# American Institute of Mathematical Sciences

December  2012, 7(4): 893-926. doi: 10.3934/nhm.2012.7.893

## Oscillatory dynamics in a reaction-diffusion system in the presence of 0:1:2 resonance

 1 Graduate school of Advanced Mathematical Science, Meiji University, Higashimita, 214-8571, Japan 2 Meteorological college, Kashiwa, 277-0852, Japan

Received  January 2012 Revised  July 2012 Published  December 2012

Oscillatory dynamics in a reaction-diffusion system with spatially nonlocal effect under Neumann boundary conditions is studied. The system provides triply degenerate points for two spatially non-uniform modes and uniform one (zero mode). We focus our attention on the 0:1:2-mode interaction in the reaction-diffusion system. Using a normal form on the center manifold, we seek the equilibria and study the stability of them. Moreover, Hopf bifurcation phenomena is studied for each equilibrium which has a Hopf instability point. The numerical results to the chaotic dynamics are also shown.
Citation: Toshiyuki Ogawa, Takashi Okuda. Oscillatory dynamics in a reaction-diffusion system in the presence of 0:1:2 resonance. Networks and Heterogeneous Media, 2012, 7 (4) : 893-926. doi: 10.3934/nhm.2012.7.893
##### References:
 [1] D. Armbruster, J. Guckenheimer and P. Holmes, Heteroclinic cycles and modulated travelling waves in system with O(2) symmetry, Physica, 29D (1988), 257-282. doi: 10.1016/0167-2789(88)90032-2. [2] J. Carr, "Applications of Center Manifold Theory," Springer, 1981. [3] J. Kaplan and J. Yorke, "Chaotic Behavior of Multi-dimensional Differential Equations and The Approximation of Fixed Points," Lecture Notes in Mathematics, 730, Springer. [4] P. Frederickson, J. Kaplan, E. Yorke and J. Yorke, The Lyapunov dimension of strange attractors, J. DIff. Eqs., 49 (1983), 185-207. doi: 10.1016/0022-0396(83)90011-6. [5] T. Ogawa, Degenerate Hopf instability in oscillatory reaction-diffusion equations, Discrete Contin. Dyn. Syst., (2007). Proceedings of the 6th AIMS International Conference, suppl., 784-793. [6] M. R. E. Proctor and C. A. Jones, The interaction of two spatially resonant patterns in thermal convection, Part 1. Exact 1:2 resonance, J. Fluid Mech., 188 (1988), 301-335. doi: 10.1017/S0022112088000746. [7] J. Porter and E. Knobloch, New type of complex dynamics in the 1:2 spatial resonance, Physica, 159D (2001), 125-154. doi: 10.1016/S0167-2789(01)00340-2. [8] Y. A. Kuznetsov, "Elements of Applied Bifurcation Theory," Springer, 1997. [9] J. Liu, F. Yi and J. Wei, Multiple bifurcation analysis and spatiotemporal patterns in a 1-D Gierer-Meinhardt model of morphogenesis, IJBC, 20 (2010), 1007-1025. doi: 10.1142/S0218127410026289. [10] I. Shimada and T. Nagashima, A numerical approach to Ergodic problem of dissipative dynamical systems, Prog. Theor. Phys., 61 (1979), 1605-1616. doi: 10.1143/PTP.61.1605. [11] T. R. Smith, J. Moehlis and P. Holmes, Heteroclinic cycles and periodic orbits for the O(2)-equivariant 0:1:2 mode interaction, Physica, 211D (2005), 347-376. doi: 10.1016/j.physd.2005.09.002. [12] Y. Morita and T. Ogawa, Stability and bifurcations of nonconstant solutions to a reaction-diffusion system with conservation mass, Nonlinearity, 23 (2010), 1387-1411. doi: 10.1088/0951-7715/23/6/007. [13] A. M. Turing, The chemical basis of morphogenesis, Phil. Trans. R. Soc. B, 237 (1952), 37-72. [14] L. Yang, M. Dolnik, A. M. Zhabotinsky and I. R. Epstein, Pattern formation arising from interactions between Turing and wave instabilities, J. Chem. Phys., 117 (2002), 7257-7265. [15] A. Vanderbauwhede and G. Iooss, Center manifold theory in infinite dimensions, Dynam. Report. Expositions Dynam. Systems (N.S.), 1, Springer, (1992), 125-163. [16] M. J. Ward and J. Wei, Hopf Bifurcation of spike solutions for the shadow Gierer-Meinhardt model, Europ. J. Appl. Math., 14 (2003), 677-711. doi: 10.1017/S0956792503005278.

