RIGOROUS CONTINUATION OF BIFURCATION POINTS IN THE DIBLOCK COPOLYMER EQUATION

. We develop general methods for rigorously computing continuous branches of bifurcation points of equilibria, speciﬁcally focusing on fold points and on pitchfork bifurcations which are forced through Z 2 symmetries in the equation. We apply these methods to secondary bifurcation points of the one-dimensional diblock copolymer model.

1. Introduction. Understanding the equilibrium structure of nonlinear partial differential equations lies at the heart of many important applied problems. Unfortunately, there are no general techniques which allow one to obtain a complete answer for any such problem. In some cases, maximum principle methods can be used to obtain partial results, in other cases one can reduce the problem to studying specific connecting orbits in associated ordinary differential equations. Both of these types of methods are restricted to the case of partial differential equations on one-dimensional domains, but even in this case many answers remain elusive.
One alternative that has been developed over the last decades is the use of computer-assisted proofs in the study of equilibrium problems of nonlinear partial differential equations. Part of the allure of these methods is that they do not depend on the order of the partial differential equation. For example, in contrast to classical maximum principle methods, which cannot generally be used for fourth-order partial differential equations, rigorous computational methods have been successfully applied to the stationary Cahn-Hilliard equation [13,22,23], the diblock copolymer model [35,36,37,38], as well as to others such as [25,40].
Almost all equilibrium problems that arise in applications involve a number of parameters, and one is generally interested in understanding how the structure of the equilibrium set changes as these parameters are varied. In this context, precise mathematical methods from bifurcation theory exist which can predict where changes in the solution set occur, i.e., they address the location of bifurcation points. In addition, they can provide complete local descriptions of emerging solution branches. Yet, these methods usually fail to provide information on the global structure of such branches. Also in this context, rigorous computational techniques have successfully been used, see for example [2,11,14,35,31] and [1, 28,29], where the latter group considered bifurcation points as well as solution branches.
In the current paper, we show that rigorous computational methods can locate curves in parameter space along which certain types of bifurcation points occur. Our focus is on studying secondary bifurcation points using rigorous computational techniques, including saddle-node bifurcations and symmetry-breaking pitchfork bifurcations. Our interest in this problem stems from recent work on the diblock copolymer equation, which for the one-dimensional domain Ω = (0, 1) is given by subject to the mass constraint µ = 1 0 u(x)dx , as well as u x (x) = u xxx (x) = 0 for x = 0, 1 .
where in the following we use the nonlinearity f (u) = u−u 3 . This partial differential equation is a model for microphase separation in diblock copolymers as described in [26]. These are formed by the reaction of two linear polymers, or blocks, containing different monomers. The function u is the relative macroscopic density of the two monomers. The parameter µ is the total mass of u. Roughly speaking, large λ represents strong short range repulsion, resulting in short range separation. Large σ indicates strong long range elasticity forces. In two and three dimensions, the interaction between the two forces results in the formation of special geometries, which induce patterns such as spots, stripes, and gyroids. In its original form it was proposed by Ohta and Kawasaki [27] and Bahiana and Oono [3]. For more information regarding its derivation and the involved parameters we refer the reader to Choksi and Ren [8,9].
In [17] we combined rigorous bifurcation-theoretic results with numerical simulations to shed light on the long-term dynamics of the diblock copolymer model in one dimension. It is well-known that the model exhibits multi-stability, i.e., the coexistence of multiple stable equilibrium solutions. Furthermore, the constant function u ≡ µ is always an equilibrium solution for the diblock copolymer model, representing a homogeneous polymer blend. For sufficiently large values of the parameter λ this stationary state loses its stability, and one is generally interested in the long-term behavior of solutions of (1) which originate close to µ. It was shown in [17] that the spatial periodicity of the long-term limit of typical solutions changes along well-defined curves in the λ-σ-parameter plane. In fact, numerical path-following computations suggested that these curves are the locations of very specific secondary bifurcation points in the diblock copolymer bifurcation diagram. The diblock copolymer model is a regular perturbation, with respect to the pa- rameter σ, of the standard Cahn-Hilliard model, which corresponds to σ = 0. The equilibrium structure of the Cahn-Hilliard equation has been completely described by  by combining phase-plane analysis with transversality arguments, and this leads to a bifurcation diagram as in the left image of Figure 1. The method of [15] relies on the fact that the stationary Cahn-Hilliard problem in one dimension can be integrated twice, reducing it to a second-order nonlinear boundary value problem. However, this is no longer the case in the diblock copolymer model -which is certainly responsible for the fact that our knowledge of its equilibrium structure is extremely limited. Nevertheless, as a regular perturbation problem the bifurcation diagram of the Cahn-Hilliard model changes in a continuous manner as σ is increased from zero. Part of this change was discussed in [17]. Namely, it was shown that as σ increases from zero, there is an infinite sequence of branch interactions near the trivial solution line as the branches successively move to the right. These local interactions have been rigorously described in [17] using analytical bifurcation-theoretic techniques. Further, the local bifurcations are observed to combine with global bifurcations from infinity, leading to different types of branch interactions at secondary bifurcations. Five of these secondary bifurcation points can be seen in the right image of Figure 1, which shows part of the diblock copolymer bifurcation diagram for σ = 6. While one of these points is a regular saddle-node type bifurcation point, the remaining four are of pitchfork type. Understanding the locations of these secondary bifurcation points in the λ-σ-parameter plane lies at the heart of understanding how the complicated diblock copolymer bifurcation diagram is obtained from the simple Cahn-Hilliard diagram. For more details, we refer the reader to the survey [37].
Motivated by the above discussion, in this paper we develop a rigorous computational method to compute the location curves of secondary bifurcation points as above. For the case of the saddle-node type bifurcation point this is accomplished by considering a standard extended system which encodes both the existence condition for the equilibrium point and the existence of a one-dimensional non-degenerate kernel of the linearization of the diblock copolymer model at this equilibrium. This extended system will lead to a well-defined continuation problem. However, for the remaining four secondary bifurcation points in the right image of Figure 1 this extended system has a degeneracy. This is due to the fact that these bifurcation points are pitchfork bifurcations which are forced by symmetries in the model. In these cases we employ a method due to Werner & Spence [39] to derive a welldefined continuation problem. While in this paper we concentrate on the case of the diblock copolymer model, similar questions arise in the study of nucleation in the Cahn-Hilliard model, as described in [4,5,6,12]. In this paper, we develop a generally applicable method for solving problems with saddle-node bifurcations and Z 2 symmetry-breaking pitchfork bifurcations. While we only apply this method in the context of the diblock copolymer equation (with µ = 0 chosen in order to allow for antisymmetric solutions), the method is directly applicable in a broad range of applications. In addition, we believe that the method can be extended to symmetry-breaking bifurcations for other symmetry classes.
After developing the general functional analytic method for rigorously verifying the existence of bifurcation points, we apply it in several specific secondary bifurcations for the one-dimensional diblock copolymer model. We do so using the method of radii polynomials. In this context, we write everything in terms of the Banach space of Fourier series. We then use a series of previously established estimates on the functional manipulation of such series. This allows us to be able to handle the rigorous estimates on the infinite-dimensional function space.
Our final results using the radii polynomial techniques consist of a set of three validated branches of saddle-node bifurcations and three validated branches of symmetry-breaking pitchfork bifurcations. Each of these cases were found numerically in [17], where it was observed numerically that there was scaling law of different bifurcation branches. From the point of view of the general techniques of the first part of this paper, these results serve as a proof of concept. From the point of view of better understanding the diblock copolymer equations, the validation of these branches of bifurcations is a first step, where our eventual goal is to combine such results with analytical methods in order to validate the scalings of bifurcation branches that we observed in the previous paper.
