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Interface oscillations in reactiondiffusion systems above the Hopf bifurcation
1.  Department of Mathematics and Statistics, Dalhousie University, Halifax, N.S. B3H 3J5 
2.  Department of Mathematics, Dalhousie University, Halifax, N.S. B3H 3J5 
3.  Department of Mathematics and Computing Science, Saint Mary's University, Canada 
References:
[1] 
M. Banerjee and S. Petrovski, Selforganised spatialpatterns and chaos in a ratiodependent predatorprey system,, Theor. Ecol., 4 (2011), 37. 
[2] 
C. M. Bender and S. A. Orszag, "Advanced Mathematical Methods for Scientists and Engineers. I. Asymptotic Methods and Perturbation Theory,", Reprint of the 1978 original, (1978). 
[3] 
K. E. Brenan, S. L.Campbell and L. R. Petzold, "Numerical Solution of InitialValue Problems in DifferentialAlgebraic Equations,", Classics in Applied Mathematics, (1996). 
[4] 
W. Chen and M. J. Ward, Oscillatory instabilities anddynamics of multispike patterns for the onedimensional GrayScott model,, European Journal of Applied Mathematics, 20 (2009), 187. 
[5] 
W. Chen and M. J. Ward, The stability and dynamics oflocalized spot patterns in the twodimensional GrayScott model,, SIAM J. Appl. Dynam. Systems, 10 (2011), 582. 
[6] 
A. Doelman, B. Sandstede, A. Scheel and G. Schneider, The dynamics of modulated wave trains,, Mem. Amer. Math. Soc., 199 (2009). 
[7] 
S.I. Ei, The motion of weakly interacting pulses in reactiondiffusion systems,, J. Dyn. Diff. Eq., 14 (2002), 85. 
[8] 
S.I. Ei, H. Ikeda and T. Kawana, Dynamics of front solutions ina specific reactiondiffusion system in one dimension,, Japan J. Indust. Appl. Math., 25 (2008), 117. 
[9] 
R. E. Goldstein, D. J. Muraki and D. M. Petrich, Interface proliferation and the growth of labyrinths in a reactiondiffusion system,, Phys. Rev. E (3), 53 (1996), 3933. 
[10] 
S. V. Gurevich, Sh. Amiranashvili and H.G. Purwins, Breathing dissipative solitons in three component reactiondiffusion system,, Physical Review E (3), 74 (2006). 
[11] 
A. Hagberg and E. Meron, Pattern formation nongradient reactiondiffusion systems: The effect of front bifurcations,, Nonlinearity, 7 (1994), 805. 
[12] 
H. Hardway and Y.X. Li, Stationary and oscillatory fronts in a twocomponent genetic regulatory network model,, Physica D, 239 (2010), 1650. 
[13] 
P. van Heijster, A. Doelman and T. J. Kaper, Pulse dynamics in a threecomponent system: Stability and bifurcations,, Physica D, 237 (2008), 3335. 
[14] 
D. Haim, G. Li, Q. Ouyang, W. D. McCormick, H. L. Swinney, A. Hagberg and E. Meron, Breathing spots in a reactiondiffusion system,, Phys. Rev. Lett., 77 (1996), 190. 
[15] 
T. Ikeda and Y. Nishiura, Pattern selection for two breathers,, SIAM J. Appl. Math., 54 (1994), 195. 
[16] 
H. Ikeda and T. Ikeda, Bifurcaiton phenomena from standingpulse solutions in some reactiondiffusion systems,, J. Dyn. Diff. Eqns., 12 (2000), 117. 
[17] 
A. Kaminaga, V. K. Vanag and I. R. Epstein, A reactiondiffusion memory device,, Angewandte Chemie, 45 (2006). 
[18] 
R. Kapral and K. Showalter, eds., "Chemical Waves and Patterns,", Kluwer, (1995). 
[19] 
B. S. Kerner and V. V. Osipov, "Autosolitons: A New Approach to Problems of SelfOrganization and Turbulence,", Fundamental Theories of Physics, 61 (1994). 
[20] 
A. J. Koch and H. Meinhardt, Biological pattern formation: From basic mechanisms to complex structures,, Rev. Modern Physics, 66 (1994), 1481. 
[21] 
S. Koga and Y. Kuramoto, Localized patterns in reactiondiffusion systems,, Progress of Theoretical Physics, 63 (1980), 106. 
[22] 
T. Kolokolnikov and M. Tlidi, Spot deformation and replication in the twodimensional BelousovZhabotinski reactiondiffusion system,, Physical Review Letters, 98 (2007). 
