February  2012, 5(1): 61-92. doi: 10.3934/dcdss.2012.5.61

Hopf dances near the tips of Busse balloons

1. 

Mathematisch Instituut, Universiteit Leiden, P.O. Box 9512, 2300 RA Leiden, Netherlands

2. 

Centrum Wiskunde en Informatica (CWI), Science Park 123, 1098 XG Amsterdam, Netherlands

3. 

Korteweg-de Vries Instituut, Science Park 904, 1098 XH Amsterdam, Netherlands

Received  July 2009 Revised  March 2010 Published  February 2011

In this paper we introduce a novel generic destabilization mechanism for (reversible) spatially periodic patterns in reaction-diffusion equations in one spatial dimension. This Hopf dance mechanism occurs for long wavelength patterns near the homoclinic tip of the associated Busse balloon ($=$ the region in (wave number, parameter space) for which stable periodic patterns exist). It shows that the boundary of the Busse balloon locally has a fine-structure of two intertwining 'dancing' (or 'snaking') Hopf destabilization curves (or manifolds) that limit on the Hopf bifurcation value of the associated homoclinic limit pulse and that have infinitely many, accumulating, intersections. The Hopf dance is first recovered by a detailed numerical analysis of the full Busse balloon in an explicit Gray-Scott model. The structure, and its generic nature, is confirmed by a rigorous analysis of singular long wave length patterns in a normal form model for pulse-type solutions in two component, singularly perturbed, reaction-diffusion equations.
Citation: Arjen Doelman, Jens D. M. Rademacher, Sjors van der Stelt. Hopf dances near the tips of Busse balloons. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 61-92. doi: 10.3934/dcdss.2012.5.61
References:
[1]

I. Aranson and L. Kramer, The world of the Ginzburg-Landau equation,, Rev. Modern Phys., 74 (2002), 99. doi: 10.1103/RevModPhys.74.99. Google Scholar

[2]

F. H. Busse, Nonlinear properties of thermal convection,, Rep. Prog. Phys., 41 (1978), 1929. doi: 10.1088/0034-4885/41/12/003. Google Scholar

[3]

R. L. Devaney, Reversible diffeomorphisms and flows,, Trans. Am. Math. Soc., 218 (1976), 89. doi: 10.1090/S0002-9947-1976-0402815-3. Google Scholar

[4]

E. J. Doedel. AUTO-07P:, Continuation and bifurcation software for ordinary differential equations,, \url{http://cmvl.cs.concordia.ca/auto}., (). Google Scholar

[5]

A. Doelman, An explicit theory for pulses in two component singularly perturbed reaction-diffusion equations,, in preparation., (). Google Scholar

[6]

A. Doelman and W. Eckhaus, Periodic and quasi-periodic solutions of degenerate modulation equations,, Physica D, 53 (1991), 249. doi: 10.1016/0167-2789(91)90065-H. Google Scholar

[7]

A. Doelman, R. A. Gardner and T. J. Kaper, Stability analysis of singular patterns in the 1-D Gray-Scott model: A matched asymptotics approach,, Physica D, 122 (1998), 1. doi: 10.1016/S0167-2789(98)00180-8. Google Scholar

[8]

A. Doelman, R. A. Gardner and T. J. Kaper, Large stable pulse solutions in reaction-diffusion equations,, Ind. Univ. Math. J., 50 (2001), 443. doi: 10.1512/iumj.2001.50.1873. Google Scholar

[9]

A. Doelman, R. A. Gardner and T. J. Kaper, A stability index analysis of 1-D patterns of the Gray-Scott model,, Memoirs AMS, 155 (2002). Google Scholar

[10]

A. Doelman, T. J Kaper and H. van der Ploeg, Spatially periodic and aperiodic multi-pulse patterns in the one-dimensional Gierer-Meinhardt equation,, Meth. Appl. An., 8 (2001), 387. Google Scholar

[11]

A. Doelman, T. J. Kaper and P. Zegeling, Pattern formation in the one-dimensional Gray-Scott model,, Nonlinearity, 10 (1997), 523. doi: 10.1088/0951-7715/10/2/013. Google Scholar

[12]

A. Doelman and H. van der Ploeg, Homoclinic stripe patterns,, SIAM J. Appl. Dyn. Syst., 1 (2002), 65. doi: 10.1137/S1111111101392831. Google Scholar

[13]

A. Doelman, B. Sandstede, A. Scheel and G. Schneider, The dynamics of modulated wave trains,, Memoirs of the AMS, 199 (2009). Google Scholar

[14]

