Homoclinic snaking: Structure and stability - Semantic Scholar

CHAOS 17, 037102 共2007兲

Homoclinic snaking: Structure and stability John Burkea兲 and Edgar Knobloch Department of Physics, University of California, Berkeley, California 94720, USA

共Received 1 March 2007; accepted 11 May 2007; published online 28 September 2007兲 The bistable Swift-Hohenberg equation exhibits multiple stable and unstable spatially localized states of arbitrary length in the vicinity of the Maxwell point between spatially homogeneous and periodic states. These states are organized in a characteristic snakes-and-ladders structure. The origin of this structure in one spatial dimension is reviewed, and the stability properties of the resulting states with respect to perturbations in both one and two dimensions are described. The relevance of the results to several different physical systems is discussed. © 2007 American Institute of Physics. 关DOI: 10.1063/1.2746816兴 Stationary spatially localized structures in one or more dimensions occur in many bistable systems of physical interest. These structures are found in a well-defined region in parameter space called the pinning or snaking region. This terminology refers to the multiplicity of stable localized structures created by the pinning of the wall of the structure to the spatial pattern within. The structure of the resulting pinning region is described in detail, and related to the primary instability responsible for the presence of the localized states. The mechanism responsible for wavelength selection within this region is described, and the stability of the resulting states with respect to two-dimensional perturbations is examined via linear stability analysis and direct numerical simulation. I. INTRODUCTION

Spatially localized states, in both one and two spatial dimensions, are common in a variety of physical systems. These include systems with variational structure, such as buckling problems,1,2 as well as problems with nonvariational structure, such as those arising in nonlinear optics,3–7 chemical systems,8,9 convection,10–15 and even in neuroscience.16,17 Recent experiments on ferrofluids in an imposed vertical magnetic field provide a particularly nice illustration of the type of behavior that may be expected.18 Oscillons, i.e., parametrically driven spatially localized oscillations,19,20 provide a closely related example.21 Much of the theory developed for understanding the origin of spatially localized states in this class of systems stems from original work on solitary waves in the Korteweg–de Vries 共KdV兲 equation with a fifth-order dispersive term 共see Ref. 22 and references therein兲. It is perhaps unexpected that similar behavior is found in such diverse systems. The reason for this can be traced to the presence of bistability in these systems between a spatially uniform trivial state and a spatially periodic state. All of these systems are in addition of at least fourth order in space, and are invariant under both translations and spatial reflections. In the present paper we explain both the physical and a兲

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the mathematical reasons behind the presence of spatially localized structures in this class of systems. We do so by focusing on the Swift-Hohenberg equation in one spatial dimension. This equation arises in a number of the applications just mentioned, and possesses three distinct advantages over other systems, in that it is 共i兲 variational in time, 共ii兲 reversible in space, and when formulated as a dynamical system in space, it 共iii兲 conserves a Hamiltonian function. These three properties lead to an attractive interpretation of the origin and properties of localized states in this system, some of which can be extended to nonvariational systems on the real line,21 and others to variational systems in more than one spatial dimension.23,24 In this article we summarize the present state of our understanding of spatially localized states and their stability properties within this equation in two particular cases.25,26 We then extend this work to consider stability with respect to two-dimensional perturbations, and identify a variety of new two-dimensional states that remain localized in one spatial dimension. Fully localized states in two dimensions have also been computed,3,23,27–29 and are discussed in detail by Champneys et al.24

II. THE SWIFT-HOHENBERG EQUATION

We study the Swift-Hohenberg equation in the form ut = ru − 共ⵜ2 + q2c 兲2u + f共u兲,

共1兲

in both one dimension 关u = u共x , t兲兴 and two dimensions 关u = u共x , y , t兲兴. Here, f共u兲 denotes nonlinear terms f共u兲 =

再

f 35 ⬅ b3u3 − b5u5 , f 23 ⬅ c2u2 − c3u3 ,

冎

共2兲

and r is the control parameter. In either case the equation is invariant under x → −x; the former case is also invariant under u → −u. In addition Eq. 共1兲 has variational structure so that in the presence of periodic boundary conditions with period 共Lx , Ly兲 it possesses a Lyapunov functional F 共which we refer to as an energy兲 given by

17, 037102-1

© 2007 American Institute of Physics

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037102-2

Chaos 17, 037102 共2007兲

J. Burke and E. Knobloch

FIG. 2. Spatial eigenvalues of the trivial fixed point in the complex ⌳ plane when 共a兲 r ⬍ 0, 共b兲 r = 0, and 共c兲 r ⬎ 0.

FIG. 1. 共a兲 A stationary localized state uᐉ共x兲 in one dimension with the nonlinearity f 35, and 共b兲 the corresponding localized stripe in two dimensions. This solution lies on the ␾ = 0 branch of localized states, between the 11th and 12th saddle-nodes. In 共a兲 x 苸 关−90, 90兴, and in 共b兲 共x , y兲 苸 关−90, 90兴 ⫻ 关−30, 30兴. Parameters: r = −0.64, qc = 1, b3 = 2, b5 = 1.

冕 冕 冕 冎

Ly/2

Lx/2

F=

dx

−Lx/2

−Ly/2

再

dy −

1 2 2 ru

+

1 2 2 关共ⵜ

+

q2c 兲u兴2

u

f共v兲dv ,

−

共3兲

0

such that ut = −␦F / ␦u. It follows that along any trajectory, the energy F decreases to a 共local兲 minimum. In particular, no Hopf bifurcations are possible and, at a fixed location in space, all time dependence ultimately dies out. Furthermore, the stationary solutions that occupy these local minima satisfy 共by definition兲 the time-independent version of Eq. 共1兲. In the following, we are interested in the existence of time-independent states uᐉ共x兲 that decay exponentially to the trivial state u0 ⬅ 0 as x → ± ⬁. These correspond in one dimension to localized states and in two dimensions to localized stripes 共Fig. 1兲. The true localized states are approximated well by large period periodic orbits provided the spatial period Lx is large compared to the width of the localized state 共see below兲. We shall also be interested in the 共linear兲 stability of such structures with respect to both onedimensional and two-dimensional perturbations. The growth rate ␤ of an infinitesimal perturbation of the form ˜u共x ; ky兲eikyye␤t to a stationary solution uᐉ共x兲 of Eq. 共1兲 satisfies the 共one-dimensional兲 eigenvalue problem ˜ 共x;ky兲 = ␤˜u共x;ky兲, 兵r − 共⳵2x − k2y + q2c 兲2 + ⳵u f关uᐉ共x兲兴其u

共4兲

and depends on both r and ky. There is always a neutrally stable 共␤ = 0兲 Goldstone mode at ky = 0 associated with translation invariance, with eigenfunction ˜u共x ; 0兲 = uᐉ⬘共x兲. Thus, stability with respect to one-dimensional 共ky = 0兲 perturbations requires that the rest of the spectrum lies on the negative real axis. For two-dimensional perturbations with a long wavelength in the y direction, Eq. 共4兲 can be solved analytically. We expand the differential operator on the left-hand side in the form L共⳵x , uᐉ , ky兲 = L0 + k2y L2 + ¯, and likewise write ˜u共x ; ky兲 = ˜u0 + k2y˜u2 + ¯, ␤ = ␤0 + k2y ␤2 + ¯. At leading order, we obtain the stability problem with respect to onedimensional perturbations, L0˜u0 = ␤0˜u0 . At

O共k2y 兲,

we obtain

共5兲

L2˜u0 + L0˜u2 = ␤2˜u0 + ␤0˜u2 .

共6兲

Since L0 is self-adjoint, it follows that

␤2 =

˜ 0,L2˜u0典 具u ˜ 0,u ˜ 0典 具u

共7兲

,

Lx/2 共¯兲dx. This calculation is readily exwhere 具¯典 ⬅ 兰−L x/2 tended to include the nonlinear terms required to describe the saturation of any long-wave two-dimensional instability.30 For larger values of ky, the linear stability problem must be solved numerically, and the analytical prediction of the saturated state is more involved.

III. SPATIAL DYNAMICS IN ONE DIMENSION

In one dimension, the time-independent version of Eq. 共1兲 is treated as a fourth-order dynamical system in x, in which the localized states of interest correspond to orbits homoclinic to the trivial fixed point u0. Whether such orbits exist depends in part on the linear stability of this fixed point in space. Near r = 0, the four spatial eigenvalues are given by ⌳ = ± iqc ± 冑− r/2qc + O共r兲.