[23] 
T. Kolokolnikov, T. Erneux and J. Wei, Mesatype patterns in the onedimensional Brusselator and their stability,, Physica D, 214 (2006), 63. 
[24] 
T. Kolokolnikov, M. Ward and J. Wei, Selfreplication of mesa patterns in reactiondiffusion systems,, Physica D, 236 (2007), 104. 
[25] 
T. Kolokolnikov, M. J. Ward and J. Wei, Slow transitional instabilities of spike patterns in the onedimensional GrayScott model,, Interfaces and Free Boundaries, 8 (2006), 185. 
[26] 
Y. Kuramoto, "Chemical Oscillations, Waves, and Turbulence,", Springer Series in Synergetics, 19 (1984). 
[27] 
I. Lengyel and I. R. Epstein, Modeling of Turing structures in the chloriteiodidemalonic acidstarch reaction system,, Science, 251 (1991), 650. 
[28] 
Y.X. Li, Tango waves in a bidomain model of fertilization calcium waves,, Physica D, 186 (2003), 27. 
[29] 
H. Meinhardt, "Models of Biological Pattern Formation,", Academic Press, (1982). 
[30] 
R. C. McKay and T. Kolokolnikov, Instability thresholds and dynamics of mesa patterns in reactiondiffusion systems,, Discrete and Continuous Dynamical Systems  B, (2011). 
[31] 
C. B. Muratov and V. Osipov, Scenarios of domain pattern formation in a reactiondiffusion system,, Phys. Rev. E, 54 (1996), 4860. 
[32] 
C. B. Muratov, Synchronization, chaos, and breakdown of collective domain oscillations in reactiondiffusion systems,, Phys. Rev. E, 55 (1997), 1463. 
[33] 
C. Muratov and V. V. Osipov, General theory of instabilities for patterns with sharp interfaces in reactiondiffusion systems,, Phys. Rev. E (3), 53 (1996), 3101. 
[34] 
J. D. Murray, "Mathematical Biology,", Biomathematics, 19 (1989). 
[35] 
M. Nagayama, K.I. Ueda and M. Yadome, Numerical approachto transient dynamics of oscillatory pulses in a bistable reactiondiffision system,, Japan J. Indust. Appl. Math., 27 (2010), 295. 
[36] 
W.M. Ni and M. Tang, Turing patterns in the LengyelEpstein system for the CIMA reaction,, Trans. AMS, 357 (2005), 3953. 
[37] 
Y. Nishiura and H. Fujii, Stability of singularly perturbed solutions to systems of reactiondiffusion equations,, SIAM J. Math. Anal., 18 (1987), 1726. 
[38] 
Y. Nishiura and M. Mimura, Layer oscillations inreactiondiffusion systems,, SIAM J. Appl. Math., 49 (1989), 481. 
[39] 
J. E. Pearson, Complex patterns in a simple system,, Science, 261 (1993), 189. 
[40] 
M. Suzuki, T. Ohta, M. Mimura and H. Sakaguchi, Breathing and wiggling motions in threespecies laterally inhibitory systems,, Physical Review E, 52 (1995), 3645. 
[41] 
M. Taki, M. Tlidi and T. Kolokolnikov, eds., Dissipative localized structures in extended systems,, Chaos Focus Issue, 17 (2007). 
[42] 
V. K. Vanag and I. R. Epstein, Localized patterns inreactiondiffusion systems,, Chaos, 17 (2007). 
[43] 
V. Volpert and S. Petrovskii, Reactiondiffusion waves in biology,, Physics of Life Reviews, 6 (2009), 267. doi: 10.1016/j.plrev.2009.10.002. 
[44] 
R. Wang, P. Keast and P. Muir, A comparison of adaptive software for 1D parabolic PDEs,, J. Comput. Appl. Math., 169 (2004), 127. 
[45] 
R. Wang, P. Keast and P. Muir, A highorder global spatially adaptive collocation method for 1D parabolic PDEs,, Applied Numerical Mathematics, 50 (2004), 239. 
[46] 
R. Wang, P. Keast and P. Muir, BACOL: BSpline Adaptive COLlocation Software for 1D Parabolic PDEs,, ACM Transacations on Mathematical Software, 30 (2004), 454. 
[47] 
M. J. Ward and J. Wei, Hopf bifurcation and oscillatory instabilities of spike solutions for the onedimensional GiererMeinhardt model,, J. Nonlinear Science, 13 (2003), 209. 
