American Institute of Mathematical Sciences

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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

Toshiyuki Ogawa 1, and  Takashi Okuda 2,

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.
Keywords: triply degenerate point, Hopf bifurcation, 0:1:2 resonance, chaotic dynamics., normal form.
Mathematics Subject Classification: Primary: 35B10, 37G05; Secondary: 37D4.
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), 257-282. doi: 10.1016/0167-2789(88)90032-2.  Google Scholar

[2]

J. Carr, "Applications of Center Manifold Theory," Springer, 1981.  Google Scholar

[3]

J. Kaplan and J. Yorke, "Chaotic Behavior of Multi-dimensional Differential Equations and The Approximation of Fixed Points,", Lecture Notes in Mathematics, 730 ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

Y. A. Kuznetsov, "Elements of Applied Bifurcation Theory," Springer, 1997. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

A. M. Turing, The chemical basis of morphogenesis, Phil. Trans. R. Soc. B, 237 (1952), 37-72. Google Scholar

[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. Google Scholar

[15]

A. Vanderbauwhede and G. Iooss, Center manifold theory in infinite dimensions, Dynam. Report. Expositions Dynam. Systems (N.S.), 1, Springer, (1992), 125-163.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

J. Carr, "Applications of Center Manifold Theory," Springer, 1981.  Google Scholar

[3]

J. Kaplan and J. Yorke, "Chaotic Behavior of Multi-dimensional Differential Equations and The Approximation of Fixed Points,", Lecture Notes in Mathematics, 730 ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

Y. A. Kuznetsov, "Elements of Applied Bifurcation Theory," Springer, 1997. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

A. M. Turing, The chemical basis of morphogenesis, Phil. Trans. R. Soc. B, 237 (1952), 37-72. Google Scholar

[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. Google Scholar

[15]

A. Vanderbauwhede and G. Iooss, Center manifold theory in infinite dimensions, Dynam. Report. Expositions Dynam. Systems (N.S.), 1, Springer, (1992), 125-163.  Google Scholar

[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.  Google Scholar

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