共8兲

The motion of these eigenvalues in the complex plane as r varies is shown in Fig. 2. For r ⬍ 0 the eigenvalues form a quartet, and u0 is hyperbolic with two stable eigenvalues and two unstable eigenvalues. In contrast, for r ⬎ 0 all the eigenvalues lie on the imaginary axis and u0 is not hyperbolic. As a result, no exponentially localized states can be present when r ⬎ 0. At r = 0, there is a pair of imaginary eigenvalues ±iqc of double multiplicity. The bifurcation at r = 0 is thus a Hopf bifurcation in a reversible system with 1:1 resonance. In the next section we review the localized states that are known to exist within the corresponding normal form, and explain how this normal form allows us to draw conclusions regarding the existence of localized states near r = 0 in Eq. 共1兲. In the present case, the spatial dynamical system conserves the Hamiltonian 2 + uxuxxx − H = − 21 共r − q4c 兲u2 + q2c u2x − 21 uxx

冕

u

f共v兲dv . 共9兲

0

Thus, orbits homoclinic to the trivial fixed point must lie in the surface H = 0. This global result complements the local eigenvalue result. In Secs. IV–VII we present in detail the analysis of the f 35 nonlinearity, while Sec. VIII summarizes the essential results in the f 23 case.

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Homoclinic snaking

uᐉ共x兲 = ⑀u1共x,X兲 + ⑀2u2共x,X兲 + ¯ .

IV. NORMAL FORM THEORY

The normal form for the reversible Hopf bifurcation with 1:1 resonance is31

With this scaling, Eq. 共1兲 at r = 0 becomes 共⳵2x + q2c 兲2共⑀u1 + ⑀2u2 + ¯ 兲

A⬘ = iqcA + B + iAP共␮ ;y,w兲, B⬘ = iqcB + iBP共␮ ;y,w兲 + AQ共␮ ;y,w兲,

共12兲

= − 关4⑀⳵xX共⳵2x + q2c 兲 + 4⑀2⳵xxXX + 2⑀2⳵XX共⳵2x + q2c 兲

共10兲

¯ − ¯AB兲, the overbar refers to where y ⬅ 兩A兩2, w ⬅ 共i / 2兲共AB complex conjugation, and in the context of spatial dynamics, the prime denotes differentiation with respect to x; ␮ is an unfolding parameter analogous to r. The functions A and B ¯ , −B ¯ 兲, and P transform under spatial reflection as 共A , B兲 → 共A and Q are polynomials with real coefficients, which we truncate to include only the first few terms: P共␮ ;y,w兲 = p1␮ + p2y + p3w + p4y 2 + p5wy + p6w2 , 共11兲 Q共␮ ;y,w兲 = − q1␮ + q2y + q3w + q4y 2 + q5wy + q6w2 . The 1:1 Hopf bifurcation from the trivial state A = B = 0 occurs at ␮ = 0; by convention, q1 ⬎ 0, so that this state is hyperbolic in the region ␮ ⬍ 0. This normal form constitutes an integrable Hamiltonian system. This is true regardless of the system from which it is derived, and in particular, it applies to systems that are not variational in time or Hamiltonian in space. Analysis of the normal form shows that small amplitude orbits homoclinic to the trivial state are present for ␮ ⬍ 0, 兩␮ 兩 Ⰶ 1, provided q2 ⬍ 0. These are the only 共small amplitude兲 homoclinic orbits that are born as ␮ passes through zero, and hence are the orbits that are of interest in the present paper. If q4 ⬎ 0, these homoclinic orbits are found only when ␮ is negative but greater than ␮D ⬅ −3q22 / 16q1q4 ⬍ 0. This curve in the 共␮ , q2兲 plane corresponds to the formation of a heteroclinic orbit connecting the trivial fixed point to the finite amplitude periodic orbit with the same energy. This result is structurally stable: if the normal form is truncated at higher order, the equation for ␮D remains valid at leading order, and the above description continues to hold. To apply these results to the Swift-Hohenberg equation in one spatial dimension, it is necessary to compute the coefficients q1, q2, and q4 in terms of the parameters of Eq. 共1兲. The direct, albeit laborious, method involves formulating the problem as a fourth-order dynamical system in space and computing the normal form transformation order by order in u.32 A somewhat simpler procedure, based on an approach used in Ref. 33, is described below, and involves reducing both the Swift-Hohenberg equation and the normal form 共10兲 to the same Ginzburg-Landau equation. Some subtleties arise, however, because the determination of q4 requires extending this method to higher order.32 A. Scaling of the time-independent Swift-Hohenberg equation

As usual we compute the normal form coefficients at the bifurcation r = 0. We introduce a small parameter ⑀ Ⰶ 1, define a large spatial scale X ⬅ ⑀x, and look for onedimensional stationary solutions of Eq. 共1兲 of the form

+ 4⑀3⳵xXXX + ⑀4⳵X4 兴共⑀u1 + ⑀2u2 + ¯ 兲 + b 3共 ⑀ u 1 + ¯ 兲 3 − b 5共 ⑀ u 1 + ¯ 兲 5 .

共13兲

Matching terms order by order in ⑀ gives O共⑀兲: 共⳵2x + q2c 兲2u1 = 0,

共14兲

O共⑀2兲: 共⳵2x + q2c 兲2u2 = − 4⳵xX共⳵2x + q2c 兲u1

共15兲

O共⑀3兲: 共⳵2x + q2c 兲2u3 = − 4⳵xX共⳵2x + q2c 兲u2 − 4⳵xxXXu1 − 2⳵XX共⳵2x + q2c 兲u1 + b3u31 ,

共16兲

O共⑀4兲: 共⳵2x + q2c 兲2u4 = − 4⳵xX共⳵2x + q2c 兲u3 − 4⳵xxXXu2 − 2⳵XX共⳵2x + q2c 兲u2 − 4⳵xXXXu1 + 3b3u21u2 ,

共17兲

O共⑀5兲: 共⳵2x + q2c 兲2u5 = − 4⳵xX共⳵2x + q2c 兲u4 − 4⳵xxXXu3 − 2⳵XX共⳵2x + q2c 兲u3 − 4⳵xXXXu2 − ⳵X4 u1 + 3b3共u1u22 + u21u3兲 − b5u51 . 共18兲 The O共⑀ , ⑀ 兲 equations are solved by 2

u1共x,X兲 = A1共X兲eiqcx + c.c.,

u2共x,X兲 = A2共X兲eiqcx + c.c., 共19兲

where A1,2共X兲 are as yet undetermined and “c.c.” denotes a complex conjugate. The ansatz u3共x,X兲 = A3共X兲eiqcx + C3共X兲e3iqcx + c.c.

共20兲

in the O共⑀3兲 equation leads to the two results 4q2c A1⬙ = − 3b3A1兩A1兩2,

C3 =

b3 3 A1 , 64q4c

共21兲

with A3 arbitrary. The ansatz u4共x,X兲 = A4共X兲eiqcx + C4共X兲e3iqcx + c.c.

共22兲

in the O共⑀4兲 equation likewise leads to 4q2c A2⬙ = 4iqcA1⵮ − 3b3共2兩A1兩2A2 + A21¯A2兲;

共23兲

the expression for C4 in terms of A1,2 is not needed in what follows. Finally, the O共⑀5兲 equation with the ansatz u5共x,X兲 = A5共X兲eiqcx + C5共X兲e3iqcx + E5共X兲e5iqcx + c.c. 共24兲 yields

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037102-4

Chaos 17, 037102 共2007兲

J. Burke and E. Knobloch

4q2c A3⬙ = 4iqcA2⵮ + A1⵳ − 3b3共2A1兩A2兩2 + ¯A1A22 + 2兩A1兩2A3

冉

+ A21¯A3兲 + −

冊

3b23

+ 10b5 A1兩A1兩4

64q4c

共25兲

after elimination of C3. It is easy to show that Eqs. 共21兲, 共23兲, and 共25兲 can now be assembled into a single equation for the amplitude Z共X , ⑀兲 ⬅ A1共X兲 + ⑀A2共X兲 + ⑀2A3共X兲 + ¯ of the eiqcx component of the solution 共12兲: 4q2c Z⬙ = − 3b3Z兩Z兩2 + 4iqc⑀Z⵮

冋 冉

+ ⑀2 Z⵳ + −

3b23 64q4c

冊 册

+ 10b5 Z兩Z兩4 + O共⑀3兲.

