Double Hopf Bifurcation in Delayed reaction–diffusion Systems

  • Yanfei Du
  • Ben NiuEmail author
  • Yuxiao Guo
  • Junjie Wei


Double Hopf bifurcation analysis can be used to reveal some complicated dynamical behavior in a dynamical system, such as the existence or coexistence of periodic orbits, quasi-periodic orbits, or even chaos. In this paper, an algorithm for deriving the normal form near a codimension-two double Hopf bifurcation of a reaction–diffusion system with time delay and Neumann boundary condition is rigorously established, by employing the center manifold reduction technique and the normal form method. The dynamical behavior near bifurcation points are proved to be governed by twelve distinct unfolding systems. Two examples are performed to illustrate our results: for a stage-structured epidemic model, we find that double Hopf bifurcation appears when varying the diffusion rate and time delay, and two stable spatially inhomogeneous periodic oscillations are proved to coexist near the bifurcation point; in a diffusive Predator–Prey system, we theoretically proved that quasi-periodic orbits exist on two- or three-torus near a double Hopf bifurcation point, which will break down after slight perturbation, leaving the system a strange attractor.


Reaction–diffusion model Delay Double Hopf bifurcation Coexistence Strange attractor 

Mathematics Subject Classification

35B32 35B41 37G05 



The authors are grateful to the handling editor and anonymous referees for their careful reading of the manuscript and valuable comments, which improve the exposition of the paper very much. This research is supported by National Natural Science Foundation of China (11701120, 11771109) and Shaanxi Provincial Education Department Grant (18JK0123).


