Homoclinic snaking in bounded domains

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<ul><li><p>Homoclinic snaking in bounded domains</p><p>S. M. Houghton1,* and E. Knobloch2,1School of Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom</p><p>2Department of Physics, University of California, Berkeley, California 94720, USAReceived 27 February 2009; revised manuscript received 27 May 2009; published 20 August 2009</p><p>Homoclinic snaking is a term used to describe the back and forth oscillation of a branch of time-independentspatially localized states in a bistable spatially reversible system as the localized structure grows in length byrepeatedly adding rolls on either side. On the real line this process continues forever. In finite domains snakingterminates once the domain is filled but the details of how this occurs depend critically on the choice ofboundary conditions. With periodic boundary conditions the snaking branches terminate on a branch of spa-tially periodic states. However, with non-Neumann boundary conditions they turn continuously into a largeamplitude filling state that replaces the periodic state. This behavior, shown here in detail for the Swift-Hohenberg equation, explains the phenomenon of snaking without bistability, recently observed in simula-tions of binary fluid convection by Mercader et al. Phys. Rev. E 80, 025201 2009.</p><p>DOI: 10.1103/PhysRevE.80.026210 PACS numbers: 47.54.r, 47.20.Ky</p><p>I. INTRODUCTION</p><p>Recent years have seen rapid developments in the theoryof spatially localized structures in reversible systems in bothone 13 and two 46 spatial dimensions. These structuresoccur in a great variety of physical systems, including non-linear optics where they are usually referred to as dissipativesolitons, reaction-diffusion systems where they take the formof spots or ensembles of spots, and convection where theyare called convectons. We refer to a recent special issue ofChaos 7 for an overview of the subject, including applica-tions, and to 8 for a discussion of open problems in thisarea.</p><p>The theory is simplest in one spatial dimension and relieson the presence of bistability between a trivial state and aspatially periodic state: on the real line spatially localizedstates first appear via a bifurcation from the trivial state anddo so simultaneously with the primary branch of spatiallyperiodic states. Typically there are two branches of spatiallylocalized states that are produced, both symmetric under re-flection xx, with either maxima or minima at x=0. In thefollowing we refer to these states in terms of their spatialphase =0, as L0,, respectively. These states are distinctand are not related by symmetry. Both L0, bifurcate in thesame direction as the periodic states, i.e., subcritically, andare initially unstable. With decreasing parameter the local-ized states grow in amplitude but shrink in extent; when theextent of the localized state approaches one wavelength andthe amplitude reaches that of the competing periodic state thebranch enters the so-called snaking or pinning region andbegins to snake back and forth. As this happens the local-ized state gradually adds rolls, symmetrically on either side,thereby increasing its length. As a result the localized stateshigh up the snaking branches resemble the finite amplitudespatially periodic state over longer and longer lengths. Typi-cally each snaking branch repeatedly gains and loses stability</p><p>via saddle-node bifurcations, producing an infinite multiplic-ity of coexisting stable states within the pinning region. Sec-ondary bifurcations to pairs of unstable branches of asym-metric states are found in the vicinity of each saddle node;these branches resemble rungs that connect the two snak-ing branches and are responsible for the snakes-and-ladders structure of the pinning region 1,4. The origin andproperties of this behavior are now quite well understood, atleast in variational systems 2,3,9,10. However, in additionto these states there are spatially localized states that re-semble holes in a background of otherwise spatially