This equation represents the Ginzburg-Landau approximation to the Swift-Hohenberg equation at r = 0. Equation 共26兲 can also be written in the form

关

A⬙ = q2A兩A兩2 + i⑀ 共3p2 − 21 q3兲A⬘兩A兩2

兴

+ 共 p2 + 21 q3兲A2¯A⬘ + ⑀2兵p3关共A⬘兲2¯A − AA⬘¯A⬘兴 + 共q4 − q3 p2 +

冉

+ −

9b3 2q2c

64q4c

冊

+ O共⑀3兲.

共32兲

␳=

9b3

, 16q4c

p2 = −

q3 = −

9b3

, 16q3c

3b3 8q3c

,

q2 = −

q4 = −

3b3 4q2c

177b23 128q6c

, 共33兲

+

5b5 2q2c

.

The remaining coefficients p1 and q1 are determined as part of the unfolding. We write r ⬅ ␮ = −⑀2␮2, where ␮2 is O共1兲 and ⑀ specifies the amplitude scaling used in Eq. 共12兲. The unfolded versions of Eqs. 共27兲 and 共32兲 through O共⑀兲 are

关2ZZ⬘¯Z⬘ + 共Z⬘兲2¯Z兴

327b23

p22兲A兩A兩4其

To compare Eqs. 共27兲 and 共32兲 one final transformation is required, i.e., Z = A + ⑀2␳A兩A兩2 + O共⑀4兲, where ␳ is to be determined. Matching all terms through O共⑀2兲, one obtains32

p3 = 0,

3i⑀b3 = − 3b3Z兩Z兩 − 共2Z⬘兩Z兩2 + Z2¯Z⬘兲 qc

再

共31兲

and this equation can be used to eliminate B from Eq. 共30兲:

2

+ ⑀2

p3 ¯ ⬘ − ¯AA⬘兲 + O共⑀3兲, A共AA 2

C. Matching and unfolding

共26兲

4q2c Z⬙

B = A⬘ − i⑀ p2A兩A兩2 + ⑀2

冎

+ 10b5 Z兩Z兩4 + O共⑀3兲

共27兲

4q2c Z⬙ = ␮2Z − 3b3Z兩Z兩2 +

by iteratively eliminating the higher-order derivatives on the right side.

i⑀ 关␮2Z − 3b3共2Z⬘兩Z兩2 + Z2¯Z⬘兲兴 qc

+ O共⑀2兲,

共34兲

关

A⬙ = q1␮2A + q2A兩A兩2 + i⑀ − 2p1␮2A⬘ B. Scaling of the normal form equation

To match the scaling of the previous section to the normal form 共10兲, we set ␮ = 0 and write 共A , B兲 = 关⑀˜A共X兲 , ⑀2˜B共X兲兴eiqcx, where X ⬅ ⑀x as before. Dropping the tildes, the polynomials P and Q in Eqs. 共11兲 become 2

3

共28兲 i ¯ ¯ Q = ⑀2q2兩A兩2 + ⑀3q3 共AB − AB兲 + ⑀4q4兩A兩4 + O共⑀5兲, 2 and the normal form 共10兲 takes the form

冋

册

i ¯ ¯ ⑀2A⬘ = ⑀2B + i⑀A ⑀2 p2兩A兩2 + ⑀3 p3 共AB − AB兲 + O共⑀5兲, 2 共29兲

冋

册 冋

i ¯ ¯ ⑀3B⬘ = i⑀2B ⑀2 p2兩A兩2 + ⑀3 p3 共AB − AB兲 + ⑀A ⑀2q2兩A兩2 2

册

共35兲 Matching terms through this order gives p1 = −

i ¯ ¯ − AB兲 + O共⑀4兲, P = ⑀ p2兩A兩 + ⑀ p3 共AB 2 2

兴

+ 共3p2 − 21 q3兲A⬘兩A兩2 + 共 p2 + 21 q3兲A2¯A⬘ + O共⑀2兲.

i ¯ ¯ + ⑀3q3 共AB − AB兲 + ⑀4q4兩A兩4 + O共⑀6兲. 2

共30兲

Equation 共29兲 yields a power series expansion for B in terms of A,

1

, 8q3c

q1 =

1 4q2c

.

共36兲

Matching to higher order is possible but requires the retention of parameter dependence in Eqs. 共10兲 beyond the linear terms. We can translate the normal form results to the corresponding ones for the Swift-Hohenberg equation 共1兲. The condition q2 ⬍ 0 for the birth of small amplitude homoclinic orbits translates into b3 ⬎ 0; it is shown below that this is the condition that the branch of periodic states bifurcates subcritically from r = 0. If we assume in addition that the quintic term in the nonlinearity provides large amplitude stabilization 共b5 ⬎ 0兲, an interval of bistability between u0 and a large amplitude periodic pattern will be present for some range of r in r ⬍ 0. This situation corresponds to q4 ⬎ 0, at least when b3 共i.e., q2兲 is sufficiently small. Thus, localized states in the Swift-Hohenberg equation are expected to emerge from the codimension-two point 共r , b3兲 = 0 into r ⬍ 0, b3 ⬎ 0, and near this codimension-two point, the normal form predicts that such states are only found above rD ⯝ −27b23 / 160b5.28,34 Moreover, along this codimension-one line there exist sta-

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037102-5

Chaos 17, 037102 共2007兲

Homoclinic snaking

tionary fronts between two semi-infinite domains: one filled with the trivial state u0 and the other with the spatially periodic pattern u P. This stationary state must lie in the surface H = 0, with H given by Eq. 共9兲. The use of normal form theory in this problem justifies the formal asymptotic reduction just described. The theory has three advantages: 共i兲 it does not require any explicit scaling assumptions, 共ii兲 it can be extended to any order in a systematic manner, and 共iii兲 it establishes a common form of the governing equations for all systems exhibiting a reversible Hopf bifurcation with 1:1 resonance. On the other hand, the resulting normal form possesses an additional symmetry, the so-called normal form symmetry S1, inherited from the translation invariance of periodic states. As a result, it predicts correctly the existence of heteroclinic orbits along the curve ␮ = ␮D but misses the heteroclinic orbits present in a neighborhood of ␮ = ␮D. To find such orbits, one has to compute terms beyond all orders; these terms break the S1 normal form symmetry, thereby restoring the symmetry of the original problem. In addition these terms lead to transversal crossing of the stable and unstable manifolds of u07,29,35 thereby broadening the curve ␮ = ␮D into a pinning region. Terms of this type were computed by Kozyreff and Chapman36 for a particular model equation, and their explicit computations shed considerable light on the properties of the normal form equation when the normal form symmetry is absent. V. LOCALIZED STATES

The leading-order amplitude equation in 共34兲 is used to approximate several branches of solutions which emerge simultaneously from the bifurcation at r = 0. These are given by A共X兲 =

冉 冊

1 ␮2 − 2qc q2

1/2

ei␾ + O共⑀兲,

共37兲

corresponding to spatially periodic states with period ␭c ⬅ 2␲ / qc near r = 0; viz.,

冉冊

1/2

u P共x兲 =

1 r qc q2

A共X兲 =

2␮2 1 − 2qc q2

and

cos共qcx + ␾兲 + O共r兲,

冉 冊

冉冑 冊

1/2

sech

X ␮ 2 i␾ e + O共⑀兲, 2qc

共38兲

共39兲

corresponding to the spatially localized states uᐉ共x兲 =

冉 冊

1 2r qc q2

1/2

冉冑 冊

sech

x −r cos共qcx + ␾兲 + O共r兲. 2qc 共40兲

In Eq. 共38兲 the phase ␾ 苸 S1 is a consequence of spatial invariance while in Eq. 共40兲 it controls the phase of the pattern within the sech envelope. Within the amplitude equation, this phase remains arbitrary to all orders in ⑀. In the following. we use the continuation package 37 AUTO to extend the results in Eqs. 共38兲 and 共40兲 away from r = 0. Without loss of generality, we choose qc = 1 and b5 = 1.

