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SIAM J. APPLIED DYNAMICAL SYSTEMS c 2014 Society for Industrial and Applied MathematicsVol. 13, No. 1, pp. 94128

Numerical Results for Snaking of Patterns over Patterns in Some 2DSelkovSchnakenberg Reaction-Diffusion Systems

Hannes Uecker and Daniel Wetzel

Abstract. For a SelkovSchnakenberg model as a prototype reaction-diffusion system on two dimensional do-mains, we use the continuation and bifurcation software pde2path to numerically calculate branchesof patterns embedded in patterns, for instance hexagons embedded in stripes and vice versa, with aplanar interface between the two patterns. We use the GinzburgLandau reduction to approximatethe locations of these branches by Maxwell points for the associated GinzburgLandau system. Forour basic model, some but not all of these branches show a snaking behavior in parameter space,over the given computational domains. The (numerical) nonsnaking behavior appears to be relatedto too narrow bistable ranges with rather small GinzburgLandau energy differences. This claimis illustrated by a suitable generalized model. Besides the localized patterns with planar interfaceswe also give a number of examples of fully localized patterns over patterns, for instance hexagonpatches embedded in radial stripes, and fully localized hexagon patches over straight stripes.

Key words. Turing patterns, pinning, snaking, GinzburgLandau approximation, Maxwell point

AMS subject classifications. 35J60, 35B32, 35B36, 65N30

DOI. 10.1137/130918484

1. Introduction. Homoclinic snaking refers to the back and forth oscillation in parameterspace of a branch of stationary localized patterns for some pattern forming PDE. Two standardmodels are the quadratic-cubic SwiftHohenberg equation (SHe)

tu = (1 + )2u+ u+ u2 u3, u = u(t, x) R, x Rd,(1.1)

and the cubic-quintic SHe

tu = (1 + )2u+ u+ u3 u5, u = u(t, x) R, x Rd,(1.2)

with suitable boundary conditions if = Rd, and where R is the linear instabilityparameter, and > 0.

In both equations the trivial solution u 0 is stable for < c := 0, where in the onedimensional (1D) case d = 1 we have a pitchfork bifurcation of periodic solutions with periodnear 2 (stripes, by trivial extension to two dimensions, also called rolls due to theirconnection to RayleighBenard convection rolls). For > 0 0, with 0 =

27/38 for (1.1)

and 0 = 0 for (1.2), the bifurcation is subcritical and the periodic branch starts with unstable

Received by the editors April 25, 2013; accepted for publication by B. Sandstede October 18, 2013; publishedelectronically January 30, 2014.

http://www.siam.org/journals/siads/13-1/91848.htmlInstitut fur Mathematik, Universitat Oldenburg, D26111 Oldenburg, Germany (hannes.uecker@uni-oldenburg.de,

daniel.wetzel@uni-oldenburg.de).

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http://www.siam.org/journals/siads/13-1/91848.htmlmailto:hannes.uecker@uni-oldenburg.demailto:daniel.wetzel@uni-oldenburg.de

Copyright by SIAM. Unauthorized reproduction of this article is prohibited.

SNAKING OF PATTERNS OVER PATTERNS 95

(a)

0.4 0.2 0 0.20

0.2

0.4

0.6

0.8

1

||u|| 2

20

60

90

500

(b) u at points 20, 60, 90, and 500 as indicated in (a)

Figure 1. Homoclinic snaking in (1.1) over a 1D domain x (12, 12) with Neumann-type boundaryconditions xu|x=12 = 3xu|x=12 = 0, = 2. Here u2 := ( 1||

|u(x, y)|2d(x, y))1/2. The branch r of

primary roll solutions (blue) bifurcates subcritically at = 0 and becomes stable at 0.53. Bifurcationpoints on the trivial branch and the roll branch are indicated by , but omitted on the homoclinic snaking branch(red), which bifurcates from the second bifurcation point on r and reconnects to r just below the fold.

small solutions r and turns around in a fold at = 0() < 0 to yield O(1) amplitude stableperiodic solutions r+. Thus, for 0 < < c there is a bistable regime of the trivial solutionand O(1) amplitude stripes.