peri-</p><p>*smh@maths.leeds.ac.ukknobloch@berkeley.edu</p><p>r</p><p>||u||L2</p><p>(a)</p><p>(c)</p><p>(b)</p><p>(e)</p><p>(d)</p><p>0</p><p>0</p><p>/2</p><p>/2 /2</p><p>/2</p><p>0.3 0</p><p>0</p><p>0.7=0</p><p>=0</p><p>=</p><p>=</p><p>(b,c)</p><p>(d,e)</p><p>0</p><p>0.7=0</p><p>0</p><p>0.7=</p><p>0</p><p>1.5=0</p><p>0</p><p>1.5=</p><p>FIG. 1. Color online a Bifurcation diagram showing theequivalent of homoclinic snaking in the Swift-Hohenberg equationwith b2=1.8 on a periodic domain with period =62. The snakingbranches L0 and L emerge from P10 in a secondary bifurcation at small amplitude and terminate on the same branch in a secondarybifurcation near the saddle node. Other spatially periodicbranches are also present but for clarity are not shown. b and cSample profiles from the =0 and = branches near the upperend of the snaking branches. d and e Sample profiles from the=0 and = branches on the lower part of the snaking branches.In this case the L0 branch connects the =0 and =0 solutions,and the L branch connects the = and = solutions. After12.</p><p>PHYSICAL REVIEW E 80, 026210 2009</p><p>1539-3755/2009/802/02621015 2009 The American Physical Society026210-1</p><p>http://dx.doi.org/10.1103/PhysRevE.80.026210</p></li><li><p>odic finite amplitude wave trains 11. These bifurcate fromthe vicinity of the saddle-node bifurcation on the branch ofspatially periodic states and are also organized into a pair ofbranches with its own snakes-and-ladders structure producedas the hole deepens and fills with longer and longer sectionsof the trivial state.</p><p>On an unbounded domain these two pairs of snakingbranches remain distinct. This is not so, however, once thedomain becomes finite 12. In this case neither set ofbranches can snake forever, and the snaking process mustterminate when the width of the localized periodic state ap-proaches the domain size and likewise for the width of thelocalized hole. Thus both pairs of branches must turn overand exit the snaking region. Typically the two pairs ofbranches connect pairwise Fig. 1 so that the small ampli-tude =0 branch now connects to the corresponding branchof holelike states and likewise for the = branch. Thisfigure, computed for the quadratic-cubic Swift-Hohenbergequation on a periodic domain of period =62, shows thatthe classic localized states enter the pinning region fromsmall amplitude on the right and exit this region at largeamplitude toward the left; however, the same figure can alsobe viewed as showing that the holelike states enter the snak-</p><p>ing region at the top from the left and exit it near the bottomtoward the right. One finds, in addition, that in domains offinite size, the branches of small amplitude localized statesno longer bifurcate directly from the trivial state but now doso in a secondary pitchfork bifurcation on the primarybranch of periodic states. The finite domain size likewiseaffects the bifurcation to the holelike states on the branch ofperiodic states: the two branches of holelike states need notbifurcate together and need not bifurcate from the primaryperiodic branch 12. Thus, depending on the spatial period,the branches of spatially localized states may or may notterminate together on a branch of periodic states, and thisbranch may or may not be the branch from which these statesbifurcate at small amplitude. However, despite these finite-size effects the behavior within the snaking region remainsessentially identical to that present on the real line, with theeffects of the finite size confined to the vicinity of the bifur-cations creating the localized and holelike states in the firstplace 12. Similar observations are made in Ref. 13 aswell. Thus the physical interpretation of the structure of thesnaking region in terms of front pinning 14 remains largelyunchanged.</p><p>The above picture, attractive as it is, relies on the use ofspatially periodic boundary conditions PBCs to approxi-mate the real line. A recent study of binary fluid convection</p><p>0.0</p><p>0.2</p><p>0.4</p><p>0.6</p><p>0.8</p><p>1.0</p><p>-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2</p><p>Amplitu</p><p>de,A</p><p>Forcing, r</p><p>(a)</p><p>(b)</p><p>(c)</p><p>(d)</p><p>P6 P5</p><p>FIG. 2. Bifurcation diagram showing the equivalent of ho-moclinic snaking with Neumann boundary conditions on a domainof length 2L=40. Only even-parity states are shown. Pitchfork bi-furcations are denoted by . Solution profiles ux at the saddlenodes ad are shown in Fig. 4.