Advanced Search
Article Contents
Article Contents

Spatiotemporal attractors generated by the Turing-Hopf bifurcation in a time-delayed reaction-diffusion system

  • * Corresponding author: Weihua Jiang

    * Corresponding author: Weihua Jiang

The authors are supported the National Natural Science Foundation of China (No.11371112)

Abstract Full Text(HTML) Figure(5) / Table(1) Related Papers Cited by
  • We study the Turing-Hopf bifurcation and give a simple and explicit calculation formula of the normal forms for a general two-components system of reaction-diffusion equation with time delays. We declare that our formula can be automated by Matlab. At first, we extend the normal forms method given by Faria in 2000 to Hopf-zero singularity. Then, an explicit formula of the normal forms for Turing-Hopf bifurcation are given. Finally, we obtain the possible attractors of the original system near the Turing-Hopf singularity by the further analysis of the normal forms, which mainly include, the spatially non-homogeneous steady-state solutions, periodic solutions and quasi-periodic solutions.

    Mathematics Subject Classification: Primary: 35B32, 35B35; Secondary: 35B36.


    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  The spatially non-homogeneous periodic attractors $H = (H_1, H_2)$, under the Case 4. Here $\rho_4 = 0.2, ~\phi_1(0) = ( 0.1-0.1\mathrm{{i}}, 0.1+0.5\mathrm{{i}})^{\mathrm{{T}}}$, $h_4 = (0.1, 0.3)^{\mathrm{{T}}}$, $w_4 = 1$, $l = 1$, $n_2 = 3$

    Figure 2.  The spatially non-homogeneous quasi-periodic attractors $H = (H_1, H_2)$, under the case 5. Here $\rho_4 = 0.2, \rho_5 = 0.1, \phi_1(0) = ( 0.1-0.1\mathrm{{i}}, 0.1+0.5\mathrm{{i}})^{\mathrm{{T}}}$, $h_4 = (0.1, 0.3)^{\mathrm{{T}}}$, $h_5 = (0.2, 0.5)^{\mathrm{{T}}}$, $w_5 = 1$, $\varpi = 0.5$, $l = 1$, $n_2 = 3$

    Figure 3.  (a) Bifurcation sets in $(\alpha_1, \alpha_2)$ plane. (b) Phase portraits in $D_1-D_6$

    Figure 4.  Two spatially inhomogeneous periodic solutions coexist in $D_3$, with $(\alpha_1, \alpha_2) = (0.05, -0.33)$. (a), (b) are the solutions $u(t, x), v(t, x)$ of (33) with the initial value functions $(\varphi(t, x), \psi(t, x)) = (u_0+0.005\sin x, v_0+0.001\sin x)$. (c), (d) are the solutions $u(t, x), v(t, x)$ of (33) with the initial value functions $(\varphi(t, x), \psi(t, x)) = (u_0-0.005\sin x, v_0-0.001\sin x)$

    Figure 5.  Two spatially inhomogeneous steady state solutions coexist in $D_5$, with $(\alpha_1, \alpha_2) = (-0.1, -0.4)$. (a), (b) are the solutions $u(t, x), v(t, x)$ of (33) with the initial value functions $(\varphi(t, x), \psi(t, x)) = (u_0+0.005\sin x, v_0+0.001\sin x)$. (c), (d) are the solutions $u(t, x), v(t, x)$ of (33) with the initial value functions $(\varphi(t, x), \psi(t, x)) = (u_0-0.005\sin x, v_0-0.001\sin x)$