  1. 1.
    An, Q., Jiang, W: Spatiotemporal attractors generated by the Turing-Hopf bifurcation in a time-delayed reaction-diffusion system. Disctete Cont. Dyn-B. (2018)
  2. 2.
    Andronov, A.A.: Application of Poincaré theorem on bifurcation points and change in stability to simple auto-oscillatory systems. C. R. Acad. Sci. Paris 189, 559–561 (1929)Google Scholar
  3. 3.
    Bajaj, A.K., Sethna, P.R.: Bifurcations in three-dimensional motions of articulated tubes. I—Linear systems and symmetry. II—Nonlinear analysis. J. Appl. Mech 49, 606–618 (1982)CrossRefGoogle Scholar
  4. 4.
    Battelino, P.M., Grebogi, C., Ott, E., Yorke, J.A.: Chaotic attractors on a 3-torus, and torus break-up. Physica D 39, 299–314 (1989)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Baurmann, M., Gross, T., Feudel, U.: Instabilities in spatially extended predator–prey systems: spatio-temporal patterns in the neighborhood of Turing–Hopf bifurcations. J. Theor. Bio. 245, 220–229 (2007)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Belair, J., Campbell, S.A., Driessche, P.V.D.: Frustration, stability, and delay-induced oscillations in a neural network model. SIAM. J. Appl. Math. 56, 245–255 (1996)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bi, P., Ruan, S.: Bifurcations in delay differential equations and applications to tumor and immune system interaction models. SIAM J. Appl. Dyn. Syst. 12, 1847–1888 (2013)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Buono, P.L., Bélair, J.: Restrictions and unfolding of double Hopf bifurcation in functional differential equations. J. Differ. Equ. 189, 234–266 (2003)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Campell, S.A., Bélair, J.: Analytical and symbolically-assisted investigation of Hopf bifurcations in delay-differential equations. Can. Appl. Math. Q. 3, 137–154 (1995)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Campell, S.A., Bélair, J., Ohira, T., Milton, J.: Limit cycles, tori, and complex dynamics in a second-order differential equations with delayed negative feedback. J. Dyn. Differ. Equ. 7, 213–236 (1995)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Campell, S.A., LeBlanc, V.G.: Resonant Hopf–Hopf interaction in delay differential equations. J. Dyn. Differ. Equ. 10, 327–346 (1998)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Chen, S., Shi, J., Wei, J.: Global stability and Hopf bifurcation in a delayed diffusive Leslie–Gower predator–prey system. Int. J. Bifurcat. Chaos 22, 331–517 (2012)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Chen, S., Yu, J.: Stability and bifurcations in a nonlocal delayed reaction–diffusion population model. J. Differ. Equations 260, 218–240 (2016)MathSciNetCrossRefGoogle Scholar
  14. 14.
    De Wit, A., Dewel, G., Borckmans, P.: Chaotic Turing–Hopf mixed mode. Phys. Rev. E 48, R4191–R4194 (1993)CrossRefGoogle Scholar
  15. 15.
    Du, Y., Guo, Y., Xiao, P.: Freely-moving delay induces periodic oscillations in a structured SEIR model. Int. J. Bifurcat. Chaos 27, 1750122 (2017)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Eckmann, J.P.: Roads to turbulence in dissipative dynamical systems. Rev. Modern Phys. 53, 643–654 (1981)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Elphick, C., Tiraopegui, E., Brachet, M.E., Coullet, P., Iooss, G.: A simple global characterization for normal forms of singular vector fields. Physica D 29, 95–127 (1987)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Faria, T.: Normal forms and Hopf bifurcation for partial differential equations with delays. Trans. Am. Math. Soc. 352, 2217–2238 (2000)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Faria, T.: Stability and bifurcation for a delayed predator–prey model and the effect of diffusion. J. Math. Anal. Appl. 254, 433–463 (2001)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Faria, T., Huang, W.: Stability of periodic solutions arising from Hopf bifurcation for a reaction–diffusion equation with time delay. Fields Inst. Commun. 31, 125–141 (2002)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Faria, T., Magalhães, L.T.: Normal forms for retarded functional differential equations with parameters and applications to Hopf bifurcation. J. Differ. Equ. 122, 181–200 (1995)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Faria, T., Magalhães, L.T.: Normal form for retarded functional differential equations and applications to Bogdanov–Takens singularity. J. Differ. Equ. 122, 201–224 (1995)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Gils, S.A.V., Krupa, M., Langford, W.F.: Hopf bifurcation with non-semisimple 1:1 resonance. Nonlinearity 3, 825–850 (1990)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Govaerts, W., Guckenheimer, J., Khibnik, A.: Defining functions for multiple Hopf bifurcations. SIAM J. Numer. Anal. 34, 1269–1288 (1997)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York (1983)CrossRefGoogle Scholar
  26. 26.
    Guo, S.: Stability and bifurcation in a reaction–diffusion model with nonlocal delay effect. J. Differ. Equ. 259, 1409–1448 (2015)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Guo, S., Ma, L.: Stability and bifurcation in a delayed reaction–diffusion equation with Dirichlet boundary condition. J. Nonlinear Sci. 26, 545–580 (2016)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Hale, J.K., Kocak, H.: Dynamics and Bifurcations. Springer, New York (1991)CrossRefGoogle Scholar
  29. 29.
    Hale, J.K., Lunel, S.M.V.: Introduction to Functional Differential Equations. Springer, New York (1993)CrossRefGoogle Scholar
  30. 30.
    Hassard, B.D., Kazarinoff, N.D., Wan, Y.H.: Theory and Applications of Hopf Bifurcation. Cambridge University Press, New York (1981)zbMATHGoogle Scholar
  31. 31.
