Hu and J. H. Wu, Global Hopf bifurcation for differential equations with state-dependent delay, J. Differential Equations, 248 (2010), 2801-2840. doi: 10.1016/j.jde.2010.03.020. Google Scholar [15] Q. W. Hu and J. H. Wu, Global continua of rapidly oscillating periodic solutions of state-dependent delay differential equations, J. Dynam. Differential Equations, 22 (2010), 253-284. doi: 10.1007/s10884-010-9162-5. Google Scholar [16] G. S. Jones, On the nonlinear differential-difference equation $f'(x) = -α f(x-1)[1+f(x)]$, J Math. Anal. Appl., 4 (1962), 440-469. doi: 10.1016/0022-247X(62)90041-0. Google Scholar [17] G. S. Jones, The existence of periodic solutions of $f'(x) = -α f(x-1)[1+f(x)]$, J. Math. Anal. Appl., 5 (1962), 435-450. Google Scholar [18] G. S. Jones, Periodic motions in Banach space and applications to functional-differential equations, Contrib. Diff. Eqns., 3 (1964), 75-106. Google Scholar [19] W. Krawcewicz and J. H. Wu, Theory of Degrees with Applications to Bifurcations and Differential Equations, Wiley-Interscience, John Wiley & Sons, Inc., New York, 1997. Google Scholar [20] Y. Kuang, On neutral-delay two-species Lotka-Volterra competitive systems, J. Austral. Math. Soc. Ser. B, 32 (1991), 311-326. doi: 10.1017/S0334270000006895. Google Scholar [21] S. Lang, Real and Functional Analysis, 3 $^{nd}$, edition Springer-Verlag, New York, 1993. Google Scholar [22] O. Lopes, Forced oscillations in nonlinear neutral differential equations, SIAM J. Appl. Math., 29 (1975), 196-207. doi: 10.1137/0129017. Google Scholar [23] J. Mallet-Paret, R. Nussbaum and P. Paraskevopoulos, Periodic solutions for functional-differential equations with multiple state-dependent time lags, Topol. Methods Nonlinear Anal., 3 (1994), 101-162. doi: 10.12775/TMNA.1994.006. Google Scholar [24] R. D. Nussbaum, A global bifurcation theorem with application to functional differential equations, J. Funct. Anal., 19 (1975), 319-338. doi: 10.1016/0022-1236(75)90061-0. Google Scholar [25] R. D. Nussbaum, Global bifurcation of periodic solutions of some autonomous functional differential equations, J. Math. Anal. Appl., 55 (1976), 699-725. doi: 10.1016/0022-247X(76)90076-7. Google Scholar [26] R. D. Nussbaum, A Hopf global bifurcation theorem for retarded functional differential equations, Trans. Amer. Math. Sot., 238 (1978), 139-164. doi: 10.1090/S0002-9947-1978-0482913-0. Google Scholar [27] E. C. Pielou, Mathematical Ecology, 2$^{nd}$, edition, Wiley Interscience, New York, 1977. Google Scholar [28] S. Rai and R. L. Robertson, Analysis of a two-stage population model with space limitations and state-dependent delay, Canad. Appl. Math. Quart., 8 (2000), 263-279. Google Scholar [29] S. Rai and R. L. Robertson, A stage-structured population model with state-dependent delay, Int. J. Differ. Equ. Appl., 6 (2002), 77-91. Google Scholar [30] F. E. Smith, Population dynamics in Daphnia magna and a new model for population growth, Ecology, 44 (1963), 651-663. doi: 10.2307/1933011. Google Scholar [31] H. Smith, Hopf bifurcation in a system of functional equations modelling the spread of infectious disease, SIAM J. Appl. Math., 43 (1983), 370-385. doi: 10.1137/0143025. Google Scholar [32] G. Vidossich, On the structure of periodic solutions of differential equations, J. Differential Equations, 21 (1976), 263-278. doi: 10.1016/0022-0396(76)90122-4. Google Scholar [33] J. H. Wu, Global continua of periodic solutions to some difference-differential equations of neutral type, Tohoku Math. J., 45 (1993), 67-88. doi: 10.2748/tmj/1178225955. Google Scholar
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