# American Institute of Mathematical Sciences

October  2012, 17(7): 2523-2543. doi: 10.3934/dcdsb.2012.17.2523

## Interface oscillations in reaction-diffusion 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

Received  October 2011 Revised  May 2012 Published  July 2012

We consider a reaction-diffusion system of the form $\left\{ \begin{array} \ u_{t}=\varepsilon^{2}u_{xx}+f(u,w)\\ \tau w_{t}=Dw_{xx}+g(u,w) \end{array} \right.$ with Neumann boundary conditions on a finite interval. Under certain generic conditions on the nonlinearities $f,g$ and in the singular limit $\varepsilon\ll1$ such a system may admit a steady state solution where $u$ has sharp interfaces. It is also known that such interfaces may be destabilized due to a Hopf bifurcation [Y. Nishiura and M. Mimura. SIAM J.Appl. Math., 49:481--514, 1989], as $\tau$ is increased beyond a certain threshold $\tau_{h}$. In this paper, we study what happens for $\tau>\tau _{h},$ or even $\tau\rightarrow\infty,$ for a solution that consists of either one or two interfaces. Under the additional assumption $D\gg1,$ using singular perturbation theory, we determine the existence of another threshold $\tau _{c}>\tau_{h}$ (where $\tau_{c}$ is allowed to be infinite) such that if $\tau_{h}<\tau<\tau_{c}$ then the system admits a solution consisting of periodically oscillating interfaces. On the other hand if $\tau>\tau_{c},$ the extent of the oscillation eventually exceeds the spatial domain size, even though very long transient dynamics can preceed this occurence. We make use of recently developed numerical software (that employs adaptive error control in space and time) to accurately compute an approximate solution. Excellent agreement with the analytical theory is observed.
Citation: Rebecca McKay, Theodore Kolokolnikov, Paul Muir. Interface oscillations in reaction-diffusion systems above the Hopf bifurcation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2523-2543. doi: 10.3934/dcdsb.2012.17.2523
##### References:
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[7] S.-I. Ei, The motion of weakly interacting pulses in reaction-diffusion systems,, J. Dyn. Diff. Eq., 14 (2002), 85. [8] S.-I. Ei, H. Ikeda and T. Kawana, Dynamics of front solutions ina specific reaction-diffusion 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 reaction-diffusion system,, Phys. Rev. E (3), 53 (1996), 3933. [10] S. V. Gurevich, Sh. Amiranashvili and H.-G. Purwins, Breathing dissipative solitons in three component reaction-diffusion system,, Physical Review E (3), 74 (2006). [11] A. Hagberg and E. Meron, Pattern formation nongradient reaction-diffusion systems: The effect of front bifurcations,, Nonlinearity, 7 (1994), 805. [12] H. Hardway and Y.-X. Li, Stationary and oscillatory fronts in a two-component genetic regulatory network model,, Physica D, 239 (2010), 1650. [13] P. van Heijster, A. Doelman and T. J. Kaper, Pulse dynamics in a three-component 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 reaction-diffusion 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 reaction-diffusion systems,, J. Dyn. Diff. Eqns., 12 (2000), 117. [17] A. Kaminaga, V. K. Vanag and I. R. Epstein, A reaction-diffusion 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 Self-Organization 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 reaction-diffusion systems,, Progress of Theoretical Physics, 63 (1980), 106. [22] T. Kolokolnikov and M. Tlidi, Spot deformation and replication in the two-dimensional Belousov-Zhabotinski reaction-diffusion system,, Physical Review Letters, 98 (2007). [23] T. Kolokolnikov, T. Erneux and J. Wei, Mesa-type patterns in the one-dimensional Brusselator and their stability,, Physica D, 214 (2006), 63. [24] T. Kolokolnikov, M. Ward and J. Wei, Self-replication of mesa patterns in reaction-diffusion systems,, Physica D, 236 (2007), 104. [25] T. Kolokolnikov, M. J. Ward and J. Wei, Slow transitional instabilities of spike patterns in the one-dimensional Gray-Scott 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 chlorite-iodide-malonic acid-starch 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 reaction-diffusion systems,, Discrete and Continuous Dynamical Systems - B, (2011). [31] C. B. Muratov and V. Osipov, Scenarios of domain pattern formation in a reaction-diffusion system,, Phys. Rev. E, 54 (1996), 4860. [32] C. B. Muratov, Synchronization, chaos, and breakdown of collective domain oscillations in reaction-diffusion systems,, Phys. Rev. E, 55 (1997), 1463. [33] C. Muratov and V. V. Osipov, General theory of instabilities for patterns with sharp interfaces in reaction-diffusion 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 reaction-diffision system,, Japan J. Indust. Appl. Math., 27 (2010), 295. [36] W.