The remainder of the paper is organized as follows. In Section 2 we present the functional-analytic setting for our studies, showing how the results of [39] can be adapted to our situation. In Section 2.1 we view the set of equilibria of a differential equation as the zero set of a nonlinear mapping in an abstract function space. We use the Lyapunov-Schmidt reduction technique to study the zero set of such a mapping. This section includes the introduction of equivariant symmetry, with the goal of being able to treat symmetry-breaking bifurcations. In Section 2.2, we then apply the Lyapunov-Schmidt method to saddle-node bifurcations. We introduce an extended system consisting of the nonlinear mapping whose zero set consists of equilibria, along with two other bifurcation conditions. We then apply our abstract approach to the extended system to prove the existence of saddle-node bifurcation points. Finally, in 2.3, we treat the case of pitchfork bifurcations. This time we make assumptions of equivariance of the nonlinear mapping, and the existence of a solution which is either symmetric or antisymmetric. This allows us to prove the existence of symmetry-breaking pitchfork bifurcation points. In Section 2.4, we demonstrate that the diblock copolymer equation satisfies all the assumptions necessary for our abstract setting. Section 3 gives specific details of how to transfer the abstract functional-analytic existence results into a rigorous computational method. In particular, we give existence results for equilibrium solutions at regular and bifurcation points, where our solutions are proved to be contained in a precisely defined small neighborhood of a function given in the form of a truncated Fourier series expansion. Section 3.1 gives the function-analytic background for rigorous bounds on manipulated functions given in terms of Fourier series expansions. Section 3.2 details the method of sequentially validating the curve of equilibrium solutions of the equation as a parameter is varied using the radii polynomial approach. Section 3.3 and Section 3.4 provide the detailed estimates and the bounds required to define the radii polynomial associated to the saddle-node bifurcations and respectively to the pitchfork bifurcations. Finally, in Section 3.5, the radii polynomial approach is combined with the Lyapunov-Schmidt results in Section 2 in order to establish branches of saddle-node and pitchfork bifurcations for the diblock copolymer equation.

2.
Extended systems for locating bifurcation points. In this section we present the functional-analytic framework which will be used to study the continuation of bifurcation points using rigorous computational techniques. As we anticipate that our methods are applicable to a wide variety of settings, we formulate the results first in a general Banach space setting, and apply them only later to the diblock copolymer model. We note that a number of these results are standard, but we include a brief version of the proofs for completeness in the efforts of keeping the paper self-contained, as well as to set the results into our notation.
2.1. Lyapunov-Schmidt reduction and equivariance. We begin by recalling the basic functional-analytic tool which can be used to study the equilibrium set of nonlinear equations in Banach spaces -the Lyapunov-Schmidt reduction. The results of this section are all well-known and stated only to keep the presentation as self-contained as possible. For more details we refer the reader to [10,41]. We study nonlinear operator equations of the form where F : R×X → Y is a smooth nonlinear mapping, and X and Y are real Banach spaces. Throughout this section, we assume the following.
Assumption 2.1 (Fredholm Property). Let X and Y denote real Banach spaces, and assume that the parameter-dependent nonlinear operator F : R × X → Y is sufficiently smooth. Suppose that the point (λ 0 , u 0 ) is a solution of (2), i.e., that we have the identity F (λ 0 , u 0 ) = 0. Finally, we assume that the Fréchet derivative L = D u F (λ 0 , u 0 ) ∈ L(X, Y ) is a Fredholm operator of index zero.
In addition, we will only consider possible bifurcation points which exhibit a one-dimensional nullspace. This leads to the following second assumption.
Assumption 2.2 (One-Dimensional Kernel). Suppose that Assumption 2.1 holds. Assume further that the linearization L = D u F (λ 0 , u 0 ) has a one-dimensional kernel. Since it has index zero, its range has co-dimension one. In this case, we have for some nonzero elements ϕ 0 ∈ X and ψ * 0 ∈ Y * , where Y * denotes the dual space of Y . Finally, letX ⊂ X andỸ ⊂ Y denote closed subspaces such that where the continuous linear projector P : Y → Y is defined via the two identities R(P ) =Ỹ and N (P ) = R(L), while the projector Q : X → X is defined via R(Q) = N (L) and N (Q) =X. Notice that both P and Q have rank one.
In the above situation, one can study the solution set of (2) in a neighborhood of (λ 0 , u 0 ) by solving an associated real-valued equation which involves only two real arguments, the so-called bifurcation equation. It is the subject of the following proposition which describes the Lyapunov-Schmidt reduction.
In other words, the solution set of b(λ, α) = 0 in a neighborhood of (λ 0 , 0) ∈ R 2 is in one-to-one correspondence to the solution set of F (λ, u) = 0 in a neighborhood of the point (λ 0 , u 0 ).
Proof. According to Assumption 2.2, solving the nonlinear problem (2) is equivalent to solving the two equations P F (λ, v + w) = 0 , as well as (3) where v = Qu and w = (I − Q)u. If we define G(λ, v, w) = (I − P )F (λ, v + w), then our assumptions readily imply that G : R × N (L) ×X → R(L). If we further introduce the abbreviations v 0 = Qu 0 and w 0 = (I − Q)u 0 , then G(λ 0 , v 0 , w 0 ) = 0, and According to our definition ofX the continuous linear operator D w G(λ 0 , v 0 , w 0 ) is therefore one-to-one and onto, hence continuous with continuous inverse -and one can apply the implicit function value theorem to solve (4) locally for w as a function of λ and v. This furnishes the function W as in the formulation of the theorem. Plugging W into (3) shows that locally near (λ 0 , u 0 ) a pair (λ, u) solves F (λ, u) = 0 if and only if (λ, v) = (λ, Qu) satisfies If we now set b(λ, α) = ψ * 0 (P F (λ, v 0 + αϕ 0 + W (λ, v 0 + αϕ 0 ))), then the result follows due to our choices of ψ * 0 and P . Table 1. Some of the partial derivatives of the bifurcation function b(λ, ν) at the point (λ 0 , 0) up to order three, together with the required partial derivatives of W .
Important for applications is the fact that one can easily derive a Taylor expansion for the bifurcation function b(λ, α) near the point (λ 0 , 0). One can show that with R(ν, α) = O( (ν, α) 4 ), where the derivatives of b can be computed explicitly as in Table 1, which also contains the necessary derivatives of W . Notice that the latter derivatives are all obtained through solving an inhomogeneous linear equation which involves the restriction of L ontoX, which in this case is an invertible mapping. The method of Lyapunov-Schmidt is one ingredient for our rigorous computational study of bifurcation points in the diblock copolymer equation. In order to treat pitchfork bifurcation points, we also make use of symmetry methods. For the purposes of this paper, we assume that symmetry occurs in the following specific form.
Definition 2.4 (Z 2 -Equivariance). In the situation of Assumption 2.1, suppose there exist bounded linear operators S X ∈ L(X) and S Y ∈ L(Y ) such that S X = I , S Y = I , S 2 X = I , S 2 Y = I , as well as F (λ, S X u) = S Y F (λ, u) for all λ ∈ R and u ∈ X .
Then we say that F is Z 2 -equivariant. Based on the symmetry operators S X and S Y one can also decompose the underlying Banach spaces into the symmetric elements and the antisymmetric elements. More precisely, for the Banach space X we define and analogously one can define the subspaces Y s and Y a of Y . Then it is straightforward to show that In our applications below, the operators S X and S Y are usually given by the same formula, even though they act on different Banach spaces. We therefore simply drop the subscripts in the following to simplify notation. It will always be clear from context whether S acts on X or on Y .