show all references

##### References:
 [1] D. Armbruster, J. Guckenheimer and P. Holmes, Heteroclinic cycles and modulated travelling waves in system with O(2) symmetry, Physica, 29D (1988), 257-282. doi: 10.1016/0167-2789(88)90032-2. [2] J. Carr, "Applications of Center Manifold Theory," Springer, 1981. [3] J. Kaplan and J. Yorke, "Chaotic Behavior of Multi-dimensional Differential Equations and The Approximation of Fixed Points," Lecture Notes in Mathematics, 730, Springer. [4] P. Frederickson, J. Kaplan, E. Yorke and J. Yorke, The Lyapunov dimension of strange attractors, J. DIff. Eqs., 49 (1983), 185-207. doi: 10.1016/0022-0396(83)90011-6. [5] T. Ogawa, Degenerate Hopf instability in oscillatory reaction-diffusion equations, Discrete Contin. Dyn. Syst., (2007). Proceedings of the 6th AIMS International Conference, suppl., 784-793. [6] M. R. E. Proctor and C. A. Jones, The interaction of two spatially resonant patterns in thermal convection, Part 1. Exact 1:2 resonance, J. Fluid Mech., 188 (1988), 301-335. doi: 10.1017/S0022112088000746. [7] J. Porter and E. Knobloch, New type of complex dynamics in the 1:2 spatial resonance, Physica, 159D (2001), 125-154. doi: 10.1016/S0167-2789(01)00340-2. [8] Y. A. Kuznetsov, "Elements of Applied Bifurcation Theory," Springer, 1997. [9] J. Liu, F. Yi and J. Wei, Multiple bifurcation analysis and spatiotemporal patterns in a 1-D Gierer-Meinhardt model of morphogenesis, IJBC, 20 (2010), 1007-1025. doi: 10.1142/S0218127410026289. [10] I. Shimada and T. Nagashima, A numerical approach to Ergodic problem of dissipative dynamical systems, Prog. Theor. Phys., 61 (1979), 1605-1616. doi: 10.1143/PTP.61.1605. [11] T. R. Smith, J. Moehlis and P. Holmes, Heteroclinic cycles and periodic orbits for the O(2)-equivariant 0:1:2 mode interaction, Physica, 211D (2005), 347-376. doi: 10.1016/j.physd.2005.09.002. [12] Y. Morita and T. Ogawa, Stability and bifurcations of nonconstant solutions to a reaction-diffusion system with conservation mass, Nonlinearity, 23 (2010), 1387-1411. doi: 10.1088/0951-7715/23/6/007. [13] A. M. Turing, The chemical basis of morphogenesis, Phil. Trans. R. Soc. B, 237 (1952), 37-72. [14] L. Yang, M. Dolnik, A. M. Zhabotinsky and I. R. Epstein, Pattern formation arising from interactions between Turing and wave instabilities, J. Chem. Phys., 117 (2002), 7257-7265. [15] A. Vanderbauwhede and G. Iooss, Center manifold theory in infinite dimensions, Dynam. Report. Expositions Dynam. Systems (N.S.), 1, Springer, (1992), 125-163. [16] M. J. Ward and J. Wei, Hopf Bifurcation of spike solutions for the shadow Gierer-Meinhardt model, Europ. J. Appl. Math., 14 (2003), 677-711. doi: 10.1017/S0956792503005278.
 [1] Dmitriy Yu. Volkov. The Hopf -- Hopf bifurcation with 2:1 resonance: Periodic solutions and invariant tori. Conference Publications, 2015, 2015 (special) : 1098-1104. doi: 10.3934/proc.2015.1098 [2] Kazuyuki Yagasaki. Existence of finite time blow-up solutions in a normal form of the subcritical Hopf bifurcation with time-delayed feedback for small initial functions. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2621-2634. doi: 10.3934/dcdsb.2021151 [3] Qigang Yuan, Jingli Ren. Periodic forcing on degenerate Hopf bifurcation. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2857-2877. doi: 10.3934/dcdsb.2020208 [4] Xin Li, Chunyou Sun, Na Zhang. Dynamics for a non-autonomous degenerate parabolic equation in $\mathfrak{D}_{0}^{1}(\Omega, \sigma)$. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 7063-7079. doi: 10.3934/dcds.2016108 [5] John Burke, Edgar Knobloch. Normal form for spatial dynamics in the Swift-Hohenberg equation. Conference Publications, 2007, 2007 (Special) : 170-180. doi: 10.3934/proc.2007.2007.170 [6] Agust Sverrir Egilsson. On embedding the $1:1:2$ resonance space in a Poisson manifold. Electronic Research Announcements, 1995, 1: 48-56. [7] Juan Sánchez, Marta Net, José M. Vega. Amplitude equations close to a triple-(+1) bifurcation point of D4-symmetric periodic orbits in O(2)-equivariant systems. Discrete and Continuous Dynamical Systems - B, 2006, 6 (6) : 1357-1380. doi: 10.3934/dcdsb.2006.6.1357 [8] Reza Mazrooei-Sebdani, Zahra Yousefi. The coupled 1:2 resonance in a symmetric case and parametric amplification model. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3737-3765. doi: 10.3934/dcdsb.2020255 [9] Hongyong Zhao, Daiyong Wu. Point to point traveling wave and periodic traveling wave induced by Hopf bifurcation for a diffusive predator-prey system. Discrete and Continuous Dynamical Systems - S, 2020, 13 (11) : 3271-3284. doi: 10.3934/dcdss.2020129 [10] Zhihua Liu, Hui Tang, Pierre Magal. Hopf bifurcation for a spatially and age structured population dynamics model. Discrete and Continuous Dynamical Systems - B, 2015, 20 (6) : 1735-1757. doi: 10.3934/dcdsb.2015.20.1735 [11] Stephen Pankavich, Nathan Neri, Deborah Shutt. Bistable dynamics and Hopf bifurcation in a refined model of early stage HIV infection. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 2867-2893. doi: 10.3934/dcdsb.2020044 [12] Tiphaine Jézéquel, Patrick Bernard, Eric Lombardi. Homoclinic orbits with many loops near a $0^2 i\omega$ resonant fixed point of Hamiltonian systems. Discrete and Continuous Dynamical Systems, 2016, 36 (6) : 3153-3225. doi: 10.3934/dcds.2016.36.3153 [13] Xavier Perrot, Xavier Carton. Point-vortex interaction in an oscillatory deformation field: Hamiltonian dynamics, harmonic resonance and transition to chaos. Discrete and Continuous Dynamical Systems - B, 2009, 11 (4) : 971-995. doi: 10.3934/dcdsb.2009.11.971 [14] Ryan T. Botts, Ale Jan Homburg, Todd R. Young. The Hopf bifurcation with bounded noise. Discrete and Continuous Dynamical Systems, 2012, 32 (8) : 2997-3007. doi: 10.3934/dcds.2012.32.2997 [15] Matteo Franca, Russell Johnson, Victor Muñoz-Villarragut. On the nonautonomous Hopf bifurcation problem. Discrete and Continuous Dynamical Systems - S, 2016, 9 (4) : 1119-1148. doi: 10.3934/dcdss.2016045 [16] John Guckenheimer, Hinke M. Osinga. The singular limit of a Hopf bifurcation. Discrete and Continuous Dynamical Systems, 2012, 32 (8) : 2805-2823. doi: 10.3934/dcds.2012.32.2805 [17] Vivi Rottschäfer. Multi-bump patterns by a normal form approach. Discrete and Continuous Dynamical Systems - B, 2001, 1 (3) : 363-386. doi: 10.3934/dcdsb.2001.1.363 [18] Todor Mitev, Georgi Popov. Gevrey normal form and effective stability of Lagrangian tori. Discrete and Continuous Dynamical Systems - S, 2010, 3 (4) : 643-666. doi: 10.3934/dcdss.2010.3.643 [19] Dario Bambusi, A. Carati, A. Ponno. The nonlinear Schrödinger equation as a resonant normal form. Discrete and Continuous Dynamical Systems - B, 2002, 2 (1) : 109-128. doi: 10.3934/dcdsb.2002.2.109 [20] Weipeng Hu, Zichen Deng, Yuyue Qin. Multi-symplectic method to simulate Soliton resonance of (2+1)-dimensional Boussinesq equation. Journal of Geometric Mechanics, 2013, 5 (3) : 295-318. doi: 10.3934/jgm.2013.5.295

2021 Impact Factor: 1.41