-M. Ni and M. Tang, Turing patterns in the Lengyel-Epstein system for the CIMA reaction,, Trans. AMS, 357 (2005), 3953. [37] Y. Nishiura and H. Fujii, Stability of singularly perturbed solutions to systems of reaction-diffusion equations,, SIAM J. Math. Anal., 18 (1987), 1726. [38] Y. Nishiura and M. Mimura, Layer oscillations inreaction-diffusion 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 three-species 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 inreaction-diffusion systems,, Chaos, 17 (2007). [43] V. Volpert and S. Petrovskii, Reaction-diffusion 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 1-D parabolic PDEs,, J. Comput. Appl. Math., 169 (2004), 127. [45] R. Wang, P. Keast and P. Muir, A high-order global spatially adaptive collocation method for 1-D parabolic PDEs,, Applied Numerical Mathematics, 50 (2004), 239. [46] R. Wang, P. Keast and P. Muir, BACOL: B-Spline Adaptive COLlocation Software for 1-D 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 one-dimensional Gierer-Meinhardt 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, Self-organised spatialpatterns and chaos in a ratio-dependent predator-prey 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 Initial-Value Problems in Differential-Algebraic Equations,", Classics in Applied Mathematics, (1996). [4] W. Chen and M. J. Ward, Oscillatory instabilities anddynamics of multi-spike patterns for the one-dimensional Gray-Scott 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 two-dimensional Gray-Scott 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 reaction-diffusion systems,, J. Dyn. Diff. Eq., 14 (2002), 85. [8] S.-I. Ei, H. Ikeda and T. Kawana, Dynamics of front solutions ina specific reaction-diffusion 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 reaction-diffusion system,, Phys. Rev. E (3), 53 (1996), 3933. [10] S. V. Gurevich, Sh. Amiranashvili and H.-G. Purwins, Breathing dissipative solitons in three component reaction-diffusion system,, Physical Review E (3), 74 (2006). [11] A. Hagberg and E. Meron, Pattern formation nongradient reaction-diffusion systems: The effect of front bifurcations,, Nonlinearity, 7 (1994), 805. [12] H. Hardway and Y.-X. Li, Stationary and oscillatory fronts in a two-component genetic regulatory network model,, Physica D, 239 (2010), 1650. [13] P. van Heijster, A. Doelman and T. J. Kaper, Pulse dynamics in a three-component 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 reaction-diffusion 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 reaction-diffusion systems,, J. Dyn. Diff. Eqns., 12 (2000), 117. [17] A. Kaminaga, V. K. Vanag and I. R. Epstein, A reaction-diffusion 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 Self-Organization 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 reaction-diffusion systems,, Progress of Theoretical Physics, 63 (1980), 106. [22] T. Kolokolnikov and M. Tlidi, Spot deformation and replication in the two-dimensional Belousov-Zhabotinski reaction-diffusion system,, Physical Review Letters, 98 (2007). [23] T. Kolokolnikov, T. Erneux and J. Wei, Mesa-type patterns in the one-dimensional Brusselator and their stability,, Physica D, 214 (2006), 63. [24] T. Kolokolnikov, M. Ward and J. Wei, Self-replication of mesa patterns in reaction-diffusion systems,, Physica D, 236 (2007), 104. [25] T. Kolokolnikov, M. J. Ward and J. Wei, Slow transitional instabilities of spike patterns in the one-dimensional Gray-Scott 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 chlorite-iodide-malonic acid-starch 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 reaction-diffusion systems,, Discrete and Continuous Dynamical Systems - B, (2011). [31] C. B. Muratov and V. Osipov, Scenarios of domain pattern formation in a reaction-diffusion system,, Phys. Rev. E, 54 (1996), 4860. [32] C. B. Muratov, Synchronization, chaos, and breakdown of collective domain oscillations in reaction-diffusion systems,, Phys. Rev. E, 55 (1997), 1463. [33] C. Muratov and V. V. Osipov, General theory of instabilities for patterns with sharp interfaces in reaction-diffusion 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 reaction-diffision system,, Japan J. Indust. Appl. Math., 27 (2010), 295. [36] W.-M. Ni and M. Tang, Turing patterns in the Lengyel-Epstein system for the CIMA reaction,, Trans. AMS, 357 (2005), 3953. [37] Y. Nishiura and H. Fujii, Stability of singularly perturbed solutions to systems of reaction-diffusion equations,, SIAM J. Math. Anal., 18 (1987), 1726. [38] Y. Nishiura and M. Mimura, Layer oscillations inreaction-diffusion 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 three-species 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 inreaction-diffusion systems,, Chaos, 17 (2007). [43] V. Volpert and S. Petrovskii, Reaction-diffusion 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 1-D parabolic PDEs,, J. Comput. Appl. Math., 169 (2004), 127. [45] R. Wang, P. Keast and P. Muir, A high-order global spatially adaptive collocation method for 1-D parabolic PDEs,, Applied Numerical Mathematics, 50 (2004), 239. [46] R. Wang, P. Keast and P. Muir, BACOL: B-Spline Adaptive COLlocation Software for 1-D 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 one-dimensional Gierer-Meinhardt 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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