For the nonlinear equation (2), equivariance has immediate consequences regarding invariance of subspaces, both for the equation and for certain derivatives of F . For example, by differentiating the identity (6) with respect to u one immediately obtains for all λ ∈ R and u, v ∈ X. If one assumes in addition v ∈ X s , then (7) furnishes the Z 2 -equivariance of the Fréchet derivative D u F (λ, u). The following lemma collects a number of similar properties, which will be useful later on. The results are stated without their straightforward proofs, see also [39].
Lemma 2.5 (Equivariance of Partial Derivatives). Suppose that Assumption 2.1 holds, and that (2) is Z 2 -equivariant as in Definition 2.4. Then the following statements hold for all parameters λ ∈ R, as long as we have u ∈ X s : The above lemma applies to the derivatives of F at any point (λ, u), as long as u ∈ X s . If we consider in particular the point (λ 0 , u 0 ) from Assumption 2.2, even more can be said. The following result shows that both the eigenfunction ϕ 0 and the element ψ * 0 are symmetric or antisymmetric. In addition, the complementsX andỸ in Assumption 2.2 can be chosen such that they respect the symmetry operations as well. Note that while most of the following result is contained in the work of Werner and Spence, we were unable to find a full treatment of the details of the symmetry preservation under linear mapping as stated below. Lemma 2.6 (Equivariance Effects of a One-Dimensional Kernel). Suppose that Assumption 2.2 holds, that (2) is Z 2 -equivariant as in Definition 2.4, and that u 0 ∈ X s . Then the following hold: (a) The eigenfunction ϕ 0 which spans the nullspace N (L) of L = D u F (λ 0 , u 0 ) is either an element of X s or of X a , i.e., we have S X ϕ 0 = ε X ϕ 0 for some ε X ∈ {±1}. (b) The element ψ * 0 which characterizes R(L) is either symmetric or antisymmetric with respect to the equivariance S * Y , i.e., we have S * (d) The projectors P and Q in Assumption 2.2 can be chosen in such a way that they commute with the symmetry actions, i.e., such that S Y P = P S Y and S X Q = QS X .
Proof. According to Lemma 2.5(b) we have both LX s ⊂ Y s and LX a ⊂ Y a , i.e., the splitting into symmetric and antisymmetric elements is respected by L. One can then easily see that This splitting furnishes 0 = Lϕ 0 = Lϕ 0,s + Lϕ 0,a , and therefore Lϕ 0,s = −Lϕ 0,a , which in turn implies that both Lϕ 0,s and Lϕ 0,a are contained in X s ∩ X a = {0}.
Combining these results, both ϕ 0,s and ϕ 0,a are in N (L), hence multiples of ϕ 0 . Together this implies that either we have ϕ 0 ∈ X s or ϕ 0 ∈ X a , which is equivalent to S X ϕ 0 = ϕ 0 or S X ϕ 0 = −ϕ 0 , respectively. This completes the proof of (a). In order to establish (b), note that since L is a Fredholm operator of index zero with one-dimensional nullspace, the same is true for its adjoint operator L * , and the closed range theorem then implies that N (L * ) = span[ψ * 0 ]. The equivariance identity S Y L = LS X immediately gives S * X L * = L * S * Y . As above, one obtains L * Y * s ⊂ X * s and L * Y * a ⊂ X * a , if the symmetric and antisymmetric elements in the dual spaces are defined via the adjoints S * Y and S * X . The remainder of the proof of (b) proceeds completely analogous to the proof of (a) above.
We now turn our attention to (c). If we assume that both L[X s ] = Y s and L[X a ] = Y a are satisfied, then one can easily see that This, however, contradicts the fact that the range R(L) has codimension one in Y . In other words, we necessarily have Y s \ L[X s ] = ∅ or Y a \ L[X a ] = ∅, or both. Now let ψ 0 denote an arbitrary element in (Y s \ L[X s ]) ∪ (Y a \ L[X a ]) = ∅. Since ψ 0 = 0 and ψ 0 ∈ R(L) we can assume without loss of generality that this element is scaled in such a way that ψ * 0 (ψ 0 ) = 1. In addition, due to ψ 0 ∈ Y s ∪ Y a there exists a sign κ ∈ {±1} such that S Y ψ 0 = κψ 0 . Then one obtains Now assume ε Y = 1. Then the above arguments readily show that Y a \L[X a ] = ∅, and this in turn implies Y s \ L[X s ] = ∅. Therefore, the subspace L[X s ] has codimension one in Y s , and L[X a ] = Y a . The case ε Y = −1 can be treated analogously and leads to the opposite configuration.
It remains to establish (d), and we begin with the statement concerning the projection P . As in the proof of (c), let ψ 0 denote an arbitrary element in (Y s \ L[X s ]) ∪ (Y a \ L[X a ]) = ∅, and we can again assume without loss of generality that ψ * 0 (ψ 0 ) = 1. In addition, one automatically obtains S Y ψ 0 = ε Y ψ 0 . Now definẽ Y = span[ψ 0 ], and let P be the associated projector as in Assumption 2.2. Then P y = ψ * 0 (y)ψ 0 , and this implies for all y ∈ Y the identity i.e., the operators P and S Y commute.
Using these results, we can now show that equivariance also leads to the following symmetry properties in the method of Lyapunov-Schmidt.
Proposition 2.7 (Equivariant Lyapunov-Schmidt Reduction). Suppose that all the assumptions of Proposition 2.3 are satisfied, and that F is Z 2 -equivariant as in Definition 2.4. Finally, assume that u 0 ∈ X s , and suppose that the projections P and Q are chosen as in Lemma 2.6(d). Then the following hold: The mapping W appearing in the the Lyapunov-Schmidt reduction in Proposition 2.3 is Z 2 -equivariant, i.e., we have where ε X and ε Y encode the symmetry properties of ϕ 0 and ψ * 0 , respectively, as shown in Lemma 2.6.
Proof. The invariance of N (L) and R(L) follows immediately from (7) and S X u 0 = u 0 . To show (b), for arbitrary λ ∈ R and v ∈ N (L) the equivariance of F , combined with the construction of W in the proof of Proposition 2.3 and Lemma 2.6(d) show Due to the uniqueness property of W guaranteed by the implicit function theorem, this furnishes where we have used the fact that S X v 0 = S X Qu 0 = QS X u 0 = Qu 0 = v 0 . This completes the proof of the lemma.
While in general it does not seem to be the case that ε X and ε Y have to be related, there are special situations when they have to coincide. One such situation is outlined in the following lemma, and it will be important for the application to the dibock copolymer model.
Our previous results give invariance conditions on symmetric spaces. However, there is no guarantee from these statements that a linear mapping preserves symmetry classes. The following result gives sufficient conditions on a linear mapping J under which we can guarantee that symmetric spaces are preserved rather than reversed. This will be critical in order to be able to state conditions for the occurence of a symmetry-breaking pitchfork bifurcation.

Lemma 2.8 (Forcing Sign Equality).
Suppose that all the assumptions of Proposition 2.3 are satisfied, and that F is Z 2 -equivariant as in Definition 2.4. Assume that u 0 ∈ X s , suppose that the projections P and Q are chosen as in Lemma 2.6(d), and let ϕ 0 and ψ * 0 define nullspace and range of the linearization L = D u F (λ 0 , u 0 ) as before. Furthermore, suppose that there is a linear mapping J : X → Y which satisfies JS X = S Y J, as well as Proof. The assumptions immediately imply Jϕ 0 , and dividing both sides by ψ * 0 Jϕ 0 = 0 furnishes the result. Notice that if Lemma 2.8 holds, then part (c) of Proposition 2.7 is particularly useful. In the nontrivial case, one has ε X = ε Y = −1, which immediately implies that the bifurcation equation b(λ, α) = 0 is odd with respect to α -and therefore forces the existence of a trivial solution. This situation will correspond to symmetrybreaking bifurcations discussed further below.