Lx/2 2 FIG. 3. Bifurcation diagram showing the norm N ⬅ 共Lx−1兰−L u dx兲1/2 of x/2 various one-dimensional stationary solutions of Eq. 共1兲 with the nonlinearity f 35, including the trivial state u0, the spatially periodic states u P共x兲, and the spatially localized states uᐉ共x兲 that make up the snakes-and-ladders structure of the pinning region r P1 ⬍ r ⬍ r P2 共shaded兲. This region straddles the Maxwell point rM ⯝ −0.6752. Only the lowest four rungs of the ladder are shown. Solid 共dotted兲 lines indicate stable 共unstable兲 solutions in one dimension; thick solid lines indicate solutions that are stable with respect to twodimensional perturbations. Parameters: qc = 1, b3 = 2, b5 = 1.

Moreover, the unfolding that led to these approximate solutions assumes that b3 = O共1兲, and we choose b3 = 2 throughout the remainder of this section. Numerical continuation of the spatially periodic branch starting from Eq. 共38兲 is complicated by the existence of a continuum of bifurcations to states with wavenumber k near the critical value of qc that bifurcate from u0 at positive values of r. Thus, there is in fact a two-dimensional manifold of finite amplitude spatially periodic states, parametrized by r and k 共in addition to the arbitrary phase ␾兲, which emerges from the trivial state. Three different one-dimensional cuts through this manifold of solutions are of particular interest. The first is the branch of solutions with wavenumber fixed to its critical value k = qc. This branch, labeled u P, is shown in Fig. 3. As anticipated, this branch undergoes a saddle-node bifurcation in r ⬍ 0. In a domain of size ␭c = 2␲ / qc, the pattern is unstable below the saddle-node and stable above; in fact, over the r range shown in the figure the states that make up the upper branch are stable in any domain of size Lx = n␭c for any finite integer n. Thus, there is bistability between u0 and u P in the range of r above the saddle-node and below r = 0. The second cut is defined by the condition 兩dF / dk兩r = 0. This condition selects the wavenumber k that minimizes the energy F at each r; i.e., the energetically most favored pattern on the real line. The third cut is defined by the condition H = 0. This surface is of interest because all homoclinic and heteroclinic connections to u0 must lie within it. The branches selected by these three approaches all emerge from the trivial state at r = 0 and behave in a qualitatively similar way away from r = 0. Figure 4 compares the wavelength ␭, the energy density F / Lx, and the Hamiltonian H for each case. This figure also shows the location of the Maxwell point defined by F = H = 0 共see below兲.

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037102-6

J. Burke and E. Knobloch

Chaos 17, 037102 共2007兲

FIG. 5. Sample localized profiles at locations indicated in Fig. 3: 共a–c兲 lie on the ␾ = 0 snaking branch, while 共d–f兲 lie on the ␾ = ␲ / 2 snaking branch.

FIG. 4. 共a兲 Wavelength ␭ relative to the critical value ␭c, 共b兲 energy density F / Lx, and 共c兲 Hamiltonian H along the “upper” branch of the three spatially periodic states discussed in the text for the parameter values used in Fig. 3. The solid line corresponds to the branch defined by k = qc, the dashed to 兩dF / dk兩r = 0, and the dash-dotted to H = 0. The Maxwell point rM is defined by F = H = 0. The symbols 〫 in 共a兲 indicate the wavelength of the pattern within the localized state, as measured between the 43rd and 44th saddle-nodes on the ␾ = 0 snaking branch.

Care must be taken when extending the localized states identified in Eq. 共40兲 to finite amplitude. The numerical routine used in this case finds one-dimensional solutions to Eq. 共1兲 on a large but finite domain Lx 共typically, 50␭c兲, subject to periodic boundary conditions. The errors introduced by approximating the homoclinic orbit with a large period orbit are small, scaling as e−兩Re ⌳兩Lx 共see Ref. 38 and references therein兲. Treatment of the phase ␾ is particularly delicate. Within the asymptotics, this phase remains arbitrary: there is no locking between the envelope and the underlying wavetrain at any finite order in ⑀. However, it is known36,39–41 that this is no longer the case once terms beyond all orders are included. These terms break the “rotational” invariance of the envelope solution, and result in a weak flow on the circle S1. This flow in turn selects specific values of the phase: ␾ = 0 , ␲ / 2 , ␲ , 3␲ / 2. As mentioned above, these terms also lead to transversal crossing of the stable and unstable manifolds of u07,29,35 thereby producing the snaking that becomes so prominent farther away from r = 0. Note that the phases ␾ = 0 , ␲ are the only two phases that preserve the symmetry 共x → −x, u → u兲, while ␾ = ␲ / 2, 3␲ / 2 are the only two that preserve the symmetry 共x → −x, u → −u兲.26 The results of following each of these four branches of localized states are shown in Fig. 3. Near the origin the amplitude of the states is small and the width of the sech envelope is large enough to contain many wavelengths of the underlying pattern 关Figs. 5共a兲 and 5共d兲兴. Away from the origin the amplitude grows and becomes comparable to the amplitude of the patterned states above the saddle-node bifurcation, and the width decreases until it is comparable to ␭c, the approximate wavelength of these states 关Figs. 5共b兲 and 5共e兲兴. Beyond this point the branches undergo a series of saddle-node bifurcations responsible for the terminology homoclinic snaking.1,2 Each saddle-node bifurcation adds a pair of oscillations to the profile uᐉ共x兲, and the saddle-node bifur-

cations asymptote rapidly to the points labeled r P1 and r P2 关Figs. 5共c兲 and 5共f兲兴. The profiles along the ␾ = 0 , ␲ branches are always even and are related by the transformation u → −u; along the ␾ = 0 branch, the midpoint 共x = 0兲 of the localized state is always a local maximum, while along the ␾ = ␲ branch, the midpoint is always a local minimum. The profiles along the ␾ = ␲ / 2, 3␲ / 2 branches are also related by u → −u but are always odd; along the ␾ = ␲ / 2 branch, the midpoint of the localized state always has maximum negative slope, while along the ␾ = 3␲ / 2 branch, the midpoint always has maximum positive slope. Secondary symmetry-breaking bifurcations on the even and odd snaking branches occur slightly below the saddlenodes near r P1 and slightly above the saddle-nodes near r P2, and result in branches of asymmetric localized states that connect the even and odd snaking branches. These form a series of “rungs” resulting in the ladder-like structure shown in Figs. 3 and 6. Each rung actually corresponds to four branches of asymmetric states 共Fig. 7兲 related by the transformations x → −x and u → −u. Figure 8共a兲 shows how the branches of localized states are connected: the four asymmetric branches connect each of the two even branches with each of the two odd branches, but the two even 共odd兲 branches are not directly connected. High up the snaking branches these secondary bifurcations rapidly approach the saddle-nodes. Note that in nonvariational systems the asymmetric solutions would drift and hence be time dependent. Thus, at each value of r within the snaking region there exists an infinite number of stationary localized profiles, each of a different width. The profile shown in Fig. 1 is from high

FIG. 6. Detail of Fig. 3 showing the rungs of asymmetric branches that connect the even and odd snaking branches.

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037102-7

Homoclinic snaking

Chaos 17, 037102 共2007兲

FIG. 7. Sample localized profiles at locations indicated in Fig. 6: 共a, f兲 lie on the ␾ = ␲ / 2 branch while 共c, d兲 lie on the ␾ = 0 branch; 共b, e兲 lie on the asymmetric rungs.

up the ␾ = 0 snaking branch and consists of a periodic state terminated at either end by a “front” of width of order ␭c that connects it to the trivial state u0. The wavelength and amplitude of the resulting localized periodic state are close to those of the extended periodic state u P. From the spatial point of view the broad localized profiles correspond to homoclinic orbits that spend a long “time” near a periodic orbit. Because of the Hamiltonian structure of the present system in space the periodic orbit approached by such a homoclinic orbit must have H = 0, a fact confirmed by measuring the wavelength of the pattern inside the localized state 关Fig. 4共a兲兴. These results suggest that within the snaking region there also exist heteroclinic orbits between u0 and the H = 0 periodic orbit. The broad localized states that exist high up each snake approximate a heteroclinic cycle consisting of two of these heteroclinic orbits, one from u0 to u P and the other from u P back to u0. Evidently at each r there are two possible front profiles, ␰1共x兲 and ␰2共x兲, connecting u0 to u P. Using just ␰1 共and its reflections x → −x, and u → −u兲, one can construct a total of four localized states corresponding to the two even and two odd solutions between, for example, the 11th and 12th saddle-nodes on the 共four兲 snaking branches. The second front ␰2 generates the corresponding solutions between, for example, the 12th and 13th saddlenodes. Finally, the asymmetric combinations generated by using ␰1 for one front and ␰2 for the other correspond to the solutions on the rungs. The numerical observations described above are consistent with the behavior predicted in Ref. 36, although allow-

FIG. 8. Schematic diagram showing the number and connectivity of the branches of localized states. In 共a兲 f = f 35, and a single rung consists of four branches connecting each constant phase branch with its two neighbors. In 共b兲 f = f 23, and each rung consists of only two branches.

