Patterns described by Discrete and Continuous Dynamical systems

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Patterns described by Discrete and Continuous Dynamical systems

Proefschrift

ter verkrijging van

de graad van Doctor aan de Universiteit Leiden, op gezag van de Rector Magnificus Dr. D. D. Breimer,

hoogleraar in de faculteit der Wiskunde en Natuurwetenschappen en die de Geneeskunde, volgens besluit van het College voor Promoties

te verdedigen op woensdag 9 juni 2004 klokke 15.15 uur

door

Jos´e Antonio Rodr´ıguez

geboren te Baracaldo (Spanje) op 23 januari 1976

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Samenstelling van de promotiecommisie:

promotor: prof. dr. ir. L.A. Peletier

referent: prof. dr. A.R. Champneys (University of Bristol)

overige leden: prof. dr. S. Verduyn Lunel prof. dr. R. van der Hout Dr. V. Rottssch¨afer

Dr. R.C.A.M. van der Vorst (Vrije Universiteit Amsterdam)

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The universe cannot be read until we have learned the language and become familiar with the characters in which it is written. It is written in mathematical language.

Galileo Galilei

If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in.

Richard Feynman

A mi familia, Bego˜na, Mari Carmen, Antonio y Patxi.

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Thomas Stieltjes Institute for Mathematics

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Preface

The subject of this thesis is the mathematical study of some special differential equations arising from different fields of science. In the introductory chapter we give a summary of the main results of the book. The other three chapters correspond to three papers, written together with prof. dr. ir. L.A. Peletier.

Chapter 2 corresponds to the paper Homoclinic orbits to a saddle-center in a fourth order differential equation, and it has been accepted for publication in the Journal of Differential Equations. Chapter 3 corresponds to the paper Fronts on a lattice, and was accepted by the journal Differential and Integral Equations.

Chapter 4 will be the content of another mathematical article to be submitted in the near future, The discrete Swift-Hohenberg equation.

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Chapter 1

Introduction

1.1 Classical second order differential equations

In the mathematical description of natural phenomena, a prominent role is played by differential equations already introduced in the times of Isaac Newton and Gottfried Wilhelm Leibnitz. Second order differential equations have been long studied since the formulation by Newton of the second law of motion.

In the field of partial differential equations (PDE’s), second order linear differential equations have been taken as models for the study of many physical phenomena, and some of them have special names like the Laplace equation, the wave equation and the heat equation.

These three equations can be written as

L^tu − ∆u = 0, (1.1)

where Lt is a linear differential operator with respect to t, with constant coeffi- cients. In the Laplace equation, L^t = 0, in the wave equation, L^t= _∂t^∂²2 and in the heat equation, Lt= _∂t^∂.

The three equations have something in common: they are linear, which means that if one has two solutions of the problem, then the sum of them, or any number of them, multiplied by a constant, is also a solution. They are also autonomous, which means that the differential equation or expressions explicitly depend on the unknown u, but not on time or space.

While these equations have been able to explain important physical phenomena or have been taken as models for different processes, in recent decades,

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particularly since the seventies, equations of a different type have been considered to describe phenomena that are shown to be impossible for equations of second order.

There exist many different generalizations of those second order equations and we do not attempt here to cover them all, not even the most important ones. It is also not our intention to investigate different general methods or ap- proaches. We are more interested in studying particular equations arising from different applications from physics (mainly) and try to study specific questions, like existence, uniqueness, stability and, specially, the qualitative properties of certain types of solutions, considered of particular interest. Some of those types are equilibria, travelling waves, periodic orbits, etc.

A slight modification of equation (1.1) is the following:

∂u

∂t − ∆u = f(u). (1.2)

Note that equation (1.2) looks similar to the heat equation, but we have added the nonlinear function f . This equation is called a reaction-diffusion equation:

the Laplacian term corresponds to diffusion and the function f corresponds to reaction. Note that roots of f correspond to homogeneus equilibria of equation (1.2). Typical choices for f are the parabolic function f (u) = u − u². Then, equation (1.2) is called a monostable equation. If f is given by the cubic function f (u) = u − u³, then equation (1.2) is called the bistable Fisher-Kolmogorov equation.

When we seek equilibria of equation (1.2), they must satisfy the equation

−∆u = f(u). (1.3)

Suppose that we consider equation (1.3) in one dimension, i.e. x ∈ . Then, phase-plane analysis can help us to describe completely all the equilibria. It can be rewritten as a second order differential system

du

dx = v, dv

dx = −f(u).

Let us define the Energy functional

E(u, v)^def= 1

2v²(x) + Z u(x)

0

f (s) ds. (1.4)

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Then, we find that the Energy is constant along orbits, i.e.

dE dx = 0.

Thus, when the initial data (u0, v0) is fixed, the orbit is a subset of the graph of E(u, v) = E0

def= E(u0, v0). Therefore, considering different values for E0∈ will give us the different possibilities for orbits. In the next plot, we show the phase- plane picture for the parabolic and the cubic nonlinearities. We draw bounded orbits with thin solid lines (corresponding with periodic orbits), unbounded orbits with dashed lines, and with thick lines the orbits that separate them. Note that for the cubic nonlinearity, two heteroclinic orbits connecting the steady states (u, v) = (±1, 0) separate the bounded and unbounded orbits, while for the quadratic nonlinearity, a homoclinic orbit to the steady state (u, v) = (1, 0) makes the separation.

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2.5

−2

−1.5

−1

−0.5 0 0.5 1 1.5 2 2.5

u

v

(a) f (u) = u − u³

−1 −0.5 0 0.5 1 1.5 2

−1.5

−1

−0.5 0 0.5 1 1.5

u

v

(b) f (u) = u − u²

Figure 1.1: Phase plane plot for different nonlinearities

The periodic orbits are uniquely determined by the Energy. In both cases we have that there exist two critical values E⁻ < E⁺ ∈ such that for every E ∈ (E⁻, E⁺) there exists a unique periodic orbit solving the equation. In the cubic case, E⁻ = 0 and E⁺ = ¹₄, while in the quadratic case, E⁻ = 0 and E⁺=¹₆.