2.2.
Saddle-node bifurcation points. We now turn our attention to the first type of bifurcation considered in this paper -the saddle-node bifurcation. In generic systems, this is the only type of bifurcation point that can be observed. For this, assume that we have a solution (λ 0 , u 0 ) of the problem (2), and that Assumption 2.2 holds. Then the condition is generically satisfied, and in this case one often refers to (λ 0 , u 0 ) as a simple saddlenode bifurcation point, see for example [39]. In fact, the left-hand side of (8) is the first term in the Taylor expansion (5) of the bifurcation equation. In combination with the implicit function theorem this shows that under condition (8), the bifurcation equation b(λ, α) = 0 can be solved for λ in a neighborhood of (λ 0 , 0). While this could lead to a situation where no bifurcation actually occurs, for example if the solution curve is monotone with respect to α, by assuming an additional generic condition one can guarantee a true saddle-node bifurcation point.
Proposition 2.9 (Existence of Saddle-Node Bifurcations). Suppose that Assumption 2.2 is satisfied and that (8) holds. If in addition the generic condition holds, then the nonlinear problem (2) undergoes a saddle-node bifurcation at (λ 0 , u 0 ).
then the bifurcating solutions exist for λ < λ 0 close to the bifurcation point, and if the ratio is negative for λ > λ 0 .
Proof. Proposition 2.3 shows that characterizing the solution set of (2) in a neighborhood of (λ 0 , u 0 ) is equivalent to solving the bifurcation equation in a neighborhood of (0, 0) ∈ R 2 , where the Taylor expansion for b is given in (5). Due to (8) and the implicit function theorem there exists a smooth function ν = h(α) which is defined in a neighborhood of 0 and whose graph contains all solutions of (10) in that neighborhood. In particular, we have h(0) = 0. Differentiating the identity b(λ 0 + h(α), α) = 0 twice with respect to α and then setting α = 0 furnishes Together with the second equation we finally obtain , which completes the proof of the lemma.
While the above result provides generic conditions that guarantee a saddle-node bifurcation, we need to reformulate these to make them amenable to a rigorous computational approach which involves Newton's method. In other words, we need to derive a nonlinear system with an isolated zero which corresponds to the bifurcation point. For this, we follow the approach in [24] and consider a suitable extended system. More precisely, we supplement the nonlinear parameter-dependent equation (2) by another equation which forces the existence of an eigenfunction v of the appropriate Fréchet derivative, together with a normalizing condition on this eigenfunction. This leads to an extended system of the form where ∈ X * is a fixed element of the dual space of X. We abbreviate this system as where F : Then the following result is analogous to [24,32]. Since in these two citations, the only operators acting on one Banach space are considered, we present the straightforward extension to mapping between different Banach spaces in detail.
Theorem 2.10 (Saddle-Node Bifurcations via Extended Systems). Suppose that Assumption 2.1 is satisfied. Then the following two statements hold.
Thus, the nonlinear problem (2) undergoes a saddle-node bifurcation at (λ 0 , u 0 ) in the sense of Proposition 2.9, if and only if the triple (λ 0 , u 0 , ϕ 0 ) is a non-degenerate zero of the nonlinear map F which defines the extended system (11).
Proof. (a) One can easily see that the Fréchet derivative of F is given by where we again use the abbreviation L = D u F (λ 0 , u 0 ).
To begin with, we show that the Fréchet derivative is one-to-one. Assume that We will now show that this implies (λ,ũ,ṽ) = (0, 0, 0). Applying ψ * 0 to the second component of the Fréchet derivative of F, together with and this in turn furnishesλ = 0. Substituting this identity into the second component one obtains Lũ = 0, i.e., there exists a constant γ ∈ R such thatũ = γϕ 0 . Applying ψ * 0 to the third component of the Fréchet derivative of F then implies implying γ = 0, andũ = γϕ 0 = 0. Now the third component reduces to Lṽ = 0, which givesṽ = ηϕ 0 . The first component then yields 0 = (ṽ) = η · (ϕ 0 ) = η, i.e., we haveṽ = 0. Thus the Fréchet derivative of F is one-to-one. It remains to show that the Fréchet derivative is onto.
Now let γ =γ denote the unique solution of the equation Then there exists a uniqueṽ ∈X such that for all η ∈ R we havẽ If we finally defineη = τ − (ṽ), then one can easily verify that and this completes the proof of (a).
(b) Now assume that there exists an ∈ X * and a ϕ 0 ∈ X such that (λ 0 , u 0 , ϕ 0 ) is a non-degenerate zero of F. According to the form of the third component of F we know that ϕ 0 ∈ N (L). Assume there is another element ϕ 1 ∈ N (L) which is linearly independent of ϕ 0 . Then one can immediately verify that for all numbers α ∈ R the linear combination and is of course also in the kernel of L. This in turn implies that for all scalars is not isolated and therefore the linearization of F at this point cannot be invertible. Thus, the operator F satisfies Assumption 2.2. Now assume that (9) is not satisfied, i.e., we have , which again violates the assumed invertibility of the Fréchet derivative. This implies the validity of (9).
The above lemma shows that generically, one can use standard path-following techniques to follow simple saddle-node bifurcation points as zeros of the system (12), as long as this system has an additional free parameter. This will be the case in our applications to the diblock copolymer model. Notice also that we do not have to separately establish the non-degeneracy conditions (8) and (9), as long as we can guarantee that the zero of the extended system is non-degenerate.
2.3. Pitchfork bifurcation points forced by symmetry. We now turn our attention to the case of pitchfork bifurcation points. Of course, such bifurcation points are non-generic in general systems. Nevertheless, our numerical path following results presented in the introduction clearly indicate that they do in fact occur in the diblock copolymer system. Unfortunately, however, at such pitchfork bifurcations the linearization of the extended system F defined in (12) in the previous section is no longer invertible -which prevents the use of Newton's method to locate them via a computer-assisted proof.
To remedy this problem we need to understand why these non-generic bifurcation points frequently do occur in partial differential equations. As it turns out, pitchfork bifurcations are usually forced through inherent symmetries. For the purposes of this paper, we assume that the symmetry occurs in the specific form introduced in Definition 2.4, i.e., via Z 2 -equivariance. It was shown in Proposition 2.7 that by choosing appropriate projections in the method of Lyapunov-Schmidt, equivariance propagates to the function W , and in some sense also to the bifurcation equation b(λ, α) = 0 through the signs ε X and ε Y introduced in Lemma 2.6. If both of these signs are negative, we are in the situation of a symmetry-breaking bifurcation, and we have the following result.

Proposition 2.11 (Existence of Symmetry-Breaking Pitchfork Bifurcations). Suppose that Assumption 2.2 is satisfied and that
that the projections P and Q are chosen as in Lemma 2.6(d), and that where ξ 0 denotes the unique solution of the equation Then the nonlinear problem (2) undergoes a pitchfork bifurcation at (λ 0 , u 0 ). Locally at this point the solution set of (2) consists of a smooth solution curve parameterized by λ, together with a parabolic curve which is tangent to ϕ 0 at (λ 0 , u 0 ). Consider the ratio where ξ 0 was defined in (15) and ζ 0 ∈ X s is defined by If the ratio is positive, then the solutions on the parabolic branch exist for λ < λ 0 close to the bifurcation point, if is negative then they exist for λ > λ 0 . If = 0, either half of the branch could lie on either side of λ 0 .

This establishes the second solution branch
In order to finish the proof, we only have to verify the tangency statement regarding the second solution branch, as well as its opening side. For this, one just has to differentiate the identity r(h(α), α) = 0 twice with respect to λ. One differentiation implies Computing the second derivative of r(h(α), α) = 0 with respect to α and then letting α = 0 finally implies where we also used Table 1. This completes the proof of the proposition.