FIG. 9. Spectrum of growth rates ␤ along the ␾ = 0 snaking branch, shown as a function of the arc length s along the branch measured from the bifurcation at the origin 共upper panel兲. Solid lines correspond to ␤共ky = 0兲; the 共even兲 amplitude and 共odd兲 phase modes are labeled by A and P, respectively. The dashed lines show the even 共W+兲 and odd 共W−兲 wall modes at ky = 0.89. The lower panel shows the branch as a function of s. The spectrum along the ␾ = ␲ branch is identical to that shown here; that of the odd snaking branches is qualitatively similar, except for the change in parity 共Ref. 26兲.

ance must be made for the difference in the models and parameter regimes. In Ref. 36 the authors consider a system which, after suitable transformations, corresponds to Eq. 共1兲 with f 23 nonlinearity. The analysis is performed in a neighborhood of the heteroclinic orbit between the trivial and spatially periodic states, with the nonlinear coefficient c2 treated as the bifurcation parameter. The analysis assumes that this parameter is close to its critical value, and hence applies near the codimension-two point analogous to 共␮ , q2兲 = 共0 , 0兲. The authors show that near this codimension-two point fronts of width ␭c␦ between the trivial and spatially periodic states exist in a O共␦−2e−␲/␦兲 neighborhood of the Maxwell point, where ␦ is a small parameter measuring the distance of c2 from its critical value, and construct stationary localized states by assembling two of these fronts back to back. This construction reproduces fully the snakes-and-ladders structure described above, and moreover predicts that the distance in parameter space between the saddle-nodes on the snaking branches and the boundary of the pinning region decreases as e−w␦, where w is the width of the localized state 共and likewise for the distance between these saddle-nodes and the secondary bifurcations to the rungs兲. The results described above confirm and extend these predictions to parameter values far from the codimension-two point where ␦ = O共1兲. In this regime, the width of the fronts is relatively narrow, of order ␭c, and the width of the pinning region is of order 1 instead of being exponentially narrow. Figure 3 also indicates the stability of the localized solutions in time, a consideration that is absent from the general theory of spatially reversible systems. The eigenvalue problem in Eq. 共4兲 at ky = 0 yields the spectrum of growth rates of infinitesimal one-dimensional perturbations of the localized states at each point along a branch 共Fig. 9兲, as well as the associated eigenfunctions. On the even 共␾ = 0 , ␲兲 and odd 共␾ = ␲ / 2 , 3␲ / 2兲 snaking branches there are three critical

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037102-8

J. Burke and E. Knobloch

FIG. 10. From top to bottom, localized solution profile high up the ␾ = 0 snaking branch, and the eigenfunctions of the amplitude, phase, and Goldstone modes. Parameters are the same as in Fig. 1共a兲.

eigenfunctions: an amplitude mode that has the same parity as the base state, a phase mode with opposite parity, and the neutrally stable Goldstone mode. All other modes are strictly stable. Near the origin, the phase modes are associated with a slow drift in phase; this drift selects the four phases, two stable 共␾ = 0 , ␲兲 and two unstable 共␾ = ␲ / 2 , 3␲ / 2兲 with respect to phase perturbations. In addition, all four branches are amplitude unstable. As the pinning region is approached, the growth rates of the amplitude and phase modes start to oscillate in sign. Each zero crossing of the growth rate of the amplitude mode corresponds to a saddle-node bifurcation on the corresponding branch, while the zero crossings of the phase modes mark the locations of secondary pitchfork bifurcations. From the figure it is clear that the stability of the localized states changes at each saddle-node, resulting in an alternating sequence of stable and unstable states along each snaking branch. Moreover, the secondary bifurcations always occur on the unstable segments of each primary branch, although they occur closer and closer to the saddle-nodes as one moves up the branch; it follows that the asymmetric states that emerge from these secondary bifurcations are always unstable. For the broad localized states that exist high up on the snaking branches, the amplitude and phase eigenvalues are nearly identical, and the corresponding eigenfunctions are localized near the fronts, as shown in Fig. 10. These stability results can be understood using the variational structure of Eq. 共1兲. Although u0 and u P always represents global extrema, a large variety of local extrema is present within the pinning region. Of the two heteroclinic orbits ␰1 and ␰2, which exist at each value of r within this region, one corresponds to a local minimum and the other a local maximum. The former is stable while the latter is unstable. Evidently any localized state consisting of two stable bound fronts is also stable, as is the case for all four snaking branches between, for example, the 11th and 12th saddlenodes. In contrast, localized states bounded on either side by a front made up of an unstable heteroclinic orbit inherit the front instability and are therefore also unstable. This is the case, for example, between the 12th and 13th saddle-nodes on all four snaking branches, and for all asymmetric states.

Chaos 17, 037102 共2007兲

FIG. 11. The pinning region r P1 ⬍ r ⬍ r P2 共light shaded兲 with solutions heteroclinic to the trivial state. Dashed lines correspond to the Maxwell points rM and rM⬘. The dark shaded region corresponds to the location of stable two-dimensional localized stripes, and is bounded by the onset of wall modes at lower b3, and wall and body modes on the left and right, respectively, at larger b3 共see inset兲. The results are based on properties of solutions between the 11th and 12th saddle-nodes on the ␾ = 0 snaking branch and provide a very good approximation to the exact stability region.

There is a simple physical explanation for the existence of an extended snaking region42 as well as for the wavelength selection within this region. In general, one might expect that a front between the trivial and patterned states will be stationary only at the Maxwell point, defined by the twin conditions F = H = 0,43,44 and that for r ⬍ rM such a front should drift so as to fill the domain with the lower energy flat state. However, this drift is frustrated by the pinning of the front to the periodic structure, resulting in stationary fronts even for r ⬍ rM; these pinned fronts are sometimes called Pomeau fronts. The same scenario holds when r ⬎ rM, where the front should drift so as to fill the domain with the patterned state. It is this pinning, therefore, that opens out the Maxwell point into a pinning region and is responsible for the existence of multiple stationary localized states in this region. Moreover, the energy difference between the u0 and u P states manifests itself in the wavelength of the periodic structure within the localized state: for r ⬍ rM the pattern is compressed relative to its energetically preferred value, while for r ⬎ rM it is stretched, in agreement with the requirement that H = 0 关Fig. 4共a兲兴. Once r ⬍ r P1, the energy difference exceeds the pinning force and the fronts unpin, leading to the gradual erosion of the periodic state and its replacement by u0. Conversely, once r ⬎ r P2, the motion of the fronts leads to symmetrical nucleation of new structures on either side of the localized state, and hence to its steady growth in time. Near r P1 and r P2, the speed of the moving front is computable analytically and the results agree well with simulations.25,45 VI. LOCALIZED STATES AS A FUNCTION OF b3

Thus far we have studied the behavior of the SwiftHohenberg equation as r varies for qc = 1, b3 = 2, and b5 = 1. Up to rescaling the solutions of Eq. 共1兲 are parametrized by 共r , b3兲. Figure 11 shows the region of existence of heteroclinic cycles between the trivial state u0 and the periodic orbit with H = 0, i.e., the pinning region. At each point within this region there exists a multiplicity of stable spatially lo-