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Apart from the Hamiltonian structure, we can define the Action as follows:

J(u)^def= Z 1

2u⁰(x)²− F u(x)

dx, (1.5)

where

F (u)^def= Z u

0

f (s)ds. (1.6)

In the definition of the potential F given by equation (1.6), one of the extrema of the interval is u and the other one is chosen 0, but other choices are also possible. The choice for 0 is the most convenient when studying solitons to the origin, as we do in the first chapter. Therefore, this convention is the one we will adopt; otherwise, we will specify an alternative definition. In any case, note that the difference between any two of them is constant and in the variational approach, we search for critical points of the action defined in equation (1.5).

Thus, the choice for the lower extremum plays no role in the discussion.

We will not describe in detail the appropriate space on which we define this functional. Plainly, it depends on the domain that is being considered and the type of solution being studied. Suppose that the domain is the whole real line, and we want to find an orbit connecting two equilibria: the usual procedure is then to find an appropriate Hilbert space H where we include the function and the derivatives (a subspace of W^1,2) and consider an affine space of the type φ(x) + H, where usually φ is an appropriate function connecting the equilibria.

The importance of this approach is that the Action is a Lyapunov function of the parabolic PDE. This has important consequences on the large time behavior of the PDE: for a big set of initial conditions, it converges to an equilibrium or an orbit connecting them. That is the strongest motivation to study equilibria of the PDE in detail.

Let us introduce an additional tool to study nonlinear parabolic partial differential equations like the Fisher-Kolmogorov equation, with different possibilities for the nonlinearity function f : a Maximum Principle. Consider the Cauchy problem for the partial differential equation (1.2) on some domain Ω. Denote the solution by u(t, x) and let v(x)^def= u(0, x) be given. Consider v1and v2two different initial conditions with v1(x) 6 v2(x) for every x ∈ Ω and let u1(t, x) and u2(t, x) be their corresponding solutions for t > 0. Then a well known Maximum Principle states that u1(t, x) 6 u2(t, x) for every t > 0 and x ∈ Ω.

The Hamiltonian structure, the variational structure and the maximum principle are basic tools that are present in Fisher-Kolmogorov type of equations and are used a great deal in the study of nonlinear parabolic partial differential

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equations. The combination of the three is also very specific for them and it determines the type of orbits we can find.

1.2 Fourth order differential equations

In this and next section we want to study nonlinear fourth order differential equations and in the last one we will study an example of a lattice differential equation. We study them because they keep some of the structures that are present in the previous equations, but at the same time we want to generalize them. This will cause the drop of some of the main properties (the maximum principle, for instance) and features and at the same time allow a larger variety of orbits that can be expected. We will not attempt to cover all, not even a large set of possible generalizations of the Fisher-Kolmogorov equation but study some specific equations and show different behavior that we can find in them.

The nonlinear parabolic partial differential equations have interesting properties and have been long studied, but they have also a relatively high number of main axioms that limits the behavior of possible orbits. To study dynamical systems that exhibit behavior of higher complexity, more complex patterns, we are forced to consider fourth order differential equations. Two such equations arising from Physics motivate our study, namely the Extended Fisher-Kolmogorov equation (1.8) and the Swift-Hohenberg equation (1.9), defined below. As we shall see, both equations possess a rich structure of solutions that can not be found in nonlinear parabolic partial differential equations. This task has been a big challenge in recent decades and new analytical tools are being developed to study this type of equations.

Let us start by a canonical formulation of an equation of fourth order in space and time.

∂u

∂t = −γ∂⁴u

∂x⁴+ β ∂²u

∂x² − f(u), t > 0, x ∈ Ω. (1.7) Here, we consider the parameters γ > 0 and β ∈ .

The introduction of this canonical equation is motivated by two well known equations. First, the Extended Fisher-Kolmogorov equation

∂u

∂t = −γ∂⁴u

∂x⁴ +∂²u

∂x²− f(u) t > 0, x ∈ Ω. (1.8)

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This is a particular case of equation (1.7) with β = 1. Note that for γ = 0, after rescaling space, equation (1.8) becomes (1.2) , so it can be viewed as the “old”

Fisher-Kolmogorov equation (1.2) with the fourth order term −γ^∂_∂x⁴^u⁴ added to it. This explains the origin of its name.

Another model is the so called Swift-Hohenberg equation, which is usually written as follows

∂u

∂t = −

1 + ∂²

∂x²

²

u + αu − u³, t > 0, x ∈ Ω. (1.9) In contrast with equation (1.8), we can not drop the fourth order term in (1.9) because the corresponding problem would be ill posed. While in the first case, the procedure to follow is to extend the known results from the parabolic equation (1.2) to equation (1.8), in the second case this procedure can not be applied to equation (1.9).

Note that for α > 1, equation (1.9) fits into the class (1.7): we rescale variables

s = (α − 1)t, y = (α − 1)¹⁴x, u =√

α − 1 v (1.10) and then the variable v verifies equation (1.7) with γ = 1 and β = −^√_α−1² < 0, and the nonlinearity function

f (v) = v³− v. (1.11)

For α < 1, equation (1.9) also fits into the class (1.7): we rescale variables s = (1 − α)t, y = (1 − α)¹⁴x, u =√

1 − α v (1.12) and then the variable v verifies equation (1.7) with γ = 1 and β = −^√_1−α² < 0, and the nonlinearity function

f (v) = v + v³. (1.13)

Note that both equations (1.8) and (1.9) fit into the class (1.7); but in the first case, the parameter β > 0 while in the second one, β < 0. This makes a relevant difference.

In both cases, the equilibria of equation (1.7) correspond to solutions of the fourth order differential equation

γ∂⁴u

∂x⁴ − β∂²u

∂x² + f (u) = 0, x ∈ Ω. (1.14)

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As in the previous section, we introduce here some of the tools that are used to study the dynamics of equation (1.14). First, we introduce the Energy functional associated to it.

E(u)^def= −γdu dx

d³u dx³ +γ

2

d²u dx²

² +β

2

du dx

2

− F (u), (1.15) where the potential function F is defined as in (1.6). As in the previous section, we have that the Energy E u(x) is constant along the orbits that satisfy equation (1.14).