[48] 
H. Yizhaq and E. Gilad, Localized structures in dry land vegetation: Forms and functions,, Chaos, 17 (2007). 
show all references
References:
[1] 
M. Banerjee and S. Petrovski, Selforganised spatialpatterns and chaos in a ratiodependent predatorprey system,, Theor. Ecol., 4 (2011), 37. 
[2] 
C. M. Bender and S. A. Orszag, "Advanced Mathematical Methods for Scientists and Engineers. I. Asymptotic Methods and Perturbation Theory,", Reprint of the 1978 original, (1978). 
[3] 
K. E. Brenan, S. L.Campbell and L. R. Petzold, "Numerical Solution of InitialValue Problems in DifferentialAlgebraic Equations,", Classics in Applied Mathematics, (1996). 
[4] 
W. Chen and M. J. Ward, Oscillatory instabilities anddynamics of multispike patterns for the onedimensional GrayScott model,, European Journal of Applied Mathematics, 20 (2009), 187. 
[5] 
W. Chen and M. J. Ward, The stability and dynamics oflocalized spot patterns in the twodimensional GrayScott model,, SIAM J. Appl. Dynam. Systems, 10 (2011), 582. 
[6] 
A. Doelman, B. Sandstede, A. Scheel and G. Schneider, The dynamics of modulated wave trains,, Mem. Amer. Math. Soc., 199 (2009). 
[7] 
S.I. Ei, The motion of weakly interacting pulses in reactiondiffusion systems,, J. Dyn. Diff. Eq., 14 (2002), 85. 
[8] 
S.I. Ei, H. Ikeda and T. Kawana, Dynamics of front solutions ina specific reactiondiffusion system in one dimension,, Japan J. Indust. Appl. Math., 25 (2008), 117. 
[9] 
R. E. Goldstein, D. J. Muraki and D. M. Petrich, Interface proliferation and the growth of labyrinths in a reactiondiffusion system,, Phys. Rev. E (3), 53 (1996), 3933. 
[10] 
S. V. Gurevich, Sh. Amiranashvili and H.G. Purwins, Breathing dissipative solitons in three component reactiondiffusion system,, Physical Review E (3), 74 (2006). 
[11] 
A. Hagberg and E. Meron, Pattern formation nongradient reactiondiffusion systems: The effect of front bifurcations,, Nonlinearity, 7 (1994), 805. 
[12] 
H. Hardway and Y.X. Li, Stationary and oscillatory fronts in a twocomponent genetic regulatory network model,, Physica D, 239 (2010), 1650. 
[13] 
P. van Heijster, A. Doelman and T. J. Kaper, Pulse dynamics in a threecomponent system: Stability and bifurcations,, Physica D, 237 (2008), 3335. 
[14] 
D. Haim, G. Li, Q. Ouyang, W. D. McCormick, H. L. Swinney, A. Hagberg and E. Meron, Breathing spots in a reactiondiffusion system,, Phys. Rev. Lett., 77 (1996), 190. 
[15] 
T. Ikeda and Y. Nishiura, Pattern selection for two breathers,, SIAM J. Appl. Math., 54 (1994), 195. 
[16] 
H. Ikeda and T. Ikeda, Bifurcaiton phenomena from standingpulse solutions in some reactiondiffusion systems,, J. Dyn. Diff. Eqns., 12 (2000), 117. 
[17] 
A. Kaminaga, V. K. Vanag and I. R. Epstein, A reactiondiffusion memory device,, Angewandte Chemie, 45 (2006). 
[18] 
R. Kapral and K. Showalter, eds., "Chemical Waves and Patterns,", Kluwer, (1995). 
[19] 
B. S. Kerner and V. V. Osipov, "Autosolitons: A New Approach to Problems of SelfOrganization and Turbulence,", Fundamental Theories of Physics, 61 (1994). 
[20] 
A. J. Koch and H. Meinhardt, Biological pattern formation: From basic mechanisms to complex structures,, Rev. Modern Physics, 66 (1994), 1481. 
[21] 
S. Koga and Y. Kuramoto, Localized patterns in reactiondiffusion systems,, Progress of Theoretical Physics, 63 (1980), 106. 
[22] 
T. Kolokolnikov and M. Tlidi, Spot deformation and replication in the twodimensional BelousovZhabotinski reactiondiffusion system,, Physical Review Letters, 98 (2007). 
[23] 
T. Kolokolnikov, T. Erneux and J. Wei, Mesatype patterns in the onedimensional Brusselator and their stability,, Physica D, 214 (2006), 63. 