In order to rigorously determine the location of symmetry-breaking bifurcation points, we would like to consider an extended system similar to the one defined in (11) and (12). Unfortunately, if we were to consider this system as is, while the point (λ 0 , u 0 , ϕ 0 ) would still be a zero, it would no longer be isolated and therefore preclude the use of any Newton-type method for its solution.
The prevalent role of symmetries in the formation of the pitchfork bifurcation allows for a simple adjustment. For this, we still consider the extended system as defined in (11) and (12). This time, however, one component of the map F r induced by the system is restricted in domain and range to subspaces of X and Y , respectively. That is, we consider the restriction of the full system F given by i.e., we restrict the argument u to the space of symmetric elements, and consider the image of the second component only in the symmetric subspace of Y . Notice that this restriction to Y s in the image is justified by Lemma 2.5(a). Then the following result holds, which is in the spirit of a related result in [39], and extended to the case of different Banach spaces in domain and range.
is an isolated non-degenerate zero. (b) Conversely, if there exists an ∈ X * and a ϕ 0 ∈ X a such that (λ 0 , u 0 , ϕ 0 ) is a zero of the map F r , and if the Fréchet derivative D (λ,u,v) F r (λ 0 , u 0 , ϕ 0 ) is invertible, then the nonlinear operator F satisfies all assumptions of Proposition 2.11. Thus, the problem (2) undergoes a symmetry-breaking pitchfork bifurcation at (λ 0 , u 0 ) in the sense of Proposition 2.11, if and only if (λ 0 , u 0 , ϕ 0 ) ∈ R × X s × X a is a nondegenerate zero of (17).
Proof. (a) As before, the Fréchet derivative of F r is given by where we use the abbreviation L = D u F (λ 0 , u 0 ). In the following, we assume that the projections P and Q have been chosen as in Lemma 2.6(d).
It remains to show that the Fréchet derivative is onto. Let (τ, y, z) ∈ R × Y s × Y be arbitrary. Since y ∈ Y s ⊂ R(L) and N (L) ∩ X s = {0} there exists a unique elementũ ∈ X s such that Lũ = y . .
(b) We now turn our attention to the proof of (b), and assume that there exists an ∈ X * and a triple (λ 0 , u 0 , ϕ 0 ) ∈ R×X s ×X a such that F r (λ 0 , u 0 , ϕ 0 ) = (0, 0, 0), and that the Fréchet derivative of F r at this point is invertible. Due to the specific form of the third component of F r we have ϕ 0 ∈ N (L). If there were another element ϕ 1 ∈ N (L) which is linearly independent of ϕ 0 , then one can immediately verify that for all α ∈ R the linear combination ϕ α = (1 − α (ϕ 1 ))ϕ 0 + αϕ 1 satisfies (ϕ α ) = 1, and it is clearly also in the kernel of L. Thus, we have F r (λ 0 , u 0 , (1 − α (ϕ 1 ))ϕ 0 + αϕ 1 ) = (0, 0, 0) for all α ∈ R, i.e., the zero (λ 0 , u 0 , ϕ 0 ) is not isolated and therefore the linearization of F r at this point cannot be invertible. Thus, the operator F r satisfies Assumption 2.2.
From our assumptions, we already know that S X u 0 = u 0 , that S X ϕ 0 = −ϕ 0 , and we can choose the projections P and Q as in Lemma 2.6(d). In addition, we have S * Y ψ * 0 = ε Y ψ * 0 for some as of yet unknown sign ε Y ∈ {±1}. Finally, since the projection I − P maps X s into Y s ∩ R(L) and since N (L) ∩ X s = {0}, Lemma 2.5(a) shows that there exists a unique ξ 0 ∈ X s which solves (15).
It remains to show that (14) is satisfied. For this, let z ∈ Y \ R(L) be arbitrary. Then the assumptions of (b) imply that there exists a (λ,ũ,ṽ) ∈ R × X s × X such that Assume for the moment that we have in factλ = 0. Then the equation for the second components of (18) implies Lũ = 0, which in turn givesũ ∈ N (L)∩X s = {0}. But in this case the last component of (18) reduces to Lṽ = z ∈ R(L), which is impossible. Thus, we necessarily have to haveλ = 0, and the second component in (18) i.e., according to (15) one obtainsũ =λξ 0 . Substituting this expression into the last component of (18) finally yields and applying ψ * 0 to both sides leads tõ λ · (ψ * 0 D λu F (λ 0 , u 0 )[ϕ 0 ] + ψ * 0 D uu F (λ 0 , u 0 )[ϕ 0 , ξ 0 ]) = ψ * 0 z = 0 , due to ψ * 0 Lṽ = 0 and z ∈ R(L). This establishes (14), and therefore all assumptions of Proposition 2.11 are satisfied. This completes the proof of the theorem. Remark 2.13 (Alternative Expressions for ξ 0 and ζ 0 ). The proof of the above theorem shows that the functions ξ 0 ∈ X s and ζ 0 ∈ X s defined in Proposition 2.11 can equivalently be defined as solutions of the two identities which no longer include the projection P .
As in the previous subsection, the above theorem shows that generically, one can use standard Newton-type techniques to determine symmetry-breaking pitchfork bifurcation points as zeros of the system (17). This time, however, the system is restricted to symmetry subspaces, and this will in fact lead to significant reductions in dimensions later on.
To close this section, let us briefly contrast the reduced system (17) to the one considered in [39]. As we mentioned earlier, this reference is concerned with a numerical method for computing the location of symmetry-breaking pitchfork bifurcations, and it employs the even further restricted system F rr (λ, u, v) = (0, 0, 0), where i.e., they restrict the argument u to the space of symmetric elements, and the argument v to the space of antisymmetric elements. From a purely computational point of view, this system leads to a reliable method. We would like to point out, however, that it cannot easily be used in a computer-assisted proof setting. As the following example shows, even if the system F rr (λ, u, v) = (0, 0, 0) has an isolated zero, this does not guarantee the existence of a symmetry-breaking pitchfork bifurcation as in Proposition 2.11. The reason for this surprising behavior lies in the fact that even for an isolated zero of this further reduced system, the Fréchet derivative D u F (λ 0 , u 0 ) does not necessarily have a one-dimensional kernel; it could be higher-dimensional. While this precludes a computer-assisted proof, it does work in a purely numerical setting, since in this setting the kernel is almost-surely onedimensional.
Example 2.14 (Degenerate Symmetry-Breaking Bifurcations). In this example we show that even if the reduced system F rr (λ, u, v) = (0, 0, 0), with F rr as in (19), has an isolated zero, this does not mean that the Fréchet derivative of the underlying map F has a one-dimensional kernel. To see this, consider the class of functions suggested in [39], which are of the form where f : R 2 → R will be specified later on. If we let X = Y = R 2 , then one can easily see that the mapping F : R × X → Y is Z 2 -equivariant with respect to the symmetries S X = S Y = 0 1 1 0 , and the symmetric and antisymmetric subspaces are given by Based on the form of F , if there is a curve (λ, x(λ)) such that f (λ, x(λ)) = 0 holds, then we automatically have F (λ, x(λ), x(λ)) = (0, 0). Consider now the specific function f (λ, y) = y − λ 1 − y 2 . Then one can easily see that f (λ, y) = 0 whenever we have λ = y − y 3 . Moreover, if we define λ ± = ±2/(3 √ 3) and x ± = ±1/ √ 3, then ∂f ∂y (λ ± , x ± ) = 0 , which in turn implies Thus, the Fréchet derivative D (x1,x2) F (λ ± , x ± , x ± ) has a two-dimensional kernel on X, and a one-dimensional kernel on X a . One can show that despite the fact that (λ ± , x ± , x ± ) are isolated solutions of the system F rr (λ, u, v) = (0, 0, 0), the bifurcation at these two points is not a pitchfork bifurcation in the sense of Proposition 2.11. Rather, at each point we have a degenerate bifurcation point. See also Figure 2. The same type of degenerate bifurcation occurs for the coupled cell reaction model with an Arrhenius reaction rate term in Example 4.1 of [39] with ε = 0.