Second, we introduce the Lagrangian action

J(u)^def= Z

Ω

(γ 2

d²u dx²

² +β

2

du dx

2

+ F u(x) )

dx. (1.16)

Critical points of the Lagrangian action in the appropriate space correspond to solutions of the differential equation (1.14). The domain of integration depends on the sort of solution being studied.

Equation (1.14) contains two parameters γ and β and holds on x ∈ Ω.

However, when we consider the unbounded domain Ω = , we can scale one parameter out. There exist several possibilities, but in the next chapter we use the following procedure: we define

q^def= − β

γ², A^def= γ^−1/4 and y = Ax, (1.17) and we define the function v(y)^def= u(x). Then, v satisfies the following differential equation

d⁴v

dy⁴+ qd²v

dy² + f (v) = 0, y ∈ . (1.18)

The differential equation (1.18) will be the starting point for our study in the next chapter.

1.2.1 Linearisation around the uniform equilibria

Later in this chapter we will discuss the dynamics of equations (1.8) and (1.9).

Let us mention that both of them can be included in a broader class of dynamical systems called Gradient systems. As we shall discuss later, if the equilibria

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are isolated, then the dynamics of the flow tend to an equilibrium. Thus, equilibria are particularly interesting solutions to be studied since they describe the asymptotic behavior of the dynamics, besides the interest that they have on their own.

Let us study the local behavior of the uniform equilibria, i.e., the solutions of equations (1.8) and (1.9) which do not depend on x. They are given by the zeros of the nonlinearity f . First, for the Extended Fisher-Kolmogorov equation, with the nonlinearity (1.11), they are given by u = ±1 and u = 0. The spectrum around u = ±1 is shown in the next sequence of pictures in Figure 1.2.

−3 −2 −1 0 1 2 3

−3

−2

−1 0 1 2 3

(a) 0 < γ < ¹₈

−3 −2 −1 0 1 2 3

−3

−2

−1 0 1 2 3

(b) γ >¹₈

Figure 1.2: Spectrum for the equation (1.8) around u = ±1

The spectrum depends on the parameter γ and two different situations are possible. In the first case, the spectrum consists of four real eigenvalues, two positive and two negative, and in the second case, four complex eigenvalues, one in each quadrant. In the first case, the equilibrium is called a real saddle, and in the second case, a saddle-focus. Note that in both cases, the spectrum is far from the imaginary axis.

In contrast, the computation of the spectrum of the linearised operator around u = 0 gives us a different picture, as in Figure 1.3(c). The spectrum consists of two purely imaginary eigenvalues and two real ones of opposite sign.

In this case, the equilibrium is called a saddle-center.

Let us make similar calculations to compute the spectrum of the uniform equilibria of the SH equation (1.9). Recall that the uniform equilibria u = ±1 exist for α > 1 while u = 0 is an equilibrium for every α ∈ . For u = ±1 and α > 1, the location of the eigenvalues of the linearised operator is shown in the

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sequence of pictures in Figure 1.3. Depending on the value of the parameter α, the equilibrium u = ±1 is either a saddle-focus (for 1 < α < 3), a center (for 3 < α < 4) or a saddle-center (for α > 4) respectively.

−3 −2 −1 0 1 2 3

−3

−2

−1 0 1 2 3

(a) A saddle-focus for α∈ (1, 3)

−3 −2 −1 0 1 2 3

−3

−2

−1 0 1 2 3

(b) A center for α ∈ (3, 4)

−3 −2 −1 0 1 2 3

−3

−2

−1 0 1 2 3

(c) A saddle-center for α >4

Figure 1.3: Spectrum for the steady SH equation around u = ±1 for α > 1

For the uniform equilibrium u = 0 we have the same sequence, depending again on the value of the parameter α: here the origin is a saddle-focus for α < 0, a center for 0 < α < 1 and a saddle-center for α > 1. Note that for α = 1, the two purely imaginary eigenvalues collapse at the origin.

1.2.2 Embedded solitons

Apart from uniform equilibria, equations (1.8) and (1.9) possess a rich structure of nonuniform equilibria. We postpone for now the description and the discussion of the extensive literature developed to study this question.

In the literature we can find a considerable number of articles about kinks connecting the equilibria u = −1 and u = 1 or pulses around these equilibria because the spectrum is far from the imaginary axis. In this way, one can establish the existence of local stable and unstable manifolds around the equilibria. Therefore, the question about existence of kinks and pulses is equivalent to the existence of intersections of these manifolds. These kind of results are relatively strong and the local behavior around the equilibria is crucial, so that they persist when varying the nonlinearities involved.

However, the question about homoclinic orbits to the equilibrium u = 0 is more delicate since it strongly depends on the nonlinearity involved and not

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only on the local behavior around u = 0. In other words, different nonlinearities with the same linearization can lead to different results.

While equilibria of equation (1.8) have been studied, equation (1.9) seems more delicate and we will focus on it. Since this equation is still difficult to handle, we will discuss a similar but simplified equation. We will state the main results of the next chapter about existence and qualitative description of homoclinic orbits converging to the origin, i.e., equilibria of (1.9) that verify

x→±∞lim u(x) = 0. (1.19)

Following the results of [17] and [46], we can think of these solitons as embedded solitons in branches of homoclinic orbits to periodic solutions. Therefore, we also discuss the existence of equilibria of (1.9) such that

x→−∞lim u(x) − φ−(x) = 0, lim

x→∞u(x) − φ⁺(x) = 0, (1.20) where φ₋(x) and φ+(x) are periodic functions which can be determined a priori.

The concept of embedded solitons comes from the following fact: for some particular cases of those periodic solutions, they are reduced to the uniform state u = 0, so that the solitons are embedded. In other words, the solitons are a particular case of homoclinic orbits to periodic solutions.

Stimulated by the results from the literature and by the difficulty of proving existence and describing the shape of the pulses for the EFK and SH equations, we develop a new approach in the next Chapter. We keep our study for the equation (1.14). But we substitute the original symmetric cubic nonlinearity (1.11) by the symmetric piecewise nonlinear function

f (u) =











u + 1 if u 6 −1

2,

− u if |u| <1 2, u − 1 if u > 1

2.