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037102-9

Homoclinic snaking

calized homoclinic states. Near the codimension-two bifurcation at 共0 , 0兲, the Maxwell point is well approximated by the heteroclinic orbit at r = rD from normal form theory.25 Near this point, the analysis in Ref. 36 is directly applicable, and demonstrates that the pinning region is in fact exponentially narrow, its width scaling as ␦−2e−␲/␦. At larger values of b3, both the location of the Maxwell point and the extent of the pinning region must be determined numerically. The bifurcation diagram in Fig. 3 corresponds to a horizontal slice through this figure at b3 = 2 and is typical of the behavior below b3 ⯝ 3.521. Above this value of b3, a new Maxwell point, labeled rM⬘, becomes dynamically important. This point corresponds to equal energy between the trivial flat state u0 and a nontrivial flat state u1 ⫽ 0. For b3 ⲏ 3.521, this new Maxwell point destroys the pinning region around rM,25,26 and the four branches of localized states created at r = 0 now snake towards rM⬘ instead of rM. However, in this case the pinning force between the two fronts at either end of the u1 intrusion decays exponentially with the length of the intrusion, and the new snakes therefore collapse to the Maxwell point rM⬘.46 The process whereby the localized states switch from one Maxwell point to another deserves further study; such transitions occur once the pinning regions corresponding to different Maxwell points start to overlap, and likely bear resemblance to other problems of this type.47 VII. LOCALIZED STRIPES IN TWO DIMENSIONS

The one-dimensional stationary localized states found in the previous section correspond to stationary localized stripes in two dimensions. Those profiles uᐉ共x兲 that are unstable in one dimension will necessarily generate localized stripes in two dimensions that are also unstable. However, the profiles that are stable in one dimension are not necessarily stable in two because of various transverse instabilities that may be present. The goal in what follows is to determine which 共if any兲 of the localized stripes are stable by solving Eq. 共4兲 for ky ⫽ 0. Stability in two dimensions requires ␤共ky兲 艋 0 for all modes and for all wavenumbers. We find that there are two types of y-dependent modes that can lead to instability. We refer to the first of these as a “wall” mode since it is associated with an eigenfunction ˜u共x ; ky兲 that is localized around the Pomeau fronts and does not effect the pattern in the interior of the localized stripe. A typical dispersion relation at a parameter value where this mode leads to instability is shown in Fig. 12共a兲. The resulting instability leads to ripples along each front as shown in Fig. 13共a兲; the length of these ripples corresponds to the wavenumber of the wall mode with maximum growth rate, typically ky ⬃ 0.9. There are in fact two wall modes, one even and one odd, corresponding to the ripples along the two fronts forming in phase or out of phase with each other. These wall modes have nearly identical growth rates, a consequence of the exponentially weak interaction between the left and right fronts. The second instability mode is a “body” mode that affects the pattern over the entire width of the localized stripe. The parity of this mode is opposite the parity of the base state, and is closely related to the well-known zigzag insta-

Chaos 17, 037102 共2007兲

FIG. 12. Dispersion relation for two-dimensional modes at two different 共r , b3兲 values, as a function of the wavenumber ky of the transverse disturbance, computed for a profile between the 11th and 12th saddle-nodes on the ␾ = 0 branch of localized states; this profile is stable in one dimension. In 共a兲 the unstable modes are the odd and even parity wall modes, while in 共b兲 the unstable mode is an odd parity body mode. Parameters: qc = 1, b5 = 1; 共a兲 r = −0.7104, b3 = 2; and 共b兲 r = −0.9183, b3 = 2.5.

bility of a one-dimensional spatially periodic structure,30 i.e., it is a long-wave instability that leads to the destabilization of the neutrally stable Goldstone mode. Consequently its growth rate is well approximated by ␤2 and can be computed from Eq. 共7兲 using ˜u0 = uᐉ⬘共x兲. Since all large ky modes necessarily decay the growth rate of the body mode always peaks at a small but finite ky, typically ky ⬃ 0.2 关Fig. 12共b兲兴. Figure 13共b兲 shows the initial stage of the growth of this mode when ␤2 ⬎ 0. A second body mode with the same parity as the base state, related to the varicose instability of the periodic structure, is typically strongly damped and will not be considered further. The two-dimensional instabilities described above 共Fig. 13兲 are related to the three previously identified critical modes present at ky = 0 共Fig. 10兲. At small values of ky the two wall modes are identified with the amplitude and phase modes, and the body mode with the Goldstone mode. The amplitude mode remains localized around the Pomeau fronts at larger values of ky and is thus always identified with one of the wall modes. The behavior of the phase and Goldstone modes, both of which have the same parity, is more complicated. When the phase mode is unstable at ky = 0, it retains its shape as ky increases, and becomes a second wall mode. Elsewhere, the damped phase mode begins to interact with the neutrally stable Goldstone modes as ky increases, leading to an avoided crossing in the dispersion relation near ky

FIG. 13. Grayscale plot showing the initial evolution of 共a兲 an unstable odd-parity wall mode, and 共b兲 an unstable odd-parity body mode. In both cases the solution evolves beyond this initial stage to finite amplitude 共see below兲. Parameters are as in Fig. 12. The domain shown is 共x , y兲 苸 关−90, 90兴 ⫻ 关−30, 30兴.

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037102-10

Chaos 17, 037102 共2007兲

J. Burke and E. Knobloch

FIG. 14. Summary of the two-dimensional stability results for localized stripes across the entire pinning region 共r P1 ⬍ r ⬍ r P2兲 at two values of b3. The results are calculated using profiles between the 11th and 12th saddlenodes on the ␾ = 0 branch of localized states, all of which are stable in one dimension. Shading indicates the existence of at least one unstable transverse mode at the corresponding value of 共r , ky兲. Stable localized stripes are found in 共a兲 between rW1 and rB, and in 共b兲 between rW1 and rW2. Parameters: qc = 1, b5 = 1; 共a兲 b3 = 2.5; and 共b兲 b3 = 2.

⬃ 0.3 关Fig. 12共a兲兴. This interaction leads to an exchange in the shape of the two modes: at larger values of ky the Goldstone mode, which has nonzero amplitude throughout the interior of the localized state, transforms itself into the second wall mode, localized around the Pomeau fronts, while the phase mode turns into the body mode. These two instability modes reduce the range of stability of the one-dimensional localized structures within the pinning region. Figure 14 summarizes the result of solving the two-dimensional eigenvalue problem 共4兲 for the localized states between the 11th and 12th saddle-nodes on the ␾ = 0 branch, spanning the entire pinning region; all these states are stable with respect to one-dimensional perturbations. The stability region is reduced to rW1 ⬍ r ⬍ rW2, where rW1, rW2 correspond to the onset of instability of a wall mode as r decreases or increases 关Fig. 14共b兲兴, or to rW1 ⬍ r ⬍ rB, where rB denotes the onset of the body mode 关Fig. 14共a兲兴. The former situation is characteristic of smaller values of b3, while the latter prevails for larger b3. The extent of the region of stability as b3 varies is shown in Fig. 11; evidently there is a critical value of b3 below which no stable twodimensional stripes exist. The unstable localized stripes evolve into several distinct states depending on the dominant mode of instability. When the unstable mode is a wall mode the fronts break up into a series of spots 关Fig. 15共a兲兴, but the stripes near the center of the localized structure may not be noticeably affected by the breakup. In this case the fronts remain pinned to the underlying periodic stripe pattern, resulting in a new class of stable localized states bounded by a pair of fronts with structure in the y direction. If the initial instability corresponds instead to a body mode, we find that the long time behavior leads to the destruction of the background flat state as the fronts drift outward. This leads either to a domain-filling zigzag pattern 关Fig. 15共b兲兴, or a labyrinthine structure that resembles a pat-

FIG. 15. Grayscale plots of several solutions with f 35 nonlinearity. The solution in 共a兲 is a steady state while the arrows in 共b, c兲 indicate that the fronts continue to propagate outward. Parameters: qc = 1, b5 = 1; 共a兲 r = −0.7105, b3 = 2; 共b兲 r = −0.9183, b3 = 2.5; and 共c兲 r = −1.227, b3 = 3. The domain shown is 共x , y兲 苸 关−120, 120兴 ⫻ 关−30, 30兴.

terned state oriented perpendicular to the original stripes 关Fig. 15共c兲兴; both states are selected dynamically by the outward propagating fronts. Similar patterns have been observed in the supercritical Swift-Hohenberg equation 共b3 = −1, b5 = 0兲 during the breakup of localized states involving a pair of symmetry-related flat states ±u1 ⫽ 0 instead of the trivial and spatially periodic states involved here.48 VIII. RESULTS FOR QUADRATIC/CUBIC NONLINEARITY

In this section we summarize the corresponding results for the f 23 nonlinearity. Because of the loss of u → −u symmetry, even-order terms are now present at various stages in the computation of the normal form coefficients. One finds25,32 p1 = −

q1 =

1

3,

8qc

1

2,

4qc

q4 = −

p2 =

q2 =

177c23 128q6c

+

9c3 16q3c

3c3 4q2c

−

−

187c22

7,

216qc

19c22

6,

18qc

5089c22c3 288q10 c

−

q3 =

78131c42 7776q14 c

p3 = − 3c3 8q3c ,

−

8c22 9q8c

共41兲

,

41c22 108q7c

,

共42兲

共43兲

while the coefficient ␳ is

␳=−

9c3 16q4c

+

355c22 216q8c

.