</p><p>-1.0</p><p>0.0</p><p>1.0</p><p>2.0</p><p>-20 -10 0 10 20</p><p>u(x)</p><p>x</p><p>(a)</p><p>-1.0</p><p>0.0</p><p>1.0</p><p>2.0</p><p>-20 -10 0 10 20</p><p>u(x)</p><p>x</p><p>(b)</p><p>FIG. 3. Sample large amplitude periodic states ux on a P6and b P5 with Neumann boundary conditions on a domain oflength 2L=40, both at r=0.</p><p>-0.4</p><p>0.0</p><p>0.4</p><p>0.8</p><p>-20 -10 0 10 20</p><p>u(x)</p><p>x</p><p>(a)</p><p>-0.4</p><p>0.0</p><p>0.4</p><p>0.8</p><p>1.2</p><p>-20 -10 0 10 20</p><p>u(x)</p><p>x</p><p>(b)</p><p>-</p><p>-20 -10 0 10 20</p><p>u(x)</p><p>x</p><p>(c)</p><p>-0.4</p><p>0.0</p><p>0.4</p><p>0.8</p><p>1.2</p><p>-20 -10 0 10 20</p><p>u(x)</p><p>x</p><p>(d)</p><p>FIG. 4. Sample two-pulse profiles ux at the saddle nodes along the figure-eight isola in Fig. 2.</p><p>S. M. HOUGHTON AND E. KNOBLOCH PHYSICAL REVIEW E 80, 026210 2009</p><p>026210-2</p></li><li><p>in a two-dimensional box with no-slip boundary conditionson all boundaries 15 reveals behavior that differs qualita-tively from the above picture: the snaking branches do notterminate on a branch of periodic states that coexist with theconduction state and instead evolve continuously into largeamplitude nonperiodic states that fill the box and play therole of the periodic state present with PBC. These statestypically include a defectlike structure in order to accommo-date the natural wavelength of the rolls, with additional de-</p><p>fectlike structures present at either lateral boundary requiredby the no-slip boundary conditions. The authors of Ref. 15refer to this situation as snaking without bistability sincethere is no large amplitude state that exists independently ofthe snaking branches.</p><p>We use these results to motivate our study in this paper ofthe Swift-Hohenberg equation with non-Neumann lateralboundary conditions. Neumann boundary conditions NBCsare special since problems with NBC on domains of length2L can be embedded in problems with PBC and period 4L16. Thus the behavior described above for problems withPBC applies equally to problems with NBC. Consequently,we focus here on the effects of a generalization of Neumannboundary conditions referred to as Robin boundary condi-tions RBCs. It should be emphasized that the change fromNeumann to non-Neumann boundary conditions has a pro-found effect already on the linear stability problem. In theformer case the null eigenfunctions critical modes are sinu-soidal, with well-defined mode number n. The branches thatbifurcate into the nonlinear regime preserve this mode num-ber, and in the following we refer to them as Pn. Moreover,as L increases the neutral curves corresponding to</p><p>0.0</p><p>0.5</p><p>1.0</p><p>1.5</p><p>2.0</p><p>0 5 10 15 20</p><p>Critical</p><p>forcing,</p><p>r c</p><p>Box half-length, L</p><p>(a)</p><p>0</p><p>0.5</p><p>1</p><p>1.5</p><p>2</p><p>0 5 10 15 20</p><p>Critical</p><p>forcing,</p><p>r c</p><p>Box half-length, L</p><p>(b)</p><p>FIG. 5. Color online Neutral curves for a =0 andb =0.5 corresponding to even red solid and odd blue dashedeigenfunctions.</p><p>0.0</p><p>0.2</p><p>0.4</p><p>0.6</p><p>0.8</p><p>1.0</p><p>1.2</p><p>-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1</p><p>Amplitu</p><p>de,A</p><p>Forcing, r</p><p>S6</p><p>S6,0</p><p>(a)</p><p>(b)</p><p>(c)</p><p>(d)(e)</p><p>(f)</p><p>FIG. 6. Bifurcation diagram showing the equivalent of ho-moclinic snaking with Robin boundary conditions 2 and =0.5 ona domain of length 2L=40. Only the =0 branch S6,0 is shown.Solution profiles ux, at the labeled points, are shown in Fig. 8.