(1.21)

Like the cubic polynomial (1.11), this function is odd and has qualitatively the same shape. The advantage of studying equation (1.14) with nonlinearity (1.21) is that we can make explicit computations and obtain very precise results about existence, multiplicity and geometric description of pulses.

Let us outline the procedure followed there. We search for even pulses that cross the line u = ¹₂ only once on the positive axis at, say, x = ζ > 0. By

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symmetry we impose the conditions u⁰(0) = u⁰⁰⁰(0) = 0 and we put u(0) = α, where α is still to be determined. We have the conservation of energy property (equation (1.15)) but applied to a new potential function F derived from the piecewise linear function (1.21). Thus, equation (1.6) can be explicitly written as follows

F (u) = Z u

0

f (s) ds =









 1

2u²+ u + 1

4 u 6 −1

2,

−1

2u² |u| <1

2, 1

2u²− u + 1

4 u > 1

2,

(1.22)

which has been chosen so that F (0) = 0. Therefore, a pulse must verify E(u) = E(0) = 0, so that u⁰⁰(0) is fixed and given by u⁰⁰(0) = ±p2F (α).

We use the fact that the orbit on x ∈ (0, ζ) and x ∈ (ζ, ∞) is described by a linear differential equation that we can solve explicitly, including some parameters. The next step is to fit all the parameters including α to obtain the solution.

Let us recall the notation used in the next chapter. We study the SH equation (1.9) via the rescaled version (1.18), with parameter q > 2 defined by (1.17). In terms of the original equation (1.9), this corresponds to the parameter interval α ∈ (1, 2). We consider the nonlinearity f given by (1.21). In our first theorem, we state the existence of a sequence of homoclinic orbits to the equilibrium u = 0.

Theorem 1.1 There exists an infinite sequence {qk > 2 : k = 1, 2, . . . } which tends to infinity such that for q = qk equation (1.18), with f given by the nonlinearity (1.21), has an even homoclinic orbit u(x) to the origin u = 0. It has the following properties:

u > 0 on and u(0) > 1 + 1

√2, (1.23)

and

u⁰(x) < 0 for 0 < x < ∞. (1.24) The plots of the first ten pulses are shown in Figure 1.4(a). In Figure 1.4(b) we plot the second derivative of the orbits u⁰⁰_k.

As was previously mentioned, we can look at the pulses as embedded in branches of homoclinic orbits to periodic solutions at x = ±∞. The periodic

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Graphs of u(x)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

u

–20 –10 10 x 20

(a) 10 solitons

u’’(k,x), k=1,3,5

–0.4 –0.2 0 0.2

u’’

–15 –10 –5 5 x 10 15

(b) Orbits for u⁰⁰k(x), k = 1, 3, 5

Figure 1.4: Homoclinic orbits to u = 0

orbits φ_±(x) can be explicitly written as cosine functions. We define ε as the amplitude of the periodic solution, ω(q) as the frequency of the periodic orbit, and again ζ(q, ε) as the unique positive point at which the orbit crosses u = ¹₂, so u(ζ) = ¹₂. Then, we have the following result

Theorem 1.2 For every k > 1, there exists an interval [ε⁻_k, ε⁺_k] around ε = 0 and an analytic function qk : [ε⁻_k, ε⁺_k] → that satisfies qk(0) = qk such that for every ε ∈ [ε⁻k, ε⁺_k] there exists an even function uk(·, ε) that solves the differential equation (1.18), with f given by (1.21), and q = qk(ε) and such that

uk(x, ε) = ε cos

ω(q) x − ζ(q, ε)

+ o(1) as x → ±∞.

In Figure 1.5(a) we show one prototype of a homoclinic solution to a periodic orbit. In Figure 1.5(b) we present the bifurcation diagram in which we plot q versus the amplitude ε.

1.2.3 Analytical methods

Equation (1.14) has been studied in detail by many people. The first results about equation (1.14) with f (u) = u − u², were established by a group of

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0 0.5 1 1.5 2

u

–20 –10 10 20

x

PSfragreplacements

εq

(a) Homoclinic orbit to periodic solution

–0.2 –0.1 0 0.1 0.2

3 4 5 6

PSfragreplacements

ε

q

(b) Bifurcation branches

Figure 1.5: (a) A homoclinic orbit to a periodic orbit. (b) Bifurcation branches of homoclinic orbits.

researchers from Bath, namely, Amick, Buffoni, Hofer and Toland in [4, 36, 11].

They study (1.18) with f given by the parabolic polynomial f (u) = u²− u.

A second community of researchers, mainly coming from Leiden University, namely J.B. van den Berg, L.A. Peletier, W.C. Troy, A.I. Rotariu-Bruma, J.

Hulshof, R.C.A.M. van den Vorst and others studied (1.14) with f given by the cubic polynomial (1.11).

In [9], it is proved that in the parameter range β = 1 and γ ∈ (0,¹₈], the set of bounded solutions allowed to exist are essentially the same as for the classical FK equation, i.e. with γ = 0. This result is generalized for other nonlinearity functions f similar to the cubic polynomial (1.11). Thus, we find that for positive γ sufficiently small, the set of bounded orbits consists of the three uniform equilibria, two monotone antisymmetric kinks and a family of periodic orbits parametrised by the Energy E ∈ −¹4, 0.

Note that the parameter range γ ∈ (0,¹8] coincides precisely with the parameter range such that the equilibrium u = ±1 is a real saddle, as shown in Figure 1.2(a).

Let us give a more geometric view of the results: the bounded solutions can be represented by a picture similar to Figure 1.1(a): while general orbits can be represented in the fourth dimensional space (u, u⁰, u⁰⁰, u⁰⁰⁰) ∈ ⁴, the bounded

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orbits are uniquely projected on the second dimensional subspace (u, u⁰) ∈ ², like for the second order differential equation (1.2).

Partial results in this direction have also been obtained by singular pertur- bation theory [30, 41] and information about the stability of these orbits has been obtained in [32, 1].

Existence and description of the shape and properties of solutions of the canonical equation (1.14) has been done in a sequence of publications by L. A.