These simple examples illustrate the general problem that we cannot determine the kernel of the linearization by only looking at the antisymmetric part of the space.

Application to the diblock copolymer equation.
In this section we demonstrate how the diblock copolymer model (1) fits into the framework described above. For more details we refer the reader to [17]. Rather than studying the model in the form described in the introduction, we apply the transformation u → µ + u so that in the following we always consider the mass constraint zero and have incorporated µ into the equation. After this transformation, finding equilibrium solutions of the evolution equation is equivalent to solving the nonlinear operator equation where F is a parameter-dependent operator F : R × X → Y on the Hölder-type spaces and with nonlinearity f (u) = u−u 3 . In order to keep our notation uniform throughout the section, we only emphasize the parameter λ in the definition of F , since this is our primary bifurcation parameter. However, the secondary parameter σ will prove to be essential in order to find location curves of bifurcation points in the diblock copolymer model. Notice also that for every choice of the parameters λ, σ, and µ the constant function u ≡ 0 satisfies (20). This solution is referred to as the trivial solution.
The bifurcation diagram of (20) in a neighborhood of the trivial solution has been completely described in [17] using a standard Lyapunov-Schmidt approach. In order to describe these results, note that the eigenvalues and eigenfunctions of the negative Laplacian −∆ on the one-dimensional domain Ω = (0, 1) and subject to homogeneous Neumann boundary conditions, are given by κ k = k 2 π 2 and ϕ k (x) = √ 2 cos kπx for k ∈ N .
Since the space X contains a mass constraint, the constant eigenfunction with eigenvalue zero will play no role in the following and has therefore been omitted. Using this notation, it was shown in [17] that the bifurcation points from the trivial solution of (20) are given by where σ ≥ 0. Since we are only interested in bifurcation points at positive λ-values, the indices k have to satisfy κ k > σ/f (µ), and one needs to assume that f (µ) > 0, i.e. the total mass µ has to be in the spinodal region. It was shown in [17], and briefly described in the introduction, that most of the solution branches of (20) which bifurcate from the trivial solution undergo a variety of secondary bifurcations. Studying these secondary bifurcation points using rigorous computational techniques is the main subject of this paper. For this, we need to first show that the nonlinear mapping F defined above satisfies all the conditions of the abstract results described so far. This will be accomplished in the following proposition.
Proposition 2.15. Consider the nonlinear operator F defined in the diblock copolymer equilibrium equation (20), and let the Banach spaces X and Y be defined as in (21). Furthermore, assume that (λ 0 , u 0 ) ∈ R + × X is arbitrary, and that σ = σ 0 > 0 and µ = µ 0 ∈ R are fixed. Then the following hold.
i.e., the functions ϕ k solve this linear differential equation subject to the zero mass constraint and Neumann boundary conditions as in (1). (c) If ψ 0 , . . . , ψ n−1 ∈ C 4, [0, 1] denote n linearly independent solutions of the adjoint problem subject to the Neumann conditions w x (0) = w xxx (0) = w x (1) = w xxx (1) = 0, but without imposing any integral constraint, then the range of L is characterized by Proof. Rather than considering X and Y directly, we first consider the standard Hölder spaces without mass constraint given by Then F : R×X → Y is a smooth nonlinear operator. In fact, standard elliptic theory shows that in this setting, the operator F is a Fredholm operator of index zero: For any choice of λ 0 , σ 0 , µ 0 ∈ R and any u 0 ∈ X the linearization L = D u F (λ 0 , u 0 ) is an elliptic operator with principal term −v xxxx . The latter operator is Fredholm with index zero, since in the associated Sobolev setting it is in fact self-adjoint. Since the additional terms in L are of lower differentiation order, they are a compact perturbation, and (a) follows for L, but with X and Y replacing X and Y . For more details we refer the reader to [41, pp. 259ff]. We now turn our attention to the operator F restricted to the spaces X and Y which impose the zero mass constraint. To begin with, it is easy to see that for any λ 0 ∈ R and u 0 ∈ X one has F (λ 0 , u 0 ) ∈ Y , i.e., considering F as an operator from R × X into Y is well-defined. Similarly, we have L(X) ⊂ Y . But even more is true. If v ∈ X is arbitrary, then due to the imposed boundary conditions and integration by parts. The latter identity, together with our assumptions λ 0 = 0 and σ 0 = 0, furnishes: This identity will prove to be essential below. Now let N (L| X ) = span[ϕ 0 , . . . , ϕ n−1 ] ⊂ X denote the nullspace of L| X considered as operator between X and Y, and let N (L * | X ) = span[ψ 0 , . . . , ψ n−1 ] ⊂ X denote the nullspace of L * | X considered as operator between X and Y. Due to our above discussion, both nullspaces have to have the same finite dimension n. Furthermore, property (24) immediately shows that N (L| X ) ⊂ X , i.e., we have dim N (L| X ) = dim N (L| X ) = n .
We now turn our attention to the characterization of the range of L. According to the closed range theorem, the operator L| X : X → Y satisfies R (L| X ) = w ∈ Y : It is clear from the form of the adjoint operator L * | X that the functions ψ k are in general not in the space X. Thus, choose α 0 , . . . , α n−1 ∈ R andψ 0 , . . . ,ψ n−1 ∈ X such that ψ k = α k +ψ k for all k = 0, . . . , n − 1. We claim that the functions ψ 0 , . . . ,ψ n−1 ∈ X are linearly independent. For this, assume that γ 0 , . . . , γ n−1 ∈ R are such that Applying L * | X on both sides, together with the fact that every constant function is an eigenfunction of L * | X with eigenvalue −λ 0 σ 0 , then furnishes which due to λ 0 σ 0 = 0 finally implies  The assumed linear independence of ψ 0 , . . . , ψ n−1 then yields γ 1 = . . . = γ n−1 = 0, and therefore also the reduced functionsψ 0 , . . . ,ψ n−1 are linearly independent. After these preparations we can now easily complete the proof of Proposition 2.15. With the help of (24) one can show that Sinceψ 0 , . . . ,ψ n−1 are linearly independent in Y , the range R(L| X ) ⊂ Y therefore has codimension n in Y , which equals the dimension of the nullspace N (L| X ) ⊂ X. This completes the proof.
The above proposition shows that despite imposing the mass constraint both in the domain and in the range of the nonlinear operator, we retain the crucial Fredholm property. We would also like to stress again the point that the solutions ψ 0 , . . . , ψ n−1 of the adjoint problem, which are used to characterize the range of L, do not in general satisfy the zero mass constraint. Finally, if in Proposition 2.15 the kernel is one-dimensional, and if we define a functional ψ * 0 ∈ Y * via ψ * 0 (w) =   Figure 1. Note that as in Figure 3, the bifurcation diagram is a double cover; corresponding to each of these four solutions, there is another solution at the same point in the bifurcation diagram. See also Theorems 3.13, 3.14, and 3.15. Proposition 2.15 enables us to apply the general results of the previous sections to the diblock copolymer model. For example, we can use the nonlinear system (11) to determine saddle-node bifurcation points via a standard Newton iteration, see also Theorem 2.10. Examples of such saddle-node bifurcation points are shown in Figure 3.