</p><p>0.00000</p><p>0.00005</p><p>0.00010</p><p>0.00015</p><p>0.00020</p><p>0.02546800 0.02546810 0.02546820</p><p>Amplitu</p><p>de,A</p><p>Forcing, r</p><p>S6</p><p>S6,0</p><p>S6,</p><p>0.00</p><p>0.05</p><p>0.10</p><p>0.15</p><p>0.20</p><p>0.25</p><p>-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1</p><p>Amplitu</p><p>de,A</p><p>Forcing, r</p><p>(l)</p><p>FIG. 7. Color online a With Robin boundary condition 2 the bifurcation at S6 becomes transcritical solid dot: the = branch S6,now bifurcates supercritically but undergoes a saddle-node bifurcation close to threshold before turning toward negative values of r, whileS6,0 bifurcates subcritically. Thus the breaking of the hidden reflection symmetry results in an apparent splitting of the P6 branch. b Thesame but on a larger scale, showing S6,0 dashed green and S6, solid red together with the two branches that become disconnected fromthem as a result of the broken hidden symmetry. Parameters: =0.5 and 2L=40.</p><p>HOMOCLINIC SNAKING IN BOUNDED DOMAINS PHYSICAL REVIEW E 80, 026210 2009</p><p>026210-3</p></li><li><p>n and n+1, say, must cross; the corresponding codimensiontwo points are key to understanding mode jumping in prob-lems of this type, i.e., the transitions from one preferredwave number to another, in the nonlinear regime 17,18.With non-Neumann boundary conditions, however, the situ-ation is quite different. The neutral modes are either odd oreven but have no well-defined wave number. It follows thatwhen neutral curves corresponding to modes of like parityodd-odd or even-even cross, they generically reconnect19, producing a quite different neutral curve topology. Onlyodd-even crossings remain structurally stable. Moreover,since the bifurcating solutions have no fixed wave numbertheir structure in the nonlinear regime may gradually evolvefrom resembling a solution with n maxima to one with n+1 maxima, say; no bifurcation is involved and no periodicsolutions are present. It follows that the snaking branchescannot terminate on a branch of periodic states and anothertermination mechanism must be present. The present paperdescribes in some detail what happens in this case. The re-sults are quite general and provide an immediate explanationof the behavior identified in Ref. 15. In addition, the resultsshould apply to any physical system with realistic lateralboundary conditions and as such should be observable notonly in numerical studies of snaking systems with realisticlateral boundary conditions but also in experiments.</p><p>This paper is organized as follows. In the next section wesummarize briefly the snaking behavior for the Swift-Hohenberg equation on periodic domains. In Sec. III we in-troduce a homotopy parameter that allows us to continuouslychange the boundary conditions away from Neumann bound-ary conditions and explore the accompanying changes in thesnaking structure. In Sec. IV we give partial results on othertypes of boundary conditions. In Sec. V we relate our find-ings to earlier work on convection in binary fluid mixtureswith thermally conducting no-slip lateral boundary condi-</p><p>tions where snaking without bistability was first identified15.</p><p>II. SWIFT-HOHENBERG EQUATION</p><p>The Swift-Hohenberg equation describes the formation ofspatially periodic patterns with a finite wave number k0 atonset. In one spatial dimension the equation takes the form</p><p>ut = ru x2 + k0</p><p>22u + fu . 1</p><p>This equation is variational and hence on finite domains allsolutions approach steady states. In the following we shall beinterested in characterizing such states. The key to the prop-erties of these states is provided by the invariance of Eq. 1under xx, uu, hereafter referred to as reversibility. Onthe real line this property is responsible for the presence of a</p><p>-0.5</p><p>0.0</p><p>0.5</p><p>1.0</p><p>1.5</p><p>-20 -10 0 10 20</p><p>u(x)</p><p>x</p><p>(a)</p><p>-</p><p>-20 -10 0 10 20</p><p>u(x)</p><p>x</p><p>(b)</p><p>-0.5</p><p>0.0</p><p>0.5</p><p>1.0</p><p>1.5</p><p>-20 -10 0 10 20</p><p>u(x)</p><p>x</p><p>(c)</p><p>-1.0-</p><p>-20 -10 0 10 20</p><p>u(x)</p><p>x</p><p>(d)</p><p>-0.5</p><p>0.0</p><p>0.5</p><p>1.0</p><p>1.5</p><p>-20 -10 0 10 20</p><p>u(x)</p><p>x</p><p>(e)</p><p>-1.0-</p><p>-20 -10 0 10 20</p><p>u(x)</p><p>x</p><p>(f)</p><p>FIG. 8. Solution profiles ux at succes...</p></li></ul>