Peletier and W. Troy [55, 56, 57, 58] and collected in [59]. There the authors make extensive use of the topological shooting method : this method is especially appropriate for finding specific types of solutions with certain symmetries. To search for odd solutions, they fix a priori u(0) = u⁰⁰(0) = 0 and put u⁰(0) = α ∈

\ {0} and they fix the Energy E. By the conservation of the Energy, u⁰⁰⁰(0) is given by

u⁰⁰⁰(0) = αβ

2 − 4E + 1 4αγ ,

so that they can parametrise the orbits by their Energy. Let the parameters β and γ be fixed and denote the solution of (1.14) by u(x, α). They study the existence of values for α such that

x→∞lim u(x, α) = 1,

to find kinks connecting the equilibria u = −1 and u = 1. To search for periodic solutions, they study the existence of some value ζ(α) such that at x = ζ we have u⁰(ζ) = u⁰⁰⁰(ζ) = 0.

To search for even solutions, they fix a priori u⁰(0) = u⁰⁰⁰(0) = 0 and u(0) = α ∈ and the Energy E. By the conservation of the Energy, u⁰⁰(0) is given by

u⁰⁰(0) = ± s2

γ

E +(α²− 1)² 4

. In this case, the existence of values for α such that

x→∞lim u(x, α) = 1,

leads to a homoclinic orbit to u = 1. By the method described above, they can also study the existence of periodic solutions.

In all the previous cases, to search for homoclinic orbits to an equilibrium, kinks and periodic solutions, the authors make use of the reversibility property:

equation (1.14) is invariant under the linear transformations

L1: u(·) → −u(·), L2: u(·) → −u(− ·). (1.25)

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This invariance property allows us to extend the orbits mentioned above to all x ∈ . This is one of the applications of finding certain types of symmetries in the equation being considered. We will make larger use of this property along the thesis.

On the other hand, imposing symmetries on the solution to be searched (either an even or an odd solution) makes the search easier, since it decreases the number of degrees of freedom. If we do not allow symmetries to be present, then a two dimensional shooting method must be used.

When the authors study equation (1.14) in [55, 56, 57, 58, 51] they use the concept of building block, that was introduced in [11]. This tool helps them to classify the orbits in terms of the localization of the extrema and to describe their shape. The analysis was originally developed in [11] for equation (1.14) with nonlinearity

f (u) = u²− u. (1.26)

In [10] these techniques were applied for the equation (1.14) with nonlinearity f given by the cubic polynomial (1.11).

In [36] the authors make use of degree theory and the antipodal mapping theorem to study the existence of orbits, including pulses to an equilibrium. In [52] the authors study this question by the shooting method and they prove nonexistence of pulses for the EFK equation for β = 1 and γ ∈ 0,¹8, but they prove the existence of pulses for nonlinearities of different type, including the nonsymmetric cubic function

f (u) = (u²− 1)(u + a), (1.27)

with a ∈ (0, 1). In both cases, the pulses are homoclinic orbits to the equilibrium u = ±1. Note that for this choice of parameters β and γ, the equilibrium u = ±1 is a real saddle, as shown in Figure 1.2(a). Thus, both the stable and unstable manifolds are two-dimensional. On the other hand, linear analysis can give us a good description of the asymptotic behavior of the pulses.

However, the question about existence of pulses to an equilibrium for different choices of parameters β and γ is more delicate and has been the object of many studies in recent years from different perspectives. In this context, we mention the work by Mielke, Holmes and O’Reilly [50] and the surveys of Champneys [13] and [14] and the references quoted therein. Concerning equations such as (1.14), we mention a result of Amick & McLeod [3], and Hammer- sley & Mazzarino [33] (see also Eckhaus [24]), which proves the nonexistence of such orbits for the function (1.26). We will also mention the work on the Extended Fisher-Kolmogorov equation done in [43, 42].

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For the cubic function f given by (1.11), Grotta-Ragazzo [61, 62] proved that once a homoclinic orbit is found for some value β^∗, there exist infinitely many such orbits for values of β in any small neighborhood around β^∗.

1.3 Pattern selection in the SH equation

1.3.1 Gradient systems

As was previously argued in Subsection 1.2.1 above, when studying PDE’s, we are not only interested in the steady-state solutions but also in the dynamics.

Equilibrium solutions are particularly interesting when they are stable in some sense (i.e., finding the appropriate space and topology) since they can be used to determine the asymptotic behavior of the PDE. But often, the flow of the PDE does not tend to an equilibrium, or it does under certain conditions, for the appropriate set of initial conditions, etc.

There exist very few systems for which this property holds, and when it does, it can be hard to prove it. But, as we mentioned before, one of those systems for which this property holds, that often occurs in physical systems, is a gradient system. The easiest example of a gradient system is a physical system where we can define a functional operator J (usually called the Action), that is decreasing (by some dissipation process) and for which the functional can not decrease indefinitely but it has a lower bound.

In a gradient system we can define an operator J on some Hilbert space such that the flow can be defined by the following equation

∂u(x, t)

∂t = − ∇J u(x, t), for x ∈ Ω, t > 0. (1.28) Note that the function J is also a Lyapunov function.

It is well known that the Fisher-Kolmogorov and the Swift-Hohenberg equation are gradient systems. In this section we will consider a finite domain Ω = (0, L), for some L > 0, and we study the Cauchy problem with initial condition u(0, x) = u0(x) for every x ∈ (0, L) and Dirichlet boundary conditions u(t, 0) = u(t, L) = 0 for every t > 0 for the FK equation. For the SH equation we add

∂²u

∂x²(t, 0) = ∂²u

∂x²u(t, L) = 0, for every t > 0. (1.29) Note that solutions of this problem correspond to 2L-periodic solutions on the real line, i.e. when we study the differential equation on x ∈ .

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The SH equation can be read as a sum of a linear differential operator and a reaction term −f^α, where

fα(u) = (1 − α)u + u³. (1.30) The action associated to the SH equation is given by

J(u, L)^def= 1 L

Z L 0

1

2(u⁰⁰)²− (u⁰)²+ Fα(u)

dx (1.31)

where

Fα(u)^def= Z u

0 −f^α(s)ds = 1 − α

2 u²+u⁴

4. (1.32)

1.3.2 The continuous model

The main motivation for this study has been the observations made in [53, 54].

