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 & 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. 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 ().

[4]

P. Frederickson, J. Kaplan, E. Yorke and J. Yorke, The Lyapunov dimension of strange attractors,, J. DIff. Eqs., 49 (1983), 185. doi: 10.1016/0022-0396(83)90011-6.

[5]

T. Ogawa, Degenerate Hopf instability in oscillatory reaction-diffusion equations,, Discrete Contin. Dyn. Syst., (2007), 784.

[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, 188 (1988), 301. 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. 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. 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. 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. 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. 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.

[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.

[15]

A. Vanderbauwhede and G. Iooss, Center manifold theory in infinite dimensions,, Dynam. Report. Expositions Dynam. Systems (N.S.), 1 (1992), 125.

[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. 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. 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 ().

[4]

P. Frederickson, J. Kaplan, E. Yorke and J. Yorke, The Lyapunov dimension of strange attractors,, J. DIff. Eqs., 49 (1983), 185. doi: 10.1016/0022-0396(83)90011-6.

[5]

T. Ogawa, Degenerate Hopf instability in oscillatory reaction-diffusion equations,, Discrete Contin. Dyn. Syst., (2007), 784.

[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, 188 (1988), 301. 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. 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. 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. 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. 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. 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.

[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.

[15]

A. Vanderbauwhede and G. Iooss, Center manifold theory in infinite dimensions,, Dynam. Report. Expositions Dynam. Systems (N.S.), 1 (1992), 125.

[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. 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]

Xin Li, Chunyou Sun, Na Zhang. Dynamics for a non-autonomous degenerate parabolic equation in $\mathfrak{D}_{0}^{1}(\Omega, \sigma)$. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 7063-7079. doi: 10.3934/dcds.2016108

[3]

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

[4]

Agust Sverrir Egilsson. On embedding the $1:1:2$ resonance space in a Poisson manifold. Electronic Research Announcements, 1995, 1: 48-56.

[5]

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 & Continuous Dynamical Systems - B, 2006, 6 (6) : 1357-1380. doi: 10.3934/dcdsb.2006.6.1357

[6]

Zhihua Liu, Hui Tang, Pierre Magal. Hopf bifurcation for a spatially and age structured population dynamics model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1735-1757. doi: 10.3934/dcdsb.2015.20.1735

[7]

Tiphaine Jézéquel, Patrick Bernard, Eric Lombardi. Homoclinic orbits with many loops near a $0^2 i\omega$ resonant fixed point of Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3153-3225. doi: 10.3934/dcds.2016.36.3153

[8]

Xavier Perrot, Xavier Carton. Point-vortex interaction in an oscillatory deformation field: Hamiltonian dynamics, harmonic resonance and transition to chaos. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 971-995. doi: 10.3934/dcdsb.2009.11.971

[9]

Ryan T. Botts, Ale Jan Homburg, Todd R. Young. The Hopf bifurcation with bounded noise. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2997-3007. doi: 10.3934/dcds.2012.32.2997

[10]

Matteo Franca, Russell Johnson, Victor Muñoz-Villarragut. On the nonautonomous Hopf bifurcation problem. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1119-1148. doi: 10.3934/dcdss.2016045

[11]

John Guckenheimer, Hinke M. Osinga. The singular limit of a Hopf bifurcation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2805-2823. doi: 10.3934/dcds.2012.32.2805

[12]

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

[13]

Vivi Rottschäfer. Multi-bump patterns by a normal form approach. Discrete & Continuous Dynamical Systems - B, 2001, 1 (3) : 363-386. doi: 10.3934/dcdsb.2001.1.363

[14]

Todor Mitev, Georgi Popov. Gevrey normal form and effective stability of Lagrangian tori. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 643-666. doi: 10.3934/dcdss.2010.3.643

[15]

Dario Bambusi, A. Carati, A. Ponno. The nonlinear Schrödinger equation as a resonant normal form. Discrete & Continuous Dynamical Systems - B, 2002, 2 (1) : 109-128. doi: 10.3934/dcdsb.2002.2.109

[16]

Emile Franc Doungmo Goufo, Melusi Khumalo, Patrick M. Tchepmo Djomegni. Perturbations of Hindmarsh-Rose neuron dynamics by fractional operators: Bifurcation, firing and chaotic bursts. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 663-682. doi: 10.3934/dcdss.2020036

[17]

David Iglesias-Ponte, Juan Carlos Marrero, David Martín de Diego, Edith Padrón. Discrete dynamics in implicit form. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1117-1135. doi: 10.3934/dcds.2013.33.1117

[18]

Zvia Agur, L. Arakelyan, P. Daugulis, Y. Ginosar. Hopf point analysis for angiogenesis models. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 29-38. doi: 10.3934/dcdsb.2004.4.29

[19]

Matt Coles, Stephen Gustafson. A degenerate edge bifurcation in the 1D linearized nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 2991-3009. doi: 10.3934/dcds.2016.36.2991

[20]

Hooton Edward, Balanov Zalman, Krawcewicz Wieslaw, Rachinskii Dmitrii. Sliding Hopf bifurcation in interval systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3545-3566. doi: 10.3934/dcds.2017152

2017 Impact Factor: 1.187

Metrics

  • PDF downloads (4)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]