W. Eckhaus and G. Iooss, Strong selection or rejection of spatially periodic patterns in degenerate bifurcations,, Physica D, 39 (1989), 124. doi: 10.1016/0167-2789(89)90043-2. Google Scholar

[15]

E. G. Eszter, "Evans Function Analysis of the Stability of Periodic Traveling Wave Solutions of the Fitzhugh-Nagumo System,", PhD thesis, (1999). Google Scholar

[16]

R. A. Gardner, On the structure of the spectra of periodic travelling waves,, J. Math. Pure Appl., 72 (1993), 415. Google Scholar

[17]

R. A. Gardner, Spectral analysis of long wavelength periodic waves and applications,, J. Reine Angew. Math., 491 (1997), 149. doi: 10.1515/crll.1997.491.149. Google Scholar

[18]

D. Iron and M. J. Ward, The dynamics of multi-spike solutions to the one-dimensional Gierer-Meinhardt model,, SIAM J. Appl. Math., 62 (2002), 1924. doi: 10.1137/S0036139901393676. Google Scholar

[19]

D. Iron, M. J. Ward and J. Wei, The stability of spike solutions to the one-dimensional Gierer-Meinhardt model,, Physica D, 150 (2001), 25. doi: i:10.1016/S0167-2789(00)00206-2. Google Scholar

[20]

T. Kolokolnikov, M. Ward and J. Wei, The existence and stability of spike equilibria in the one-dimensional Gray-Scott model: The pulse-splitting regime,, Physica D, 202 (2005), 258. doi: 10.1016/j.physd.2005.02.009. Google Scholar

[21]

T. Kolokolnikov, M. J. Ward and J. Wei, The existence and stability of spike equilibria in the one-dimensional Gray-Scott model: The low feed-rate regime,, Stud. Appl. Math., 115 (2005), 21. doi: i:10.1111/j.1467-9590.2005.01554. Google Scholar

[22]

B. J. Matkowsky and V. A. Volpert, Stability of plane wave solutions of complex Ginzburg-Landau equations,, Quart. Appl. Math., 51 (1993), 265. Google Scholar

[23]

A. Mielke, The Ginzburg-Landau equation in its role as modulation equation,, in, (2002), 759. doi: 10.1016/S1874-575X(02)80036-4. Google Scholar

[24]

D. S. Morgan, A. Doelman and T. J. Kaper, Stationary periodic patterns in the 1D Gray-Scott model,, Meth. Appl. Anal., 7 (2002), 105. Google Scholar

[25]

C. B. Muratov and V. V. Osipov, Traveling spike autosolitons in the Gray-Scott model,, Physica D, 155 (2001), 112. doi: 10.1016/S0167-2789(01)00259-7. Google Scholar

[26]

C. Muratov and V. V. Osipov, Stability of the static spike autosolitons in the Gray-Scott model,, SIAM J. Appl. Math., 62 (2002), 1463. doi: 10.1137/S0036139901384285. Google Scholar

[27]

Y. Nishiura and D. Ueyama, A skeleton structure for self-replication dynamics,, Physica D, 130 (1999), 73. doi: 10.1016/S0167-2789(99)00010-X. Google Scholar

[28]

Y. Nishiura and D. Ueyama, Spatio-temporal chaos for the Gray-Scott model,, Physica D, 150 (2001), 137. doi: 10.1016/S0167-2789(00)00214-1. Google Scholar

[29]

W.-M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states,, Notices AMS, 45 (1998), 9. Google Scholar

[30]

M. Oh and K. Zumbrun, Stability of periodic solutions of conservation laws with viscosity: Analysis of the Evans function,, Arch. Rational Mech. Anal., 166 (2003), 99. doi: 10.1007/s00205-002-0216-7. Google Scholar

[31]

J. E. Pearson, Complex patterns in a simple system,, Science, 261 (1993), 189. doi: 10.1126/science.261.5118.189. Google Scholar

[32]

H. van der Ploeg and A. Doelman, Stability of spatially periodic pulse patterns in a class of singularly perturbed reaction-diffusion equations,, Indiana Univ. Math. J., 54 (2005), 1219. doi: 10.1512/iumj.2005.54.2792. Google Scholar

[33]

V. Petrov, S. K. Scott and K. Showalter, Excitability, wave reflection, and wave splitting in a cubic autocatalysis reaction-diffusion system,, Phil. Trans. Roy. Soc. Lond., 347 (1994), 631. doi: 10.1098/rsta.1994.0071. Google Scholar

[34]

J. D. M. Rademacher, B. Sandstede and A. Scheel, Computing absolute and essential spectra using continuation,, Physica D, 229 (2007), 166. doi: 10.1016/j.physd.2007.03.016. Google Scholar