共44兲

In this case the condition q2 ⬍ 0 corresponds to c22 ⬎ 27c3q4c / 38, and implies again that the bifurcation at r = 0 is subcritical. With c3 ⬎ 0, we find again that q4 is positive, at least when q2 is small. Thus, for the f 23 nonlinearity, localized states emerge from the codimension-two point 共r , c22兲 = 共0 , 27c3q4c / 38兲. Moreover, the analytical results 共38兲 and 共40兲 remain valid near r = 0. The three spatially periodic branches found by continuing the result 共38兲 to larger amplitudes behave in a qualitatively similar fashion to the f 35 case 关Fig. 16兴. However, due to the reduced symmetry of the problem the beyond all

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037102-11

Chaos 17, 037102 共2007兲

Homoclinic snaking

FIG. 16. 共a兲 Wavelength ␭ relative to the critical value ␭c, 共b兲 energy density F / Lx, and 共c兲 Hamiltonian H along the “upper” branch of three spatially periodic states for the f 23 nonlinearity. The Maxwell point rM is defined by F = H = 0. Notation is as in Fig. 4. Parameters: qc = 1, c2 = 1.64, c3 = 1.

orders terms select only two phases of localized states, ␾ = 0 , ␲, when the state 共40兲 is continued in r ⬍ 0. The resulting bifurcation diagram is very similar to that shown in Fig. 3, and again consists of two intertwined snaking branches connected by a series of rungs. However, in the present case the ␾ = 0 and ␾ = ␲ states are distinct, unrelated by u → −u, and each rung connecting these branches is only doubly degenerate 关Fig. 8共b兲兴. The resulting pinning region is shown in Fig. 17 in the 共r , c22兲 parameter plane. Once again, the stability interval of the localized stripes within this region is reduced by the presence of two-dimensional instabilities of the type already discussed, although in the present case rB always lies below rW2 and hence always forms the upper stability boundary. Outside of the stability interval 共but still inside the pinning region兲, the evolution of the two-dimensional instability reveals new features, related to the presence of the quadratic term in the equation. Once again the wall mode causes the breakup of the fronts into spots but these now evolve into a hexagonal pattern. This pattern may spread inward, eliminating the stripes, as well as outward, eliminating the uniform

FIG. 17. The pinning region r P1 ⬍ r ⬍ r P2 共light shaded兲 of solutions heteroclinic to the trivial state. Dashed lines correspond to the Maxwell points rM and rM⬘. The dark shaded region indicates localized stripes that are stable in two dimensions. The results are based on properties of solutions between the 11th and 12th saddle-nodes on the ␾ = 0 snaking branch and provide a very good approximation to the exact stability region.

FIG. 18. Grayscale plots of several solutions with the f 23 nonlinearity. In 共a兲 the arrows indicate that the hexagons continue to invade both the inner striped regions and the outer flat region. In 共b兲 the solution is a steady state. In 共c兲 the fronts continue to propagate outward. Parameters: qc = 1, c3 = 1; 共a兲 r = −0.2175, c2 = 1.64; 共b兲 r = −0.4749, c2 = 2; and 共c兲 r = −0.5990, c2 = 2.6. The domain shown is 共x , y兲 苸 关−120, 120兴 ⫻ 关−30, 30兴.

background state 关Fig. 18共a兲兴. At some values of the parameters the front only spreads inward, producing a stable, stationary localized band of hexagons 关Fig. 18共b兲兴; such stripes have also been identified in Ref. 24, where it is shown that they exhibit the type of snaking illustrated in Fig. 3. In contrast, when the stripe is destabilized by a growing body mode it evolves into outwardly propagating structures oriented perpendicular to the original pattern 关Fig. 18共c兲兴. Thus, in this case a long wavelength zigzag instability ultimately leads to an O共␭c兲 spacing in the outward propagating pattern as well as the zigzag structure associated with both orientations of the pattern. Patterns of this type are selected dynamically by the outward propagating fronts. IX. APPLICATIONS

The results described above shed light on the behavior observed, either experimentally or numerically, in a number of systems of interest. We mention several in this section. A. Systems modeled by the Swift-Hohenberg and related equations

Although the Swift-Hohenberg equation was originally derived in the context of infinite Prandtl number convection it is a basic 共albeit variational兲 equation that exhibits a finite wavenumber instability at onset 共r = 0兲. This property distinguishes it from a number of envelope or phase equations such as the Kuramoto-Sivashinsky equation which are typically derived using a long-wave expansion. For this reason, it provides a convenient model of spontaneous steady state instabilities with a finite wavenumber. Such instabilities are often referred to as Turing instabilities although no implication as to the physical origin of the instability is intended. However, since the equation is of fourth order in space as well as reversible it comes as no surprise that much of the behavior described here is also found in two-species reaction-diffusion equations, such as the Brusselator or GrayScott models.9 In particular, the self-replication observed in the Gray-Scott model, and reproduced in experiments,8 has been explained by the evolution of the system slightly above

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the pinning region, where the patterned state is favored over the homogeneous flat state. The closer the system is to r P2 the longer are the “gestation” periods between successive nucleation events. In the Gierer-Meinhardt model the breakup of two-dimensional homoclinic stripes and the resulting patterns resemble the results described above for the Swift-Hohenberg equation.49 Buckling problems1,2 offer a ready application of the theory, as does the Korteweg–de Vries equation for long wavelength water waves with higher order dispersion.30 These applications give rise directly to the Swift-Hohenberg equation, although the equation is parametrized differently. For example, in buckling problems the load enters as a parameter in front of the second derivative term, while in the KdV equation the parameter is the speed of the wave and enters in the constant term only. Thus, each of these applications corresponds to a different slice through the parameter space of the system and in particular through the pinning region.1 This fact may result in the absence of some types of behavior in some applications, but the totality of behavior is identical to that described here. Weakly bistable optical systems with weak dispersion are modeled by the cubic Swift-Hohenberg equation with a constant term.3 As a result the homogeneous state is parameter-dependent, but a simple shift of the variables transforms the problem into the quadratic-cubic case studied here. The multiplicity of localized states found in this system is thus identical to that described here. When the system is more strongly bistable it becomes nonvariational but the observed structures5,6 continue to resemble the behavior of the Swift-Hohenberg equation in two dimensions.23,24 The behavior observed in recent experiments on a ferrofluid in an imposed normal magnetic field18 have also been interpreted in terms of the 共quadratic-cubic兲 SwiftHohenberg equation. The complete equations describing this system predict a finite wavenumber at onset of the primary instability. This instability is subcritical and leads to bistability between the undeformed surface and a hexagonal pattern of peaks. In a specific regime a finite amplitude perturbation can produce a stable localized peak, or several peaks, that coexist with the stable flat state. Indeed, a large multiplicity of localized states consisting of different collections of peaks has been constructed in this way, all at the same value of the applied magnetic field. This behavior is characteristic of the pinning region in equations such as the Swift-Hohenberg equation in one and two dimensions.23,24 B. Convectons

Ever since their discovery in magnetoconvection, localized convective structures called convectons13 have been of great interest. Fundamentally, this is because one expects a fluid uniformly heated from below to respond similarly at all locations. While this is true of pure fluid convection in which bistability is absent this is not true of bistable systems such as magnetoconvection or convection in a binary mixture. Recent results12 show how closely the behavior of a waterethanol mixture heated from below 共Fig. 19兲 resembles that of the Swift-Hohenberg equation, despite the fact that here the pinning region is present in a parameter regime in which