Similarly, we can use symmetry methods to establish the location of symmetrybreaking pitchfork bifurcation points via the solution of the extended system (16) using Newton's method, see also Theorem 2.12. For this, of course, we need to specify suitable symmetry operations. Consider for example the symmetry operators S X and S Y defined via i.e., via the same formula in both X and Y . Since X ⊂ Y is embedded via the identity map, we can apply Lemma 2.8 with J = I. This implies that if ϕ 0 ∈ R(L), then the signs ε X and ε Y in Lemma 2.6 necessarily have to agree. In the right column of Figure 4, two equilibrium solutions of the diblock copolymer model are depicted in orange, along with the kernel functions of the Fréchet derivative D u F (λ 0 , u 0 ), which are shown in blue. These solutions are symmetry-breaking pitchfork bifurcation points which lie on the second and fourth bifurcation branches from the trivial solution curve shown in the right image of Figure 1. The equilibria are symmetric elements with respect to the above symmetry S, and the associated kernel functions are antisymmetric. Thus these solutions satisfy the symmetry conditions needed for a symmetry-breaking pitchfork bifurcation, and in both of these cases, Proposition 2.15 will enable us to use Theorem 2.12 to rigorously prove their existence.
But what about the pitchfork bifurcation points on the first, third and fifth branches shown in the right image of Figure 1? In these cases, the solutions on the branch are no longer symmetric elements. If instead we define a new symmetry operator via then the pitchfork bifurcation points are again symmetric elements. The corresponding solutions for the first and third branch are shown in the left column of Figure 4 along with their associated kernel functions. Since the kernel function in the upper left panel is antisymmetric with respect to this new symmetry, this is a symmetry-breaking bifurcation point which can be treated using Theorem 2.12, but with respect to the new symmetry. In contrast, the solution and kernel function of the third branch solution in the lower left panel are both symmetric with respect to the new symmetry. This implies that our symmetry-breaking pitchfork bifurcation result no longer applies. In fact, this bifurcation is not a Z 2 -symmetry-breaking bifurcation, but rather a symmetry-breaking bifurcation with a different equivariance group. Thus a different approach is needed to treat this bifurcation, and we will address this in a future paper.
3. Rigorous verification of bifurcations. As already mentioned in Section 2.4, Proposition 2.15 enables us to study saddle-node and pitchfork bifurcation points via Theorem 2.10 and Theorem 2.12, respectively. In particular, by Theorem 2.10, proving existence of a non-degenerate isolated zero of (12) implies the existence of a saddle-node bifurcation, and by Theorem 2.12, proving existence of a nondegenerate isolated zero of (17) implies the existence of a pitchfork bifurcation.
In this section, we use the tools of rigorous computing (e.g. see [33]) to verify existence of bifurcations in the diblock copolymer equation. In order to do so, we begin by presenting the equivalent of (11) in the space of Fourier coefficients. We note that the reformulation of a function space problem in terms of an infinite series is standard as both a numerical and analytical technique, but it also has a long history as a method of computer-assisted proof, such as for example in Lanford's proof of the Feigenbaum conjectures [18]. Note that while our general results in previous sections establish conclusions in the the more general case of Hölder continuity, since we are considering the space of Fourier coefficients, our results in this section have much more additional regularity than just Hölder continuity. Namely, we are able to rigorously establish real analytic bifurcating solutions.
Consider the steady states diblock copolymer equation (20) with f (u) = u − u 3 . Our results apply for all values of µ, but in order for the mass constraint to allow for both symmetric and asymmetric solutions, in our numerics, we fix µ = 0. That is Plugging the cosine Fourier expansion a k cos(kπx), (a k ∈ R, a −k = a k , a 0 = 0) (26) in F (λ, u) = 0 leads to k∈Z g k e ikπx = 0, with where and The relations a k ∈ R, a 0 = 0 and a −k = a k imply the new relations g k ∈ R, g −k = g k and g 0 = 0. Hence, we only need to solve g k = 0 for k ≥ 1. Set g def = (g k ) k≥1 . Now consider the kernel equation associated to the steady states diblock copolymer equation:

Plugging the cosine Fourier expansion of
where As above, the relations h k ∈ R, h −k = h k and h 0 = 0 implies that we only need to . Recall the definition of the extended operator F given in (11). We re-order the entries and consider Moving to Fourier space, and recalling (27) and (30), the functional equation F(λ, u, v) = 0 becomes, for a fixed parameter value σ ∈ R, where where the condition η(b) = 0 implies that we fix the k 0 -th component of the eigenvector to be equal to a fixed valueb k0 , as η(b) = 0 is equivalent to b k0 =b k0 The corresponding element in the dual space is The Banach space on which we study the zeroes of f defined in (31) is endowed with the norm where δ > 0 is a weight and where is equipped with a "weighted ell one norm" for some fixed weight ν ≥ 1.
We solve the problem f (x) = 0 using the field of rigorous numerics (e.g. see [33]). This requires recalling some basic tools from functional analysis. Since all the results of the section are classical and can be found for instance in [16], we omit the proofs.
3.1. Functional-analytic background. We note that 1 ν defined in (35) is a Banach space and moreover has the property of being a Banach algebra under discrete convolution defined as More explicitly, we have the following.
Then, given a ∈ 1 ν and c ∈ ∞ ν −1 , we conclude by (39) that This bound is used to estimate linear operators of the following type. Denote by B(V, W ) the space of bounded linear operators acting between a Banach space V and a Banach space W , and denote · B(V,W ) the operator norm.
Lemma 3.3. Given ν ≥ 1, k ≥ 1 and a ∈ 1 ν , the function l k a : 1 with h ∈ 1 ν , is a bounded linear functional, and Fix a truncation mode to be m. N ∈ N and letᾱ = (ᾱ 1 , . . . ,ᾱ N , 0, 0 Then, for all h ∈ 1 ν such that h 1,ν ≤ 1, 3.2. The general rigorous computational method. We now present a general method to prove existence and compute global smooth solution curves of the equation f (x, σ) = 0 in the Banach space X as defined in (33). The method is based on the radii polynomial approach, first introduced in [11]. The version of the approach we use here is a direct application of the work [7] and is strongly influenced by the rigorous branch following method of [34]. The idea is to compute a set of numerical approximations {x 0 , . . . ,x j } of f (x, σ) = 0 at the parameter values {σ 0 , . . . , σ j } by considering a finite dimensional projection, to use the approximations to construct a global continuous curve of piecewise linear interpolations between thex i 's and to apply the uniform contraction theorem on tubes centered at each segment to conclude about the existence of a unique smooth solution curve of f = 0 nearby the piecewise linear curve of approximations.
where g (m) (x (m) , σ) ∈ R m−1 (resp. h (m) ) corresponds to the finite part of g (resp. h) of size m − 1. We have that f (m) : R 2m−1 × R → R 2m−1 . Assume that using a standard parameter continuation (i.e. a predictor-corrector algorithm based on Newton's method) in σ, we compute a set {x 0 , . . . ,x j } of approximations at the parameter values {σ 0 , . . . , σ j } respectively that defines a piecewise linear approximation curve (see Figure 5). The next step is to show existence of a unique smooth solution curve C of f = 0 nearby the piecewise linear curve of approximations, as portrayed in Figure 5. This task is twofold. First, one shows the existence of a unique portion of solution curve C (i) in a small tube centered at the segment }. This is done in Theorem 3.5. Second, one shows that is a global continuous solution curve of f (x, σ) = 0.

Uniform Contraction and the Radii Polynomial Approach.