The authors are interested in the asymptotic behavior of the dynamics for fixed initial data and parameter α and varying parameter L. They restrict their search for symmetric initial data, so u0satisfies the following equation:

u(x) = u(L − x), for every x ∈ (0, L). (1.33) By the form of the equation (1.9), the symmetry given by (1.33) holds for every t > 0. Therefore, they are interested in symmetric equilibria.

The authors find a finite series of Gaps, i.e., a sequence of intervals of the type [L⁻_k(α), L⁺_k(α)] with k = 1, 2, . . . , n such that u(x, t) → 0 as t → ∞ if L ∈ [L⁻_k(α), L⁺_k(α)]. However, outside these Gaps, the limit behavior is a nontrivial equilibrium. The purpose is to describe the selected pattern for different values of L, the qualitative properties, and the local bifurcation diagram around the extrema of the Gaps.

Besides the analytical approach, they are interested in a more global picture, motivated by the numerical experiments. They study the map L → u∞(·) L, where u_∞(·) represents the final pattern on (0, L). One of the relevant issues observed is the existence of discontinuities in this map. More specifically, they fix α and u0 and they draw the action or the norm of the final pattern versus L, and they find that the plot is discontinuous at certain values of L.

Our intention in this section is to study more carefully a simplified model, which is able to capture these phenomena and, at the same time, which is sufficiently simple so that it can be carefully analyzed. In this way we try to understand the relevant aspects of the “complex” continuous model by studying the “simplified” discrete model in detail.

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1.3.3 The discrete model

As mentioned above, our intention is to simplify equation (1.9). As an introduction, we will start with the simplification of the classical Fisher-Kolmogorov equation (1.2), for which the dynamics is well known so that we can easily compare the continuous and the discrete case. Then, we will follow a similar procedure for the SH equation (1.9). In both equations, we consider the domain Ω = (0, L).

For equation (1.2), we rescale the variables x = Lx^∗ and t = L²t^∗. This transforms equation (1.2) into the following equation on the unit interval I = (0, 1).

∂u

∂t^∗ = ∆u + σf (u), x^∗∈ (0, 1), t^∗> 0, (1.34) with σ = L². Here, the reaction term is given by f (u) = u − u³. Note that we have adopted a different convention than in (1.11). The operator ∆ represents the Laplacian operator with respect to x^∗. From now on, we will drop the asterisks when we refer to this equation.

We discretize equation (1.34) by defining a finite lattice of N + 1 nodes as follows

un(t)^def= u(nh, t), for n = 0, . . . , N + 1, with h = 1

N. (1.35) We define the discrete Laplacian operator by

∆un

def= un+1− 2un+ u_n−1. (1.36) Thus, the discrete Fisher-Kolmogorov system is defined as follows

dun(t)

dt = ∆un(t) + σf un(t), n = 1, . . . , N − 1. (1.37) For equation (1.9), we rescale space x = Lx^∗. This transforms equation (1.9) into the following equation on the unit interval I = (0, 1).

∂u

∂t^∗ = −γ²∆²u − 2γ∆u − fα(u), x^∗∈ (0, 1) (1.38) with γ = _L¹2 and fα given by (1.30). Here, the operator ∆ represents the Laplacian operator with respect to x^∗. From now on, we will drop the asterisks when we refer to this equation.

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We discretize equation (1.38) by defining again the finite lattice (1.35) of N + 1 nodes. We use the discrete bi-Laplacian operator

∆²un

def= ∆ ∆un = un+2− 4uⁿ⁺¹+ 6un− 4un−1+ u_n−2, (1.39) so that the discrete Swift-Hohenberg system is defined as follows

dun(t)

dt = −γ²∆²un(t) − 2γ∆uⁿ(t) − f^α un(t), n = 1, . . . , N − 1, (1.40) where we have defined the virtual nodes

u₋₁(t) = −u¹(t) and uN +1(t) = −uN −1(t), for every t > 0. (1.41) Note that we need to introduce these extra nodes for the system (1.40) to be well defined. The choice given by (1.41) is the natural one for the boundary conditions (1.29).

We will denote vectors in ^{N −1} by u, and given a function f defined from to , we will write

f (u) = f (u1), f (u2), . . . , f (u_{N −1})T

. With this notation equation (1.37) can be rewritten as

u⁰(t) = −Au(t) + σf(u), (1.42)

where A = (aij) is the symmetric positive definite matrix with ajj = 2 and aj,j+1 = −1, aj+1,j = −1, and all other entries equal to 0. Its eigenvalues are given by λj, where

λj= 2

1 − cos πj N

, for j = 1, . . . , N − 1.

We can define the action J of the discrete Fisher-Kolmogorov system (1.37) and the discrete Swift-Hohenberg system (1.40). First, we introduce the operators J1 and J2

J1(u)^def= 1

2u^TAu =1 2

N

X

n=1

(un− un−1)², (1.43)

J2(u)^def= 1

2u^TA²u = 1 2

N −1

X

n=1

(∆un)². (1.44)

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Then, the action associated to (1.37) is given by

JF(u)^def= J1(u) − σ

N −1

X

n=1

F (un), (1.45)

where the potential F is defined by (1.6). Similarly, the action associated to (1.40) is given by

JS(u)^def= γ²J2(u) − 2γJ¹(u) +

N −1

X

n=1

Fα(un), (1.46)

where the potential Fα is defined by (1.32). The discrete FK system (1.37) is a Gradient system for every σ > 0 while the discrete SH system (1.40) is a Gradient system for every α ∈ and γ ∈ . Therefore, this property holds for both the discrete and continuous equations.

An extra property holds for both cases. Define the following linear operators in ^{N −1}.

L1(u1, u2, . . . , u_{N −2}, u_{N −1}) = (−u¹, −u², . . . , −uN −2, −uN −1), L2(u1, u2, . . . , u_{N −2}, u_{N −1}) = ( u_{N −1}, u_{N −2}, . . . , u2, u1), L3(u1, u2, . . . , u_{N −2}, u_{N −1}) = (−uN −1, −uN −2, . . . , −u², −u¹).