[35]

J. D. M. Rademacher and A. Scheel, Instabilities of wave trains and Turing patterns in large domains,, Int. J. Bif. Chaos, 17 (2007), 2679. doi: 10.1142/S0218127407018683. Google Scholar

[36]

J. D. M. Rademacher and A. Scheel, The saddle-node of nearly homogeneous wave trains in reaction-diffusion systems,, J. Dyn. Diff. Eq., 19 (2007), 479. doi: 10.1007/s10884-006-9059-5. Google Scholar

[37]

W. N. Reynolds, J. E. Pearson and S. Ponce-Dawson, Dynamics of self-replicating patterns in reaction diffusion systems,, Phys. Rev. Lett., 72 (1994), 2797. doi: 10.1103/PhysRevLett.72.2797. Google Scholar

[38]

B. Sandstede, Stability of travelling waves,, in, (2002), 983. doi: 10.1016/S1874-575X(02)80039-X. Google Scholar

[39]

B. Sandstede and A. Scheel, Absolute and convective instabilities of waves on unbounded and large dounded domains,, Physica D, 145 (2000), 233. doi: 10.1016/S0167-2789(00)00114-7. Google Scholar

[40]

B. Sandstede and A. Scheel, On the stability of periodic travelling waves with large spatial period,, J. Diff. Eq., 172 (2001), 134. doi: 10.1006/jdeq.2000.3855. Google Scholar

[41]

A. Shepeleva, On the validity of the degenerate Ginzburg-Landau equation,, Math. Methods Appl. Sci., 20 (1997), 1239. doi: 10.1002/(SICI)1099-1476(19970925)20:14<1239::AID-MMA917>3.0.CO;2-O. Google Scholar

[42]

A. Shepeleva, Modulated modulations approach to the loss of stability of periodic solutions for the degenerate Ginzburg-Landau equation,, Nonlinearity, 11 (1998), 409. doi: 10.1088/0951-7715/11/3/002. Google Scholar

[43]

M. J. Smith and J. A. Sherratt, The effects of unequal diffusion coefficients on periodic travelling waves in oscillatory reaction-diffusion systems,, Physica D, 236 (2007), 90. doi: 10.1016/j.physd.2007.07.013. Google Scholar

[44]

M. J. Ward and J. Wei, Hopf bifurcations and oscillatory instabilities of spike solutions for the one-dimensional Gierer-Meinhardt model,, J. Nonl. Sc., 13 (2003), 209. doi: 10.1007/s00332-002-0531-z. Google Scholar

[45]

J. Wei and M. Winter, Existence and stability of multiple-spot solutions for the Gray-Scott model in $\RR^2$,, Physica D, 176 (2003), 147. doi: 10.1016/S0167-2789(02)00743-1. Google Scholar

[46]

J. Wei and M. Winter, Existence and stability analysis of asymmetric patterns for the Gierer-Meinhardt system,, J. Math. Pures Appl. (9), 83 (2004), 433. doi: 10.1016/j.matpur.2003.09.006. Google Scholar

show all references

References:
[1]

I. Aranson and L. Kramer, The world of the Ginzburg-Landau equation,, Rev. Modern Phys., 74 (2002), 99. doi: 10.1103/RevModPhys.74.99. Google Scholar

[2]

F. H. Busse, Nonlinear properties of thermal convection,, Rep. Prog. Phys., 41 (1978), 1929. doi: 10.1088/0034-4885/41/12/003. Google Scholar

[3]

R. L. Devaney, Reversible diffeomorphisms and flows,, Trans. Am. Math. Soc., 218 (1976), 89. doi: 10.1090/S0002-9947-1976-0402815-3. Google Scholar

[4]

E. J. Doedel. AUTO-07P:, Continuation and bifurcation software for ordinary differential equations,, \url{http://cmvl.cs.concordia.ca/auto}., (). Google Scholar

[5]

A. Doelman, An explicit theory for pulses in two component singularly perturbed reaction-diffusion equations,, in preparation., (). Google Scholar

[6]

A. Doelman and W. Eckhaus, Periodic and quasi-periodic solutions of degenerate modulation equations,, Physica D, 53 (1991), 249. doi: 10.1016/0167-2789(91)90065-H. Google Scholar

[7]

A. Doelman, R. A. Gardner and T. J. Kaper, Stability analysis of singular patterns in the 1-D Gray-Scott model: A matched asymptotics approach,, Physica D, 122 (1998), 1. doi: 10.1016/S0167-2789(98)00180-8. Google Scholar