[24] 
T. Kolokolnikov, M. Ward and J. Wei, Selfreplication of mesa patterns in reactiondiffusion systems,, Physica D, 236 (2007), 104. 
[25] 
T. Kolokolnikov, M. J. Ward and J. Wei, Slow transitional instabilities of spike patterns in the onedimensional GrayScott model,, Interfaces and Free Boundaries, 8 (2006), 185. 
[26] 
Y. Kuramoto, "Chemical Oscillations, Waves, and Turbulence,", Springer Series in Synergetics, 19 (1984). 
[27] 
I. Lengyel and I. R. Epstein, Modeling of Turing structures in the chloriteiodidemalonic acidstarch reaction system,, Science, 251 (1991), 650. 
[28] 
Y.X. Li, Tango waves in a bidomain model of fertilization calcium waves,, Physica D, 186 (2003), 27. 
[29] 
H. Meinhardt, "Models of Biological Pattern Formation,", Academic Press, (1982). 
[30] 
R. C. McKay and T. Kolokolnikov, Instability thresholds and dynamics of mesa patterns in reactiondiffusion systems,, Discrete and Continuous Dynamical Systems  B, (2011). 
[31] 
C. B. Muratov and V. Osipov, Scenarios of domain pattern formation in a reactiondiffusion system,, Phys. Rev. E, 54 (1996), 4860. 
[32] 
C. B. Muratov, Synchronization, chaos, and breakdown of collective domain oscillations in reactiondiffusion systems,, Phys. Rev. E, 55 (1997), 1463. 
[33] 
C. Muratov and V. V. Osipov, General theory of instabilities for patterns with sharp interfaces in reactiondiffusion systems,, Phys. Rev. E (3), 53 (1996), 3101. 
[34] 
J. D. Murray, "Mathematical Biology,", Biomathematics, 19 (1989). 
[35] 
M. Nagayama, K.I. Ueda and M. Yadome, Numerical approachto transient dynamics of oscillatory pulses in a bistable reactiondiffision system,, Japan J. Indust. Appl. Math., 27 (2010), 295. 
[36] 
W.M. Ni and M. Tang, Turing patterns in the LengyelEpstein system for the CIMA reaction,, Trans. AMS, 357 (2005), 3953. 
[37] 
Y. Nishiura and H. Fujii, Stability of singularly perturbed solutions to systems of reactiondiffusion equations,, SIAM J. Math. Anal., 18 (1987), 1726. 
[38] 
Y. Nishiura and M. Mimura, Layer oscillations inreactiondiffusion systems,, SIAM J. Appl. Math., 49 (1989), 481. 
[39] 
J. E. Pearson, Complex patterns in a simple system,, Science, 261 (1993), 189. 
[40] 
M. Suzuki, T. Ohta, M. Mimura and H. Sakaguchi, Breathing and wiggling motions in threespecies laterally inhibitory systems,, Physical Review E, 52 (1995), 3645. 
[41] 
M. Taki, M. Tlidi and T. Kolokolnikov, eds., Dissipative localized structures in extended systems,, Chaos Focus Issue, 17 (2007). 
[42] 
V. K. Vanag and I. R. Epstein, Localized patterns inreactiondiffusion systems,, Chaos, 17 (2007). 
[43] 
V. Volpert and S. Petrovskii, Reactiondiffusion waves in biology,, Physics of Life Reviews, 6 (2009), 267. doi: 10.1016/j.plrev.2009.10.002. 
[44] 
R. Wang, P. Keast and P. Muir, A comparison of adaptive software for 1D parabolic PDEs,, J. Comput. Appl. Math., 169 (2004), 127. 
[45] 
R. Wang, P. Keast and P. Muir, A highorder global spatially adaptive collocation method for 1D parabolic PDEs,, Applied Numerical Mathematics, 50 (2004), 239. 
[46] 
R. Wang, P. Keast and P. Muir, BACOL: BSpline Adaptive COLlocation Software for 1D Parabolic PDEs,, ACM Transacations on Mathematical Software, 30 (2004), 454. 
[47] 
M. J. Ward and J. Wei, Hopf bifurcation and oscillatory instabilities of spike solutions for the onedimensional GiererMeinhardt model,, J. Nonlinear Science, 13 (2003), 209. 
[48] 
H. Yizhaq and E. Gilad, Localized structures in dry land vegetation: Forms and functions,, Chaos, 17 (2007). 
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