    Hethcote, H.W., Lewis, M.A., Driessche, P.V.D.: An epidemiological model with a delay and a nonlinear incidence rate. J. Math. Biol. 27, 49–64 (1989)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Hopf, E.: Abzweigung einer periodischen lösung eines differential systems. Berichen Math. Phys. Kl. Säch. Akad. Wiss. Leipzig 94, 1–22 (1942)Google Scholar
  33. 33.
    Hsu, S.B., Huang, T.W.: Global stability for a class of predator–prey systems. SIAM J. Appl. Math. 55, 763–783 (1995)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Ji, J., Li, X., Luo, Z.: Two-to-one resonant Hopf bifurcations in a quadratically nonlinear oscillator involving time delay. Int. J. Bifurcat. Chaos 22, 1250060 (2012)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Kielhöfer, H.: Bifurcation Theory: An Introduction with Applications to Partial Differential Equations. Springer, New York (2011)zbMATHGoogle Scholar
  36. 36.
    Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory. Springer, New York (2011)Google Scholar
  37. 37.
    Lewis, G.M., Nagata, W.: Double Hopf bifurcations in the differentially heated rotating annulus. SIAM J. Appl. Math. 63, 1029–1055 (2003)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Lin, X., So, J.W.H., Wu, J.: Centre manifolds for partial differential equations with delays. P. Roy. Soc. Edinb. A 122, 237–254 (1992)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Luongo, A., Paolone, A.: Perturbation methods for bifurcation analysis from multiple nonresonant complex eigenvalues. Nonlinear Dyn. 14, 193–210 (1997)MathSciNetCrossRefGoogle Scholar
  40. 40.
    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)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Meixner, M., De Wit, A., Bose, S., Schöll, E.: Generic spatiotemporal dynamics near codimension-two Turing–Hopf bifurcations. Phys. Rev. E 55, 6690–6697 (1997)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Poincaré, H.: Les Méthodes Nouvelles de la Mécanique Céleste. Cauthier-Villars, Paris (1892)zbMATHGoogle Scholar
  43. 43.
    Reddy, D.V.R., Sen, A., Johnston, G.L.: Time delay effects on coupled limit cycle oscillators at Hopf bifurcation. Physica D 129, 15–34 (1999)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Revel, G., Alonso, D.M., Moiola, J.L.: Interactions between oscillatory modes near a 2:3 resonant Hopf–Hopf bifurcation. Chaos 20, 113–129 (2010)CrossRefGoogle Scholar
  45. 45.
    Revel, G., Alonso, D.M., Moiola, J.L.: Numerical semi-global analysis of a 1:2 resonant Hopf–Hopf bifurcation. Physica D 247, 40–53 (2013)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Ruan, S., Xiao, D.: Global analysis in a predator–prey system with nonmonotonic functional response. SIAM J. Appl. Math. 61, 1445–1472 (2000)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Ruelle, D., Takens, F.: On the nature of turbulence. Commun. Math. Phys. 20, 167–192 (1971)MathSciNetCrossRefGoogle Scholar
  48. 48.
    Song, Y., Wei, J.: Local Hopf bifurcation and global periodic solutions in a delayed predator–prey system. J. Math. Anal. Appl. 301, 1–21 (2005)MathSciNetCrossRefGoogle Scholar
  49. 49.
    Song, Y., Zhang, T., Peng, Y.: Turing–Hopf bifurcation in the reaction–diffusion equations and its applications. Commun. Nonlinear Sci. Numer. Simul. 33, 229–258 (2016)MathSciNetCrossRefGoogle Scholar
  50. 50.
    Steen, P.H., Davis, S.H.: Quasiperiodic bifurcation in nonlinearly-coupled oscillators near a point of strong resonance. SIAM J. Appl. Math. 42, 1345–1368 (1982)MathSciNetCrossRefGoogle Scholar
  51. 51.
    Su, Y., Wei, J., Shi, J.: Hopf bifurcations in a reaction–diffusion population model with delay effect. J. Differ. Equ. 247, 1156–1184 (2009)MathSciNetCrossRefGoogle Scholar
  52. 52.
    Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer, New York (2003)zbMATHGoogle Scholar
  53. 53.
    Wu, J.: Theory and Applications of Partial Functional-Differential Equations. Springer, New York (1996)CrossRefGoogle Scholar
  54. 54.
    Xiao, D.: Bifurcations of a ratio-dependent predator–prey system with constant rate harvesting. SIAM J. Appl. Math. 65, 737–753 (2005)MathSciNetCrossRefGoogle Scholar
  55. 55.
    Xiao, Y., Chen, L.: An SIS epidemic model with stage structure and a delay. Acta Math. Appl. Sin. E. 18, 607–618 (2002)MathSciNetCrossRefGoogle Scholar
  56. 56.
    Xu, X., Wei, J.: Turing–Hopf bifurcation of a class of modified Leslie–Gower model with diffusion. Discrete Continuous Dyn. Syst. Ser. B 23, 765–783 (2018)MathSciNetCrossRefGoogle Scholar
  57. 57.
    Yan, X., Li, W.: Stability of bifurcating periodic solutions in a delayed reaction–diffusion population model. Nonlinearity 23, 1413–1431 (2010)MathSciNetCrossRefGoogle Scholar
  58. 58.
    Yi, F., Wei, J., Shi, J.: Bifurcation and spatiotemporal patterns in a homogenous diffusive predator–prey system. J. Differ. Equ. 246, 1944–1977 (2009)CrossRefGoogle Scholar
  59. 59.
    Yu, P.: Analysis on double Hopf bifurcation using computer algebra with the aid of multiple scales. Nonlinear Dyn. 27, 19–53 (2002)MathSciNetCrossRefGoogle Scholar
  60. 60.
    Yu, P., Bi, Q.: Analysis of non-linear dynamics and bifurcations of a double pendulum. J. Sound Vib. 217, 691–736 (1998)MathSciNetCrossRefGoogle Scholar
  61. 61.
    Yu, P., Yuan, Y., Xu, J.: Study of double Hopf bifurcation and chaos for oscillator with time delay feedback. Commun. Nonlinear Sci. Numer. Simul. 7, 69–91 (2002)MathSciNetCrossRefGoogle Scholar
  62. 62.
    Zhang, Y., Xu, J.: Classification and computation of non-resonant double Hopf bifurcations and solutions in delayed van der Pol–Duffing system. Int. J. Nonlinear Sci. Numer. Simul. 6, 67–74 (2005)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.College of Arts and SciencesShaanxi University of Science and TechnologyXi’anChina
  2. 2.Department of MathematicsHarbin Institute of Technology at WeihaiWeihaiChina
  3. 3.School of Mathematics and Big DataFoshan UniversityFoshanChina

Personalised recommendations