[8]

A. Doelman, R. A. Gardner and T. J. Kaper, Large stable pulse solutions in reaction-diffusion equations,, Ind. Univ. Math. J., 50 (2001), 443. doi: 10.1512/iumj.2001.50.1873. Google Scholar

[9]

A. Doelman, R. A. Gardner and T. J. Kaper, A stability index analysis of 1-D patterns of the Gray-Scott model,, Memoirs AMS, 155 (2002). Google Scholar

[10]

A. Doelman, T. J Kaper and H. van der Ploeg, Spatially periodic and aperiodic multi-pulse patterns in the one-dimensional Gierer-Meinhardt equation,, Meth. Appl. An., 8 (2001), 387. Google Scholar

[11]

A. Doelman, T. J. Kaper and P. Zegeling, Pattern formation in the one-dimensional Gray-Scott model,, Nonlinearity, 10 (1997), 523. doi: 10.1088/0951-7715/10/2/013. Google Scholar

[12]

A. Doelman and H. van der Ploeg, Homoclinic stripe patterns,, SIAM J. Appl. Dyn. Syst., 1 (2002), 65. doi: 10.1137/S1111111101392831. Google Scholar

[13]

A. Doelman, B. Sandstede, A. Scheel and G. Schneider, The dynamics of modulated wave trains,, Memoirs of the AMS, 199 (2009). Google Scholar

[14]

W. Eckhaus and G. Iooss, Strong selection or rejection of spatially periodic patterns in degenerate bifurcations,, Physica D, 39 (1989), 124. doi: 10.1016/0167-2789(89)90043-2. Google Scholar

[15]

E. G. Eszter, "Evans Function Analysis of the Stability of Periodic Traveling Wave Solutions of the Fitzhugh-Nagumo System,", PhD thesis, (1999). Google Scholar

[16]

R. A. Gardner, On the structure of the spectra of periodic travelling waves,, J. Math. Pure Appl., 72 (1993), 415. Google Scholar

[17]

R. A. Gardner, Spectral analysis of long wavelength periodic waves and applications,, J. Reine Angew. Math., 491 (1997), 149. doi: 10.1515/crll.1997.491.149. Google Scholar

[18]

D. Iron and M. J. Ward, The dynamics of multi-spike solutions to the one-dimensional Gierer-Meinhardt model,, SIAM J. Appl. Math., 62 (2002), 1924. doi: 10.1137/S0036139901393676. Google Scholar

[19]

D. Iron, M. J. Ward and J. Wei, The stability of spike solutions to the one-dimensional Gierer-Meinhardt model,, Physica D, 150 (2001), 25. doi: i:10.1016/S0167-2789(00)00206-2. Google Scholar

[20]

T. Kolokolnikov, M. Ward and J. Wei, The existence and stability of spike equilibria in the one-dimensional Gray-Scott model: The pulse-splitting regime,, Physica D, 202 (2005), 258. doi: 10.1016/j.physd.2005.02.009. Google Scholar

[21]

T. Kolokolnikov, M. J. Ward and J. Wei, The existence and stability of spike equilibria in the one-dimensional Gray-Scott model: The low feed-rate regime,, Stud. Appl. Math., 115 (2005), 21. doi: i:10.1111/j.1467-9590.2005.01554. Google Scholar

[22]

B. J. Matkowsky and V. A. Volpert, Stability of plane wave solutions of complex Ginzburg-Landau equations,, Quart. Appl. Math., 51 (1993), 265. Google Scholar

[23]

A. Mielke, The Ginzburg-Landau equation in its role as modulation equation,, in, (2002), 759. doi: 10.1016/S1874-575X(02)80036-4. Google Scholar

[24]

D. S. Morgan, A. Doelman and T. J. Kaper, Stationary periodic patterns in the 1D Gray-Scott model,, Meth. Appl. Anal., 7 (2002), 105. Google Scholar

[25]

C. B. Muratov and V. V. Osipov, Traveling spike autosolitons in the Gray-Scott model,, Physica D, 155 (2001), 112. doi: 10.1016/S0167-2789(01)00259-7. Google Scholar

[26]

C. Muratov and V. V. Osipov, Stability of the static spike autosolitons in the Gray-Scott model,, SIAM J. Appl. Math., 62 (2002), 1463. doi: 10.1137/S0036139901384285. Google Scholar

[27]

Y. Nishiura and D. Ueyama, A skeleton structure for self-replication dynamics,, Physica D, 130 (1999), 73. doi: 10.1016/S0167-2789(99)00010-X. Google Scholar