Let us define what is required to prove existence of some portion of curve C (i) . Without loss of generality, let us introduce the idea to prove the existence of C (0) that is the piece of curve close to the segment [x 0 ,x 1 ]. For any s in [0, 1], we set where A is an injective linear operator approximating the inverse of Df (x 0 , σ 0 ). Denotex 0 = (λ 0 ,ā 0 ,b 0 ). In the sequel, we use the notation x = (x 1 , x 2 , x 3 ) with x 1 = λ, x 2 = a and x 3 = b. To obtain the operator A, assume that • the Jacobian matrix Df (m) (x 0 , σ 0 ) has been computed; • an approximate inverse A (m) of Df (m) (x 0 , σ 0 ) has been computed; with A (m) 11 The operator A which acts as an approximate inverse for where where δ i,j equals 1 if i = j and 0 otherwise.
Combining the above, A is a linear operator which acts on x = (x 1 , x 2 , x 3 ) ∈ X component-wise as with (Ax) 1 ∈ R and (Ax) i ∈ 1 ν , for i = 2, 3. Moreover, define the linear operator which acts on c = (c 1 , c 2 , c 3 ) component-wise as where for i, j = 2, 3, A † i,j c j is defined component-wise by The operator A † acts as an approximation for D x f (x 0 , σ 0 ), and is used in the radii polynomial approach, which we now present. A more detailed proof can be found for instance in [19,20].
Define the radii polynomial If there exists r 0 > 0 such that p(r 0 ) < 0, Furthermore, these are the only solutions in the tube s∈[0,1] B r0 (x s ).
3.3.1. The bound Y 0 . Recall that the bound Y 0 satisfies the inequality (51). Denote ξ s = (λ s ,ā s ,b s , σ s ) and ξ = (λ, a, b, σ). Since ξ s = ξ 0 +s(ξ 1 −ξ 0 ), then using Taylor's theorem where Using interval arithmetic, one can easily compute bounds Y which we denote as z = (z 1 , z 2 , z 3 ). Note that z 1 = 0 and that Using Corollary 3.4, we get that for 1 ≤ k < m, The following elementary result is useful to bound the tail quantities involved in z 2 and z 3 .
To verify rigorously a pitchfork bifurcation, we use the general approach of Section 3.2, as in the case of saddle-node bifurcations. All estimates are similar as in the case of saddle-node bifurcations, expect that in this case we consider the norm · X (i) given in (68), where the index i depends on the symmetry of the bifurcation. This implies that all expansions are the same, but the norms are replaced by (68). Using these new expansions, we define the equivalent radii polynomial which is the same as the one defined in (66), except that the bounds Z 0 , Z 1 , Z 2 , Z (1) 2 and Z (2) 2 defining it are slightly different, as they come from a different problem.

3.5.
Results. In this section, we are finally ready to state our results, which consist of a set of three validated branches of saddle-node bifurcations and three validated branches of symmetry-breaking pitchfork bifurcations. As mentioned in the introduction, each of these cases were found numerically in [17], where it was observed numerically that there was a common scaling law for different bifurcation branches. The choice of bifurcation points was made in a somewhat arbitrary way as secondary bifurcation point for small λ that still satisfied the conditions of our theorem. We emphasize that the validation of these branches of bifurcations serves two purposes: first, they illustrate our general functional analytic methods, methods which can easily be adapted to other equations. Second, for the diblock copolymer equation, this is a first step towards our eventual goal of validating the scalings of branches that we have observed in the previous paper. We state each of these results as a theorem, which is presented along with an associated table giving the detailed information on the Fourier coefficients found for each solution. The tables are presented for completeness and preciseness, but we do not expect that a reader would find the data particularly illuminating unless it was being used as a basis for future work. For all the computer-assisted proofs, the weight δ in the definition the norm of the Banach spaces is fixed to be equal to the numerical valueλ 0 . This choice of weight δ is used to compensate for the fact that the variable λ sometimes is several orders of magnitude larger than the a 1,ν and b 1,ν . All proofs are performed using MATLAB codes available at [21] and use the interval arithmetic package INTLAB [30].
3.5.1. Saddle-Node Bifurcation Points. We begin by presenting three sample theorems of existence of saddle-node bifurcations at the fixed parameter value σ = 6, which we proved using the radii polynomial (66) and Theorem 3.5. Note that the direction of the bifurcation (i.e. the sign of in Proposition 2.11) was not important for our purposes, so we did not compute it. In order to do so, we would have to validate the sign of the involved reals, which would not be hard, but would involve a norm bound for the involved derivatives.
The saddle-node bifurcation point of Theorem 3.7 is portrayed in Figure 3.
The saddle-node bifurcation point of Theorem 3.8 is portrayed in the left part of Figure 6.
The saddle-node bifurcation point of Theorem 3.9 is portrayed in the right part of Figure 6. Proof. The proof is similar to the proof of Theorem 3.7. In this case ν = 1.01, m = 121, and the MATLAB script script proof sn pt2.m computes the coefficients of the radii polynomials with interval arithmetic and the Fourier coefficients associated to the numerical approximationx 0 ∈ R 2m−1 = R 241 are given in the MATLAB datafile pt2 sn.mat.
Using the radii polynomial approach, we proved the following three results. The branch of saddle-node bifurcations proven in Theorem 3.10 is portrayed in red in Figure 7. The branch of saddle-node bifurcations proven in Theorem 3.11 is portrayed in green in Figure 7.   Figure 7. Three global C ∞ branches of saddle-node bifurcation points of the diblock copolymer equation. The red (respectively green, blue) branch is proven in Theorem 3.10 (respectively Theorem 3.11, Theorem 3.12).

Pitchfork Bifurcation Points.
We begin by presenting three theorems of existence of pitchfork bifurcations at the fixed parameter value σ = 6, which we proved using the radii polynomial (70) together with Theorem 3.5.
The bifurcation point of Theorem 3.13 is portrayed in the top left plot of Figure 4.
The bifurcation point of Theorem 3.14 is portrayed in the top right plot of Figure 4.
Proof. The proof is similar to the proof of Theorem 3.13. In this case ν = 1.1, m = 29, and the MATLAB script proof thm2 pitchfork.m computes the coefficients of the radii polynomials with interval arithmetic and the Fourier coefficients associated to the numerical approximationx 0 ∈ R 3(m−1)/2 = R 43 are given in the MATLAB datafile pt 105.mat.
The bifurcation point of Theorem 3.15 is portrayed in the bottom right plot of Figure 4.
Proof. The proof is similar to the proof of Theorem 3.13. In this case ν = 1.01, m = 57, and the MATLAB script proof thm3 pitchfork.m computes the coefficients of the radii polynomials with interval arithmetic and the Fourier coefficients associated to the numerical approximationx 0 ∈ R 3(m−1)/2 = R 85 are given in the MATLAB datafile pt 174.mat.
In the last three theorems, we have established the existence of pitchfork bifurcation points. The next three theorems establish that each of these points lies within a branch in the two-parameter λ − σ-plane. The branch of pitchfork bifurcations of Theorem 3.16 is portrayed in red in Figure 8. The branch of pitchfork bifurcations of Theorem 3.17 is portrayed in green in Figure 8.    Figure 9. The cosine Fourier coefficients of the saddle-node bifurcation point from Theorem 3.7. We showā k for k ≥ 1. Note that all even coefficients are 0. Figure 10. The cosine Fourier coefficients of the saddle-node bifurcation point from Theorem 3.8. We showā k for k ≥ 1. Note that all even coefficients are 0.  Figure 11. The cosine Fourier coefficients of the pitchfork bifurcation point from Theorem 3.13. We showā k for k ≥ 1. Note that all even coefficients are 0.  Figure 13. The cosine Fourier coefficients of the pitchfork bifurcation point from Theorem 3.15. We showā k for k ≥ 2. Note that all other coefficients are 0.