FIG. 19. Bifurcation diagram showing the dimensionless convective heat transport as a function of the Rayleigh number R in a Lx = 60 periodic domain. The conduction state loses instability at Rc = 1760.8. Steady spatially periodic convection 共SOC兲 acquires stability at a parity-breaking bifurcation marking the destruction of a branch of spatially periodic traveling waves 共TWs兲. Above the threshold, small amplitude dispersive chaos is present 共solid dots兲, which leads into the pinning region 共1774⬍ R ⬍ 1781兲 containing a multiplicity of stable localized states of both even and odd parity. 共After Ref. 12.兲

the trivial 共conduction兲 state is already unstable to a Hopf mode. As a result at values of r just below r P1, a convecton gradually erodes and collapses to an unstable state that then regrows another convecton, resulting in irregular relaxation oscillations.12 On the other hand, the transition to the patterned or fully convecting state just above r P2 follows the behavior seen in the cubic-quintic Swift-Hohenberg equation and occurs via repeated nucleation of rolls on either side of the convecton. For some parameter values adjacent convectons may be separated by the conduction state, while in others the ”space” between them may fill with traveling waves,11 much as observed in certain optical systems.4 C. Neural systems

Networks of neural systems are described by integrodifferential equations. When the kernel has a suitable form these equations can be Fourier-transformed, resulting in a partial differential equation. It is of interest that the equations that result also exhibit the phenomena associated with the presence of a pinning or snaking region, including multiple spatially localized states in both one and two spatial dimensions.16,17 X. DISCUSSION

In this paper we have described the origin and properties of spatially localized structures in the one-dimensional Swift-Hohenberg equation with either a quadratic-cubic nonlinearity or a cubic-quintic nonlinearity. To do this we have employed the technique of spatial dynamics to formulate the search for such structures as a search for homoclinic orbits to the trivial state. Such orbits can be located in a neighborhood of a reversible Hopf bifurcation with 1:1 resonance that occurs at r = 0. At this bifurcation point a branch of periodic

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states bifurcates subcritically, simultaneously with either two or four branches of spatially localized states. We have described a relatively simple technique for computing the normal form coefficients from the Swift-Hohenberg equation required by the theory, and have used the analytical approximations to the localized states obtained from this theory to initialize the branch-following computations that allowed us to follow the 共unstable兲 branches of localized states into what we have called the snaking or pinning region. In this region the primary branches of localized states undergo a series of saddle-node bifurcations at which they repeatedly gain and lose stability. As one follows the primary branches the localized states broaden, nucleating structures at either end in such a way as to preserve the parity of the state. Thus, high up on a snaking branch, the localized structure approximates a heteroclinic cycle, consisting of a pair of back-to-back heteroclinic connections, one from the trivial state to a periodic state, and the other back to the trivial state. In addition, we have seen that the primary branches are connected by a series of secondary branches of localized states with no symmetry, forming a characteristic snakes-andladders structure. The analysis of the Swift-Hohenberg equation is greatly aided by the fact that it is variational in time, and 共in one dimension兲 Hamiltonian in space. The Maxwell point is defined by the two conditions F = H = 0. We have seen that near this point there exist heteroclinic cycles to periodic orbits that must lie within the H = 0 surface, and showed that the wavenumber of that orbit does indeed agree well with the measured wavenumber of the structure within the localized state. This wavenumber differs from the wavenumber of the periodic state at onset, and also from the wavenumber selected energetically. We have suggested an intuitive explanation of this result based on pinning. This explanation predicts that for r below the Maxwell point the periodic structure should be compressed and vice versa for r above the Maxwell point. We therefore attribute the appearance of the snaking or pinning region to the broadening of the Maxwell point caused by the pinning of the fronts bounding localized states to the periodic structure within, and understand the snakesand-ladders structure as arising from the resulting frustration. There is much that we have not covered, even in the one-dimensional case. Of interest are the transitions resulting from the collision or overlap of different pinning regions associated with different Maxwell points, as well as the great variety of multipulse states, consisting of widely separated but nonetheless weakly bound localized states of the type we have described here.50,51 Additional interesting behavior arises when the solutions in the pinning region asymptote to a periodic state whose stability changes inside the pinning region, i.e., when the heteroclinic cycle that describes spatially broad but nonetheless localized states undergoes a change in stability as r varies within the pinning region. Since the stability properties 共in time兲 of this cycle are inherited by the nearby homoclinic states we expect these to lose stability, perhaps resulting in fission of broad localized states into shorter, more compact states. The behavior of the Swift-Hohenberg equation is dynamically very simple. This is a consequence of the varia-

Chaos 17, 037102 共2007兲

tional structure of the equation, which implies that the long time behavior of the system consists of steady states or, on the real line, of traveling fronts. In particular, no Hopf bifurcations are possible, and no sustained oscillations are present. However, since each stable state corresponds to a local minimum of the energy F, it is clear that the energy landscape in the pinning region is exceedingly complex, and in particular, the depths of the corresponding potential wells are in general unequal. As a result the addition of random fluctuations to the system leads to “tunneling” from a higher minimum to a lower minimum, implying that the fronts bounding states with higher energy unpin due to fluctuations, and the system jumps repeatedly to more and more stable states.28,45,52 Thus, fluctuations reduce dramatically the multiplicity of 共metastable兲 states that coexist within the pinning region, and may lead to the annihilation of the pinning region altogether. In spatially reversible systems that are not variational the pinning region is still associated with the formation of a heteroclinic cycle, but this cycle can no longer be computed by searching for a Maxwell point, since no analog to the energy F can be defined. In addition, the system may not be Hamiltonian in space. Despite these differences, the phenomenology identified in the Swift-Hohenberg equation persists. In particular, the notion of pinning still explains the presence of a broad pinning region, although it is not clear what the optimum choice is for the wavenumber of an extended pattern or how it differs for the localized states. In addition, the dynamics associated with depinning, i.e., outside of the pinning region where no stationary localized states exist, typically follow the behavior familiar from variational systems.11,12 There are, however, other possibilities as well. Particularly dramatic are the relaxation oscillations observed in binary fluid convection just below the pinning region.11,12 Behavior of this type cannot take place in variational systems. In the one-dimensional Swift-Hohenberg equation the localized structures always grow by adding to the structure from the outside, by nucleating additional “rolls” at the location of the two bounding fronts. However, this is not the only way localized structures can grow in length, and we describe other mechanisms in a forthcoming paper.53 In this paper we have extended existing results on the Swift-Hohenberg equation in one spatial dimension25,26 by examining the stability of the resulting localized stripes with respect to two-dimensional perturbations. We have found that in general stable two-dimensional localized stripes may exist, but that the region of stability is smaller than for stable one-dimensional localized states due to transverse instabilities. These instabilities are of two basic types: wall modes localized at the bounding fronts that lead to the breakup of each front into cellular structures along the boundary of the stripe, and a body mode that leads to a zigzag-like structure throughout the interior of the localized state. As in the classical zigzag instability of a roll pattern, the zigzag mode comes in in order to reduce the internal wavelength of the pattern, and is fundamentally a long-wave mode. Which of these states is preferred depends of course on the parameters of the problem, as does the fate of the state after long time.

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Of particular interest are the states that remain localized in the x direction but develop structure in the y direction. In the cubic-quintic Swift-Hohenberg equation we have seen that these consist of an inner striped region separated from the outer uniform region by a single layer of spots, while in the quadratic-cubic Swift-Hohenberg equation the inner region may be entirely filled with a hexagonal pattern. No doubt other possibilities exist as well. For these structures one can still think in terms of a pinning mechanism that binds the fronts to the underlying pattern, while outside of this region the fronts drift. The types of patterns left in the wake of this drift are selected dynamically, as opposed to quasistatically, and hence differ from the patterns one might otherwise expect. In particular, one expects and sees patterns with defects which correspond to local minima of the energy, but do not anneal. The study of this dynamic pattern selection is beyond the scope of the present paper.

ACKNOWLEDGMENTS

This work was supported by NSF under grants DMS0305968 and DMS-0605238. We are grateful to M. Sprague and A. Yochelis for helpful discussions. 1

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