[28]

Y. Nishiura and D. Ueyama, Spatio-temporal chaos for the Gray-Scott model,, Physica D, 150 (2001), 137. doi: 10.1016/S0167-2789(00)00214-1. Google Scholar

[29]

W.-M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states,, Notices AMS, 45 (1998), 9. Google Scholar

[30]

M. Oh and K. Zumbrun, Stability of periodic solutions of conservation laws with viscosity: Analysis of the Evans function,, Arch. Rational Mech. Anal., 166 (2003), 99. doi: 10.1007/s00205-002-0216-7. Google Scholar

[31]

J. E. Pearson, Complex patterns in a simple system,, Science, 261 (1993), 189. doi: 10.1126/science.261.5118.189. Google Scholar

[32]

H. van der Ploeg and A. Doelman, Stability of spatially periodic pulse patterns in a class of singularly perturbed reaction-diffusion equations,, Indiana Univ. Math. J., 54 (2005), 1219. doi: 10.1512/iumj.2005.54.2792. Google Scholar

[33]

V. Petrov, S. K. Scott and K. Showalter, Excitability, wave reflection, and wave splitting in a cubic autocatalysis reaction-diffusion system,, Phil. Trans. Roy. Soc. Lond., 347 (1994), 631. doi: 10.1098/rsta.1994.0071. Google Scholar

[34]

J. D. M. Rademacher, B. Sandstede and A. Scheel, Computing absolute and essential spectra using continuation,, Physica D, 229 (2007), 166. doi: 10.1016/j.physd.2007.03.016. Google Scholar

[35]

J. D. M. Rademacher and A. Scheel, Instabilities of wave trains and Turing patterns in large domains,, Int. J. Bif. Chaos, 17 (2007), 2679. doi: 10.1142/S0218127407018683. Google Scholar

[36]

J. D. M. Rademacher and A. Scheel, The saddle-node of nearly homogeneous wave trains in reaction-diffusion systems,, J. Dyn. Diff. Eq., 19 (2007), 479. doi: 10.1007/s10884-006-9059-5. Google Scholar

[37]

W. N. Reynolds, J. E. Pearson and S. Ponce-Dawson, Dynamics of self-replicating patterns in reaction diffusion systems,, Phys. Rev. Lett., 72 (1994), 2797. doi: 10.1103/PhysRevLett.72.2797. Google Scholar

[38]

B. Sandstede, Stability of travelling waves,, in, (2002), 983. doi: 10.1016/S1874-575X(02)80039-X. Google Scholar

[39]

B. Sandstede and A. Scheel, Absolute and convective instabilities of waves on unbounded and large dounded domains,, Physica D, 145 (2000), 233. doi: 10.1016/S0167-2789(00)00114-7. Google Scholar

[40]

B. Sandstede and A. Scheel, On the stability of periodic travelling waves with large spatial period,, J. Diff. Eq., 172 (2001), 134. doi: 10.1006/jdeq.2000.3855. Google Scholar

[41]

A. Shepeleva, On the validity of the degenerate Ginzburg-Landau equation,, Math. Methods Appl. Sci., 20 (1997), 1239. doi: 10.1002/(SICI)1099-1476(19970925)20:14<1239::AID-MMA917>3.0.CO;2-O. Google Scholar

[42]

A. Shepeleva, Modulated modulations approach to the loss of stability of periodic solutions for the degenerate Ginzburg-Landau equation,, Nonlinearity, 11 (1998), 409. doi: 10.1088/0951-7715/11/3/002. Google Scholar

[43]

M. J. Smith and J. A. Sherratt, The effects of unequal diffusion coefficients on periodic travelling waves in oscillatory reaction-diffusion systems,, Physica D, 236 (2007), 90. doi: 10.1016/j.physd.2007.07.013. Google Scholar

[44]

M. J. Ward and J. Wei, Hopf bifurcations and oscillatory instabilities of spike solutions for the one-dimensional Gierer-Meinhardt model,, J. Nonl. Sc., 13 (2003), 209. doi: 10.1007/s00332-002-0531-z. Google Scholar

[45]

J. Wei and M. Winter, Existence and stability of multiple-spot solutions for the Gray-Scott model in $\RR^2$,, Physica D, 176 (2003), 147. doi: 10.1016/S0167-2789(02)00743-1. Google Scholar

[46]

J. Wei and M. Winter, Existence and stability analysis of asymmetric patterns for the Gierer-Meinhardt system,, J. Math. Pures Appl. (9), 83 (2004), 433. doi: 10.1016/j.matpur.2003.09.006. Google Scholar

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