(1.47)

Note that L3= L1L2= L2L1. The unique fixed point of L1 is the origin, while the fixed points of the other operators are given by

Π2

def= u ∈ ^{N −1}: uj= u_{N −j} for every j = 1, . . . , N − 1 , Π3

def= u ∈ ^{N −1}: uj= −uN −j for every j = 1, . . . , N − 1 . (1.48) Invariance of the flow with respect to Π2in the discrete model corresponds to the property (1.33) for every t > 0 in the continuous model. Similarly, invariance of the flow with respect to Π3 in the discrete model corresponds to the following property holding for every t > 0 in the continuous model

u(x) = −u(L − x), for every x ∈ (0, L). (1.49) The elements of Π2 will be called symmetric and the elements of Π3 will be called antisymmetric.

From now on, we study in detail the case N = 4. In this case, the symmetric equilibria can be represented as points in ². Let us summarize the main results

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about equilibria of the discrete FK equation (1.37) in the following theorem. In Figure 1.6 we draw the bifurcation diagram of the symmetric equilibria. There we represent the norm of the equilibria ku(σ)k versus the parameter σ. In dashed line we draw the unstable equilibria and in solid line the stable ones.

Theorem 1.3 Let N = 4. The symmetric equilibria of equation (1.37) have the following properties:

1. For σ ∈0, 2 −√

2, the origin is the unique symmetric equilibrium. For σ > 2 −√

2, the origin is unstable and a pair of nontrivial symmetric equilibria ±u^∗(σ) bifurcate from the origin. They are located in the first and third quadrant and they are the global symmetric mimimisers of JF

defined by (1.45) for every σ > 2 −√ 2.

2. For σ > 2 +√

2, a second pair of nontrivial symmetric equilibria ±ζ(σ) bifurcate from the origin. They are saddles located in the second and fourth quadrant.

3. There exists ˜σ > 2 +√

2 such that for σ > ˜σ, two pairs of nontrivial symmetric equilibria emerge from a saddle-node bifurcation. They are located in the second and fourth quadrant. Therefore, there exist symmetric initial data for which the flow converges to an equilibrium that has not one sign.

Property 1 mirrors the continuous FK model, while Property 3 is qualitatively specific for the discrete FK model.

Our main interest is focused on the discrete SH system (1.40). Like in [53, 54], for fixed α and every N we are able to find a sequence of Gaps such that u(t) → 0 if the parameter γ lies in one of those Gaps and it tends to a nontrivial equilibrium otherwise.

We find that for given N , for α > 0 sufficiently small, there exists at least one Gap. This case is relatively simple, since all the nontrivial equilibria bifurcate from the origin and the final state φ_∞(u0, γ) is continuous with respect to γ.

Moreover, the values of γ for which the bifurcation occurs is determined by the linearized operator around the origin and the shape is determined by the corresponding eigenvector. But when pushing α higher, Gaps start to dissapear and new phenomena occur. It is rather difficult to describe the process for high N and we will consider N = 4.

We introduce the following critical values of γ γ₁^±(α) ^def=

1 +^√₂² (1 ±√

α) . γ₃^±(α) ^def=

1 −^√2²

(1 ±√ α) .

(1.50)

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0 5 10 15 20 0

0.5 1 1.5 2

||u||

PSfrag replacements

σ

Figure 1.6: Bifurcation diagram of the symmetric equilibria

For every N , there exist N − 1 pairs of critical values: the odd ones correspond to symmetric eigenvectors and the even critical values with antisymmetric eigenvectors. For this reason, we consider γ₁^± and γ₃^±. Note that γ₁⁻(α) > γ₃⁺(α) iff α ∈ 0,¹2. Therefore, we can distinguish two parameter ranges for α: 0 < α < ¹₂ and ¹₂ < α < 1. In Figure 1.7, the critical values correspond to the bifurcation points of the diagram and they are ordered as follows: γ₃⁻< γ₃⁺< γ₁⁻ < γ₁⁺.

We start by considering the interval 0 < α < ¹₂. Take γ > γ₁⁺. Then, the origin is the unique equilibrium. However, when γ drops below γ₁⁺ two nontrivial solutions ±u¹(γ) bifurcate from the origin, one into the first quadrant, and one into the third. They continue to exist until γ reaches γ₁⁻, when they merge with the trivial solution again. For γ ∈ [γ3⁺, γ₁⁻] the origin remains the only equilibrium state, but at γ₃⁺ two nontrivial solutions u3(γ) bifurcate again from the origin, this time into the second and the fourth quadrant. They too, return to the trivial solution, at γ₃⁻. For γ below γ₃⁻ the trivial solution is the unique equilibrium state. The situation is illustrated in a bifurcation diagram in Figure 1.7.

Theorem 1.4 Let α ∈ 0,¹₂ and γ > 0.

(a) If γ /∈ γ1⁻, γ₁⁺ ∪ γ3⁻, γ₃⁺, then the origin is the unique equilibrium state,

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0 0.5 1 1.5 2 2.5 3 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

PSfrag replacements

γ

kuk

Figure 1.7: Graph of kuk versus γ for α = 0.3

and it atracts all orbits.

(b) If γ ∈ γ1⁻, γ₁⁺, then there exist two equilibrium states ±u1(γ), one in the first quadrant, and one in the third. The origin is now unstable and ±u1

are stable.

(c) If γ ∈ γ3⁻, γ₃⁺, then there exist two equilibrium states ±u³(γ), one in the second quadrant, and one in the fourth. The origin is unstable and ±u³are also unstable.

Next, let us study the interval ¹₂ < α < 1. Again, the origin is the unique equilibrium for γ > γ₁⁺ and when γ drops below γ₁⁺ two nontrivial solutions

±u¹(γ) bifurcate from the origin. But then, at γ⁺₃ two nontrivial solutions u3(γ) bifurcate again from the origin and before they both return to the trivial solution, two pairs of new solutions emerge by a saddle-node bifurcation at some γ_∗⁺ ∈ ¹2, γ₃⁺. For 0 < γ < ¹₂ the bifurcation phenomena is reversed and the eight nontrivial solutions gradually dissapear. This is described in the following theorem. For convenience, we put the critical values of γ on a γ-axis. Note

1/2 0

(b) (c) (d) (d) (c) (b) (a)

(a)

PSfrag replacements

γ₃⁻ γ₁⁻ γ_∗⁻ γ_∗⁺ γ₃⁺ γ₁⁺ Figure 1.8: Critical values of γ.

