Hopf dances near the tips of Busse balloons

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.

[1]  Björn Sandstede,et al.  The Dynamics of Modulated Wave Trains , 2009 .

[2]  R. Devaney.  Reversible diffeomorphisms and flows , 1976 .

[3]  Arjen Doelman,et al.  Periodic and quasi-periodic solutions of degenerate modulation equations , 1991 .

[4]  Arjen Doelman,et al.  Stability of spatially periodic pulse patterns in a class of singularly perturbed reaction-diffusion equations. , 2005 .

[5]  Wiktor Eckhaus,et al.  Strong selection or rejection of spatially periodic patterns in degenerate bifurcations , 1989 .

[6]  Michael J. Ward,et al.  The existence and stability of spike equilibria in the one-dimensional Gray-Scott model: the pulse-splitting regime , 2005 .

[7]  Arjen Doelman,et al.  Pattern formation in the one-dimensional Gray - Scott model , 1997 .

[8]  F. Busse,et al.  Non-linear properties of thermal convection , 1978 .

[9]  Daishin Ueyama,et al.  Spatio-temporal chaos for the Gray—Scott model , 2001 .

[10]  Daishin Ueyama,et al.  A skeleton structure of self-replicating dynamics , 1997 .

[11]  Matthias Winter,et al.  Existence and stability analysis of asymmetric patterns for the Gierer-Meinhardt system , 2004 .

[12]  Arnd Scheel,et al.  The Saddle-Node of Nearly Homogeneous Wave Trains in Reaction–Diffusion Systems , 2007 .

[13]  Cyrill B. Muratov,et al.  Stability of the Static Spike Autosolitons in the Gray--Scott Model , 2002, SIAM J. Appl. Math..

[14]  Michael J. Ward,et al.  The stability of spike solutions to the one-dimensional Gierer—Meinhardt model , 2001 .

[15]  Björn Sandstede,et al.  On the Stability of Periodic Travelling Waves with Large Spatial Period , 2001 .

[16]  Robert Gardner,et al.  Large stable pulse solutions in reaction-diffusion equations , 2001 .

[17]  Michael J. Ward,et al.  The Dynamics of Multispike Solutions to the One-Dimensional Gierer--Meinhardt Model , 2002, SIAM J. Appl. Math..

[18]  Arjen Doelman,et al.  Spatially periodic and aperiodic multi-pulse patterns in the one-dimensional Gierer-Meinhardt equation , 2001 .

[19]  I. Aranson,et al.  The world of the complex Ginzburg-Landau equation , 2001, cond-mat/0106115.

[20]  J. E. Pearson.  Complex Patterns in a Simple System , 1993, Science.

[21]  B. Sandstede,et al.  Chapter 18 - Stability of Travelling Waves , 2002 .

[22]  Reynolds,et al.  Dynamics of self-replicating patterns in reaction diffusion systems. , 1994, Physical review letters.

[23]  A. Shepeleva,et al.  Modulated modulations approach to the loss of stability of periodic solutions for the degenerate Ginzburg-Landau equation , 1998 .

[24]  Bernard J. Matkowsky,et al.  Stability of plane wave solutions of complex Ginzburg-Landau equations , 1993 .

[25]  Edgardo Gabriel Eszter.  Evans function analysis of the stability of periodic travelling wave solutions associated with the Fitzhugh -Nagumo system , 1999 .

[26]  Robert Gardner,et al.  A stability index analysis of 1-D patterns of the Gray-Scott model , 2002 .

[27]  Matthias Winter,et al.  Existence and stability of multiple-spot solutions for the Gray–Scott model in R2 , 2003 .

[28]  Cyrill B. Muratov,et al.  Traveling spike autosolitons in the Gray-Scott model , 2001 .

[29]  Jonathan A. Sherratt,et al.  The effects of unequal diffusion coefficients on periodic travelling waves in oscillatory reaction–diffusion systems , 2007 .

[30]  Kevin Zumbrun,et al.  Stability of Periodic Solutions¶of Conservation Laws with Viscosity:¶Analysis of the Evans Function , 2003 .

[31]  Arjen Doelman,et al.  Homoclinic Stripe Patterns , 2002, SIAM J. Appl. Dyn. Syst..

[32]  Alexander Mielke,et al.  The Ginzburg-Landau Equation in Its Role as a Modulation Equation , 2002 .

[33]  Arnd Scheel,et al.  Instabilities of Wave Trains and Turing Patterns in Large Domains , 2007, Int. J. Bifurc. Chaos.

[34]  A. Shepeleva,et al.  On the validity of the degenerate Ginzburg-Landau equation , 1997 .

[35]  Björn Sandstede,et al.  Computing absolute and essential spectra using continuation , 2007 .

[36]  Robert Gardner,et al.  Stability analysis of singular patterns in the 1-D Gray-Scott model I: a matched asymptotics approach , 1998 .

[37]  Wei-Ming Ni,et al.  DIFFUSION, CROSS-DIFFUSION, AND THEIR SPIKE-LAYER STEADY STATES , 1998 .

[38]  Björn Sandstede,et al.  Absolute and Convective Instabilities of Waves on Unbounded and Large Bounded Domains , 2022 .

[39]  Valery Petrov,et al.  Excitability, wave reflection, and wave splitting in a cubic autocatalysis reaction-diffusion system , 1994, Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences.

[40]  R. Gardner,et al.  Spectral analysis of long wavelength periodic waves and applications. , 1997 .

[41]  Arjen Doelman,et al.  Stationary periodic patterns in the 1D Gray–Scott model , 2000 .

[42]  Thomas F. Fairgrieve,et al.  AUTO 2000 : CONTINUATION AND BIFURCATION SOFTWARE FOR ORDINARY DIFFERENTIAL EQUATIONS (with HomCont) , 1997 .

[43]  R. Gardner,et al.  ON THE STRUCTURE OF THE SPECTRA OF PERIODIC TRAVELING WAVES , 1993 .

[44]  Juncheng Wei,et al.  Hopf Bifurcations and Oscillatory Instabilities of Spike Solutions for the One-Dimensional Gierer-Meinhardt Model , 2003, J. Nonlinear Sci..