In the simplest case the localized patterns then consist of 1D stripes over the homogeneousbackground u = 0, and in each pair of turns in the snake the localized pattern grows byadding a stripe on both sides, and this continues forever over the infinite line. See, e.g.,[BK06, BK07, BKL+09] for seminal results in this setting and [CK09, DMCK11] for detailedanalysis using a GinzburgLandau formalism and beyond all order asymptotics. In finitedomains snaking cannot continue forever, and instead branches typically connect primarystripe branches, with either the same wave number or wave numbers near each other; see,e.g., [HK09, BBKM08, Daw09, KAC09] for detailed results and Figure 1 for illustration.

In two dimensions, by rotational invariance, depending on the domain and boundary con-ditions, multiple patterns may bifurcate from u 0 at = 0, for instance straight stripes andso-called hexagons. There are a number of studies of localized patterns for (1.1) and (1.2) overtwo dimensional (2D) domains [LSAC08, ALB+10], often combining analysis and numerics,and in more complicated systems like fluid convection [LJBK11]. See also [LO13] for recentresults on snaking of localized hexagon patches in a 2-component reaction-diffusion system intwo dimensions. However, all these works essentially consider patterns over a homogeneousbackground. Already in [Pom86] it is pointed out that pinned fronts connecting stripes andhexagons may exist in reaction-diffusion systems as a codimension 0 phenomenon in parameterspace, i.e., for a whole interval of parameters. Similar ideas were also put forward in [MNT90]with the 2D quadratic-cubic SHe (1.2) as an example; see, in particular, [MNT90, AppendixC]. Such a pinned front is observed in [HMBD95] for (1.2) by time integration, but so far nostudies of branches of stationary solutions involving different 2D patterns seem available.

Here we start with a standard model problem for predator (u) prey (v) reaction-diffusion

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Copyright by SIAM. Unauthorized reproduction of this article is prohibited.

96 HANNES UECKER AND DANIEL WETZEL

systems, namely

tU = DU +N(U, ), N(U, ) =

(u+ u2v u2v

),(1.3)

with U = (u, v)(t, x, y) R2, diffusion matrix D = ( 1 00 d ), d fixed to d = 60, and bifurcationparameter R+. The reaction term N of (1.3) is a special version of

N(U, ) =

(u+ u2v + b+ av u2v av

),

known as the Selkov [Sel68] (a 0, b = 0) and Schnakenberg [Sch79] (a = 0, b 0) model.For simplicity, here we set a = b = 0.

In particular, we consider the stationary system

(1.4) DU +N(U, ) = 0.

The unique spatially homogeneous solution of (1.4) is w = (, 1/). We write (1.4) in theform

tw = L()w +G(w),(1.5)

with w = U w and L(, ) = J() + D, where J is the Jacobian of N in w. Fork = (m,n) R2 we have L(, )ei(mx+ny) = L(k, )ei(mx+ny), where L(k, ) = J() Dk2,k :=

m2 + n2, and thus we also write L(k, ). The eigenvalues of L are given by (k, ) =

(k, ) =trL(k,)

2

( trL(k,)2 )2 detL(k, ). Following [Mur89, Chapter 14] we find that

in c =d

38 3.2085 we have +(k, c) = 0 for all vectors k R2 of lengthkc =

2 1 0.6436, and all other (k, c) < 0; i.e., there is a Turing bifurcation at

c, with critical wave vectors k R2 in the circle |k| = kc. The most prominent Turingpatterns near bifurcation are stripes and hexagons, which modulo rotational invariance canbe expanded as

U = w + 2(A cos(kcx) +B cos

(kc2

(x+

3y

))+B cos

(kc2

(x

3y

)))+ h.o.t.

(1.6)

= w + 2(A cos(kcx) + 2B cos

(kc2x

)cos

(kc2

3y

))+ h.o.t.,

where R2 is the critical eigenvector of L(kc, c), and h.o.t. stands for higher order terms.The amplitudes 2A, 2B R (where the factor 2 has been introduced for consistency withsection 3) of the corresponding Turing pattern depend on , with A = B = 0 at bifurcation.The stripes bifurcate in a supercritical pitchfork, the hexagons bifurcate transcritically, andthe subcritical part of the hexagons is usually called the cold branch, as here u has aminimum in the center of each hexagon.

In sections 2 and 4 we present some numerically calculated Turing patterns for (1.4),including so-called mixed modes, and, moreover, some branches of stationary solutions whichinvolve different patterns, namely

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SNAKING OF PATTERNS OVER PATTERNS 97

(a) planar fronts between stripes and hexagons, and associated localized patterns, e.g.,hexagons localized in one di

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