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that for γ = 1/2, the system of ODE’s consists of two independent differential equations.

Theorem 1.5 Let α ∈ ¹2, 1 and γ > 0. Then, we identify 6 critical values of γ:

0 < γ₃⁻(α) < γ₁⁻(α) < γ_∗⁻(α) < γ_∗⁺(α) < γ₃⁺(α) < γ⁺₁(α), such that:

(a) If γ /∈ γ3⁻, γ₁⁺, the origin is the unique equilibrium state and it attracts all orbits.

(b) If γ_∗⁻ < γ < γ₁⁺, a pair of nontrivial equilibria ±u exist, one in the first quadrant and one in the third. At γ = γ⁻_∗, u has a saddle-node bifurcation and at γ⁺₁, it has a supercritical bifurcation from the origin. u is a stable equilibrium and it is the minimizer for ¹₂< γ < γ₁⁺.

(c) If γ₁⁻ < γ < γ₃⁺, a pair of nontrivial equilibria ±y exist. For γ1⁻ < γ < ¹₂, they are located in the second quadrant (respectively the fourth) and for

1

2 < γ < γ₃⁺, they are located in the first quadrant (respectively, the third).

They are all saddles.

(d) If γ₃⁻< γ < γ_∗⁺, a pair of nontrivial equilibria ±z^∗ exist, one in the second quadrant and one in the fourth. At γ = γ_∗⁺, it has a saddle-node bifurcation and at γ = γ₃⁻ it has a supercritical bifurcation from the origin. The points

±z^∗ are stable and they are minimizers for γ₃⁻< γ < ¹₂.

(e) If γ_∗⁻< γ < γ⁺_∗, a pair of nontrivial equilibria ±z^∗∗ exist. For γ₁⁻< γ < ¹₂, they are located in the second quadrant (respectively the fourth) and for

1

2 < γ < γ₃⁺, they are located in the third quadrant (respectively, the first).

They are saddles.

A simple bifurcation theorem shows that around the critical values γ₁⁻, γ₁⁺, γ₃⁻, γ₃⁺there exist local bifurcation branches. These branches can be continued as long as the conditions for the Implicit Function Theorem hold, that is, as long as the appropriate determinant does not vanish. Applying this argument to this case, we find by a combination of analytical and numerical arguments that one branch can be extended smoothly along the interval [γ₁⁻, γ₃⁺], and the other one can be extended continuousy but not smoothly along [γ₃⁻, γ₁⁺].

This is summarised in the following proposition:

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0 0.2 0.4 0.6 0.8 1 0

0.5 1 1.5

||u||

PSfrag replacements

γ

Figure 1.9: Graph of kuk versus γ for α = 0.7

Proposition 1.1 There exists an smooth parametrization h1: [0, 1] → [γ1⁻, γ₃⁺]×

3 with components s → (γ, u1, u2, u3) and a continuous (but not smooth) parametrization h2: [0, 1] → [γ3⁻, γ₁⁺] × ³ such that

(a) h1(0) = (γ₁⁻, 0, 0, 0), h1(1) = (γ₃⁺, 0, 0, 0), and ^∂γ_∂s is continuous and positive on s ∈ (0, 1).

(b) h2(0) = (γ₃⁻, 0, 0, 0), h2(1) = (γ₁⁺, 0, 0, 0), and ^∂γ_∂s is discontinuous on s ∈ (0, 1).

(c) h1(·) and h²(·) represent all the symmetric equilibria of the discrete SH equation, up to symmetry. More specifically, let (γ, u) be a nontrivial symmetric equilibrium of the discrete SH equation. Then, there exists a unique s ∈ (0, 1) and a unique i ∈ {1, 2} such that either hi(s) = (γ, u) or hi(s) = (γ, −u).

In Figure 1.9, we can see the bifurcation diagram of the norm of all the equilibria.

Here, we show ku(γ)k versus γ for α = 0.7. The equilibria bifurcating from the origin correspond to γ₃⁻ < γ₁⁻< γ₁⁺. The values γ_∗⁻ < γ_∗⁺ are those for which the branch of equilibria changes stability.

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0 1 2 3 4

−1

−0.5 0 0.5 1

x

u

(a) γ = 4

0 1 2 3 4

−1

−0.5 0 0.5 1

x

u

(b) γ = 3

0 1 2 3 4

−1

−0.5 0 0.5 1

x

u

(c) γ = 0.48

0 1 2 3 4

−1

−0.5 0 0.5 1

x

u

(d) γ = 0.47

0 1 2 3 4

−1

−0.5 0 0.5 1

x

u

(e) γ = 0.05

0 1 2 3 4

−1

−0.5 0 0.5 1

x

u

(f) γ = 0.01

Figure 1.10: Dynamics for α = 0.7 and different values of γ.

Let us finish this section with some remarks about the final profile. Let the initial value u0 ∈ ³, u0 6= (0, 0, 0) be fixed and symmetric, and consider the map

γ → φ∞(u0, γ), for every γ > 0. (1.51) As we have mentioned above, in the continuous model there were observed discontinuities with respect to γ. Now we are able to describe the reason for this for the discrete model. Let us restrict to the case α ∈ ¹2, 1. Then, as described in Theorem 1.5, there exist four different stable symmetric equilibria ±u and

±z^∗ on γ ∈ (γ_∗⁻, γ_∗⁺). Moreover, φ_∞(u0, γ⁻₁) = ±u and φ∞(u0, γ₃⁺) = ±z^∗ for almost every initial u0, since they are the only stable equilibria. Therefore, there exists some γ ∈ (γ_∗⁻, γ_∗⁺) where the final profile switches and the map (1.51) has at least one discontinuity.

We show this evolution of the final profile in Figure 1.10. We fix the initial condition u(0) = sin ^π₄ , sin ^π₂ , sin ^π₄

=

√1

2, 1,^√¹₂

and α = 0.7 and we plot φ_∞(u0, γ) for different values of γ in thick line. As γ decreases,

Patterns described by Discrete and Continuous Dynamical systems