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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 |
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. |
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