    Table 1.  The correspondence between the planner and original system

    Planar system (29) The original system (1)
    $E_1$ Constant steady state $(0, 0)$
    $E_2$ Spatially homogeneous periodic solution
    $E_3$ Non-constant steady state
    $E_4$ Spatially non-homogeneous periodic solution
    Periodic solution Spatially non-homogeneous quasi-periodic solution
     | Show Table
    DownLoad: CSV
  •   M. Baurmann , T. Gross  and  U. Feudel , Instabilities in spatially extended predator-prey systems: Spatio-temporal patterns in the neighborhood of turing-hopf bifurcations, J. Theoret. Biol., 245 (2007) , 220-229.  doi: 10.1016/j.jtbi.2006.09.036.
      J. Carr, Applications of Centre Manifold Theory, vol. 35, Springer-Verlag, New York-Berlin, 1981. doi: 10.1007/978-1-4612-5929-9.
      V. Castets, E. Dulos, J. Boissonade and P. D. Kepper, Experimental evidence of a sustained standing turing-type nonequilibrium chemical pattern, Phys. Rev. Lett., 64 (1990), 2953. doi: 10.1103/PhysRevLett.64.2953.
      S. S. Chen, Y. Lou and J. J. Wei, Hopf bifurcation in a delayed reaction-diffusion-advection population model, J. Differential Equations, 264 (2018), 5333-5359, arXiv: 1706.02087. doi: 10.1016/j.jde.2018.01.008.
      T. Faria , Normal forms and Hopf bifurcation for partial differential equations with delays, Trans. Amer. Math. Soc., 352 (2000) , 2217-2238.  doi: 10.1090/S0002-9947-00-02280-7.
      T. Faria  and  L. T. Magalhaes , Normal forms for retarded functional differential equations with parameters and applications to Hopf bifurcation, J. Differential Equations, 122 (1995) , 181-200.  doi: 10.1006/jdeq.1995.1144.
      J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, vol. 42 of Applied Mathematical Sciences, Springer New York, 1983. doi: 10.1007/978-1-4612-1140-2.
      K. P. Hadeler  and  S. G. Ruan , Interaction of diffusion and delay, Discrete Contin. Dyn. Syst. Ser. B, 8 (2007) , 95-105.  doi: 10.3934/dcdsb.2007.8.95.
      J. R. Huang , Z. H. Liu  and  S. G. Ruan , Bifurcation and temporal periodic patterns in a plant-pollinator model with diffusion and time delay effects, J. Biol. Dyn., 11 (2017) , 138-159.  doi: 10.1080/17513758.2016.1181802.
      R. E. Kooij , J. T. Arus  and  A. G. Embid , Limit cycles in the holling-tanner model, Publ. Mat., 41 (1997) , 149-167.  doi: 10.5565/PUBLMAT_41197_09.
      I. Lengyel  and  I. R. Epstein , Modeling of turing structures in the chlorite-iodide-malonic acid-starch reaction system, Science, 251 (1991) , 650-652.  doi: 10.1126/science.251.4994.650.
      W. T. Li , G. Lin  and  S. G. Ruan , Existence of travelling wave solutions in delayed reaction diffusion systems with applications to diffusion competition systems, Nonlinearity, 19 (2006) , 1253-1273.  doi: 10.1088/0951-7715/19/6/003.
      X. D. Lin , J. W. H. So  and  J. H. Wu , Centre manifolds for partial differential equations with delays, Proc. Roy. Soc. Edinburgh, 122 (1992) , 237-254.  doi: 10.1017/S0308210500021090.
      P. K. Maini , K. J. Painter  and  H. N. P. Chau , Spatial pattern formation in chemical and biological systems, J. Chem. Soc. Faraday Trans., 93 (1997) , 3601-3610.  doi: 10.1039/a702602a.
      R. H. Martin  and  H. L. Smith , Reaction-diffusion systems with time delays: Monotonicity, invariance, comparison and convergence, J. Reine Angew. Math., 413 (1991) , 1-35.  doi: 10.1515/crll.1991.413.1.
      J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, Springer, 2003.
      Q. Ouyang  and  H. L. Swinney , Transition from a uniform state to hexagonal and striped turing patterns, Nature, 352 (1991) , 610-612.  doi: 10.1038/352610a0.
      C. V. Pao , Dynamics of nonlinear parabolic systems with time delays, J. Math. Anal. Appl., 198 (1996) , 751-779.  doi: 10.1006/jmaa.1996.0111.
      C. V. Pao , Convergence of solutions of reaction-diffusion systems with time delays, Nonlinear Anal., 48 (2002) , 349-362.  doi: 10.1016/S0362-546X(00)00189-9.
      A. Rovinsky  and  M. Menzinger , Interaction of turing and hopf bifurcations in chemical systems, Phys. Rev. A, 46 (1992) , 6315-6322.  doi: 10.1103/PhysRevA.46.6315.
      E. Sáez  and  E. González-Olivares , Dynamics of a predator-prey model, SIAM J. Appl. Math., 59 (1999) , 1867-1878.  doi: 10.1137/S0036139997318457.
      L. A. Segel  and  J. L. Jackson , Dissipative structure: An explanation and an ecological example, J. Theoret. Biol., 37 (1972) , 545-559.  doi: 10.1016/0022-5193(72)90090-2.
      H. B. Shi  and  S. G. Ruan , Spatial, temporal and spatiotemporal patterns of diffusive predator-prey models with mutual interference, IMA J. Appl. Math., 80 (2015) , 1534-1568.  doi: 10.1093/imamat/hxv006.
      C. C. Travis  and  G. F. Webb , Existence and stability for partial functional differential equations, Trans. Amer. Math. Soc., 200 (1974) , 395-418.  doi: 10.2307/1997265.
      A. M. Turing , The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. London Ser. B, 237 (1952) , 37-72.  doi: 10.1098/rstb.1952.0012.
      H. B. Wang  and  W. H. Jiang , Hopf-pitchfork bifurcation in van der Pol's oscillator with nonlinear delayed feedback, J. Math. Anal. Appl., 368 (2010) , 9-18.  doi: 10.1016/j.jmaa.2010.03.012.
      A. D. Wit , D. Lima , G. Dewel  and  P. Borckmans , Spatiotemporal dynamics near a codimension-two point, Phys. Rev. E, 54 (1996) , 261-271.  doi: 10.1103/PhysRevE.54.261.
      J. H. Wu, Theory and Applications of Partial Functional Differential Equations, Springer, 1996. doi: 10.1007/978-1-4612-4050-1.
      J. H. Wu  and  X. F. Zou , Traveling wave fronts of Reaction-Diffusion systems with delay, J. Dynam. Differential Equations, 13 (2001) , 651-687.  doi: 10.1023/A:1016690424892.
      X. F. Xu  and  J. J. Wei , Bifurcation analysis of a spruce budworm model with diffusion and physiological structures, J. Differential Equations, 262 (2017) , 5206-5230.  doi: 10.1016/j.jde.2017.01.023.
      R. Yang  and  Y. L. Song , Spatial resonance and Turing-Hopf bifurcations in the Gierer-Meinhardt model, Nonlinear Anal. Real World Appl., 31 (2016) , 356-387.  doi: 10.1016/j.nonrwa.2016.02.006.
      X. Q. Zhao, Dynamical Systems in Population Biology, Springer, Cham, 2017. doi: 10.1007/978-3-319-56433-3.
  • 加载中




Article Metrics

HTML views(739) PDF downloads(397) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint