STEADY STATE BIFURCATIONS FOR THE KURAMOTO-SIVASHINSKY EQUATION: A COMPUTER ASSISTED PROOF

. We apply the method of self-consistent bounds to prove the existence of multiple steady state bifurcations for Kuramoto-Sivashinski PDE on the line with odd and periodic boundary conditions.


1.
Introduction. The aim of this paper is to present all necessary tools that with computer assistance allow us to produce a rigorous steady state bifurcation diagram for dissipative PDEs. For the purpose of this introduction, by a dissipative PDE we mean an infinite-dimensional system that is well approximated by its Galerkin projections.
Our approach is based on the concept of the self-consistent a priori bounds developed in [23,24,26], which when applied to steady states can be thought as an improvement of the Cesari method [3] (see [23] for more details). Informally speaking, the self-consistent a priori bounds are used to control rigorously the difference between a PDE and its Galerkin projections. Then we apply finite dimensional tools, mainly the local degree and the Implicit Function Theorem, to infer information about the steady states and their bifurcations. In an application of the proposed method to a concrete dissipative PDE, one needs to check many inequalities (up to several hundred for the Kuramoto-Sivashinsky equation in one spatial dimension). This task, while practically impossible for a human, can be rigorously accomplished by a computer program using interval arithmetic [18,19].
In this paper we apply our method to the Kuramoto-Sivashinsky equation (which we will now denote as the KS equation), which was introduced in [16,21] in the context of wave propagation. The KS equation is given by We assume odd and periodic boundary conditions u(t, x) = u(t, x + 2π), u(t, x) = −u(t, −x).
The KS equation is well studied in the literature [5,6,9,10,11,12,13,14,15,22]. It serves as one of the model PDE examples for which it was shown rigorously that 96 PIOTR ZGLICZYŃSKI the dynamics is finite dimensional [9,13], but there are virtually no rigorous results about the details of the dynamics away from the zero solution.  [1], where a different but equivalent form of this equation was used. The relation between α and µ is given by µ = 4/α.
As a test for our method we would like to prove that the non-rigorous steady state bifurcation diagram presented in [12, Fig 1a] and [15, Fig. 3.2, 3.4, 3.5] is correct. In this paper we did not get this far. Our main results about the steady state bifurcations for the KS equation can be formulated as follows. (See [15,12,22] (also Section 6.1) for the explanation of the names of steady states branches.) Theorem 1.1. There are pitchfork symmetry breaking bifurcations for equations (1)(2) for the following values of ν: • For ν ∈ 0.247833 + 2 · 10 −6 · (−1, 1), both unimodal branches collide with the negative bimodal branch (Point iP 1 in Fig. 1). • For ν ∈ 0.177336 + 2 · 10 −7 · (−1, 1), there is a creation of the bi-tri branch off the positive bimodal branch (iP 2 in Fig. 1). • For ν ∈ 0.075627151 + 5 · 10 −9 · (−1, 1), there is a creation of the giant branch off the negative bimodal branch (P 1 in Fig. 1). For the following values of ν there are the bifurcations consisting of intersections of two branches of steady states.
To summarize, we were able to establish the existence and determine the (in)stability of most main steady state bifurcations involving the zero solution branch and the unimodal, bimodal, and trimodal branches.
The paper is organized in two main parts: an abstract part, where we state and prove abstract theorems, and a details part, where we provide all necessary formulas, which allow us (with computer assistance) to verify the assumptions of the abstract theorems for the KS equation with odd and periodic boundary conditions.
To construct a rigorous steady state bifurcation diagram for a dissipative PDE, one needs to solve the following problems.
1: How does one establish the local uniqueness for regular steady states ? 2: How does one obtain the regularity and compute the derivatives of steady states with respect to the parameters? 3: How does one handle the bifurcation point in an infinite dimensional situation when the bifurcation point is not given explicitly and the spectral data are hard to obtain?
It is apparent that problem 1 is much easier than the other ones. In Section 3 we present the method which allows us to establish the local uniqueness for regular steady states. It works for all the fixed points for the KS equation, whose existence was established in [23]. The main result in this section is Theorem 3.7. In Section 4, we discuss the criteria to establish the (asymptotic) stability and instability of the fixed point. The main result is Theorem 4.2. Again we were able to rigorously establish the stability/instability of all fixed points for the KS equation from [23].
The problem 2 is harder than 1, but its solution is required also in the solution of 3. In Section 5, it is proven that whenever we can prove for the KS equation the existence and uniqueness for a parameter range using the self-consistent a priori bounds, then the dependence on the parameters is C ∞ . Sections 5 and 10 contain, respectively, the description of an effective algorithm and all necessary formulas for the KS equation, which allow us to compute rigorously the derivatives of the steady states with respect to the parameters and the derivatives of the functions appearing in the Liapunov-Schmidt method (see [4]), which will be essential in the solution of problem 3.
To solve problem 3, we use an approach which is a combination of the methods presented in [4] and the self-consistent a priori bounds. Section 6 contains the abstract theorems and Section 11 the computational details related to the proof of the existence of the symmetry breaking pitchfork bifurcation off the bimodal branches (most of primary bifurcations are of this type for the KS equation) and intersections of two steady state branches (see Theorem 1.1). Companion file bifdata.txt [27] contains the relevant numerical data from the proof of the stability/instability of two exemplary fixed points and the bifurcations listed in Theorem 1.1.
Sections 7 to 11 contain the formulas necessary for the application of the theory developed in Sections 3, 4, 5 and 6 to the KS equations. Our intention is to give enough details so that the formulas given here together with the ones in [23,26] can serve as a documentation of our program.
From the inspection of the bifurcation diagrams in [15,12] it is quite clear that tools given in this paper should be sufficient for providing a rigorous steady states bifurcation diagram over a reasonable range of ν, because apparently all steady state bifurcations appear to be either folds (on regular branches) or the symmetry breaking bifurcations and intersections discussed in this paper. Hence linking our method with a continuation approach should be enough for the task. This is the approach taken by Maier-Pappe and co-authors in [17], where using the method of self-consistent a priori bounds the authors were able to continue the branches of steady states for the Cahn-Hillard equation on the square (they did not treat bifurcations).
The first draft version of this paper was prepared in 2003 and then later in 2005. Arioli and Koch in [1] using different tools were able to produce the full steady state bifurcation diagram for the KS equation with odd and periodic boundary conditions. Arioli's and Koch's method differs considerably from ours since they use the 'integral formulation' of the stationary KS equation and rely heavily on the abstract PDE theory. In contrast, our approach is rather ODE-type as we prove everything relying mainly on the isolation concept originating in the Conley index theory (see references in [23]).
1.1. The Kuramoto-Sivashinsky equation in the Fourier domain. Throughout this paper we look at a dissipative PDE as an infinite ladder of ordinary differential equations. To be more specific the KS equation with odd and periodic boundary conditions can be reduced (see [23]) to the following infinite system of ordinary differential equations for the coefficients of the Fourier expansion of a n a k−n + 2k ∞ n=1 a n a n+k k = 1, 2, 3, . . . .
2. The method of the self-consistent a priori bounds. Consider a Hilbert space H, {e i } an orthogonal basis, and X k = span(e 1 , . . . , e k ). For x ∈ H, by x i we will denote the i th coordinate and for any function we define F i (x) the same way. For any n, by P n denote the projection onto the subspace spanned by X n = {e 1 , e 2 , . . . , e n } and by Y n = (I − P n )H, we will denote the orthogonal complement of X n in H.
Assume that F : D(F ) → H, D(F ) ⊆ H. We investigate the equation form the self-consistent a priori bounds for Equation (5).
we have form the topologically self-consistent a priori bounds for equation (5) if W and {x ± k } are the self-consistent a priori bounds for Equation (5) and the following condition holds.
C5: For all x ∈ ∂N and for all u ∈ We define a modified projection operator P * k for k ≥ m by P * k (x) = P k (x) + (I − P k )y. (8) Observe that for k > M we have P * k = P k . The main reason to introduce the projection P * k is to have the following property: Theorem 2.3. Assume that N ⊆ W and {x ± k } are the topologically self-consistent a priori bounds for Equation (5). Assume that Consider a set Obviously , and hence by C3 it follows that N k ⊆ dom (F ) and P k F is continuous on N k .

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It is easy to see from C4 that this definition does not depend on the choice of x.
We will compute the degree deg(P k F, int N k , 0) using a homotopy linking P k F with the map G defined as follows.
The homotopy H is given by the following conditions.
Let us observe that H(0, x) = P k F (x) and H(1, x) = G(x). It remains to show that deg(H(t, ·), int N k , 0) is defined and does not depend on t.
For this it is enough to show that Observe that if x ∈ ∂N k , then one of the following conditions is satisfied.
Assume (14) holds. Since P m ((1 − t)x + tP * m (x)) = P m (x) ∈ ∂N , the condition C5 implies P m H(t, x) = 0, and hence P k H(t, x) = 0. Now assume (15). Without any loss of generality we can assume that x i = x + i and α i = 1. The proof for the other cases is analogous. This means that From the multiplicative property of the local degree and our assumption about the degree of P m F it follows that This means that there exists a point x k ∈ int N k , such that P k F (x k ) = 0. Passing to the limit as in the proof of Theorem 2.16 in [23] we obtain x * such that F (x * ) = 0.
3. The issue of local uniqueness. Assume H is a real Hilbert space, F : k , a + k ] form the self-consistent a priori bounds, and that W is convex. (In fact we need only the conditions C1, C2, and C3).
We need two additional conditions about the derivatives of F . F1: For every i and j, ∂Fi ∂xj : V → R is continuous.
where c ij ∈ ∂Fi ∂xj (V ) I , which is defined by Moreover, if x → y then c ij → ∂Fi ∂xj (y). Proof. Let us fix an i and for any n we consider the map F i : X n → R. We have for any x, y ∈ X n Let us take any x, y ∈ V , and then from (17) it follows immediately that F i (P n x) − F i (P n y) = n j=1 1 0 ∂F i ∂x j (P n y + t(P n x − P n y))dt · (x j − y j ). (18) We want now to pass to the limit n → ∞ in the above equation. From C3 it follows that F i (P n x) → F i (x) and F i (P n y) → F i (y). We will show that Let us fix any > 0. Observe that from F2 it follows immediately that there exists n 0 such that for n ≥ n 0 we have 1 0 ∂F i ∂x j (P n y + t(P n x − P n y))dt · (x j − y j ) < .

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Hence for n ≥ n 0 we obtain To finish the proof of (19) observe that from F1 and the compactness of V it follows that for each j the functions [0, 1] t → ∂Fi ∂xj (P n (y + t(x − y))) converge uniformly on [0, 1] to the function [0, 1] t → ∂Fi ∂xj (y + t(x − y)). Hence we obtain To finish the proof observe that Then F is an injection on V . In particular, V contains at most one zero of F .
Proof. Let i 0 be such that for all i From Lemma 3.1 it follows that 3.1. Block decomposition. For Lemma 3.2 to apply, the matrix ∂F ∂x has to be dominated by the diagonal terms. This can be achieved by an approximate diagonalization in case of real eigenvalues, but to handle complex eigenvalues we need a slight generalization of Lemma 3.2. To this end following [26] we introduce the notion of the block decomposition of H. 1: For the block decomposition of H we adopt the following notation, which makes a distinction between the blocks and one dimensional subspaces spanned by e i . For the blocks we use H (i) = e i1 , . . . , e i k , where (i) = (i 1 , . . . , i k ). The symbol H i will always mean the subspace generated by e i . For a one-dimensional block (i) we adopt the following convention: the only element of (i) will be denoted by the same letter, i.
For a given block decomposition of H and a block (i), we set For any x ∈ H by x (i) we will denote the projection of x onto H (i) . For any l and (i) = (i 1 , . . . , i k ) we will say that (i) ≤ l if i s ≤ l for all s = 1, . . . , k and we say that (i) > l if i s > l for all s = 1, . . . , k. Let us denote by B the set of multi-indices defined by our block decomposition, i.e., we have On each component H (i) we will use the norm induced from H. By P (i) we will denote an orthogonal projection onto H (i) . By Lin(H (i) , H (j) ) we denote the set of all linear maps from H (i) to H (j) equipped with the operator norm |A| = max |v|=1,v∈H (i) |Av|.
Assume that we have a block decomposition of H and conditions F1, F2 are satisfied. Then it is easy to see that the following two conditions are satisfied.
F1': For every (i) ∈ B and (j) ∈ B, the map is continuous. F2': Let n (i)(j) = max x∈V ∂F (i) ∂x (j) (x) . Then for every (i) ∈ B and every x, y ∈ V , the series Definition 3.5. For any linear map A ∈ Lin(R n , R m ) we define inf(A) = min |v|=1,v∈R n |Av|.
For a matrix valued function A(x) we set We have the following easy lemma.

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Lemma 3.6. Let A ∈ Lin(R n , R n ) be an isomorphism. Then The following theorem is a direct generalization of Lemma 3.2 and the proof is essentially the same. inf Then F is an injection on V . In particular, V contains at most one zero of F .

4.
The issues of stability and instability.

4.1.
A criterion for the instability. For a matrix valued function A(x) we set µ sup (C).

Remark 1.
It is easy to see that µ sup (A) coincides with the logarithmic norm µ(A) based on the Euclidean norm, which was used in [26].
In the theorem below we use the notion of an isolating block (which has an origin in the Conley index theory) N and its exit set N − . We refer the reader to [23] for the precise definition.
be the self-consistent a priori bounds and N ⊆ V be an isolating block for a fixed point p. We assume that the logarithmic norms for Galerkin projections are all uniformly bounded on V .
Assume that we have a block decomposition (in the sense of Def. 3.3) and let the block (i 0 ) consists of all exit directions. Assume that F1 and F2 are satisfied. Let then p is unstable, i.e., there exists > 0 such that for any δ > 0 there exist x δ such that |x δ − p| < δ and t δ > 0 such that . From Lemma 3.1 if follows that for x ∈ Z and any block (j) the following condition is satisfied.
The assumption about logarithmic norms implies that we have the existence and the uniqueness of classical solutions in V (see [24,26]).
Assume that we have an orbit For any (i) we have From the above inequalities it follows that for any (i) = (i 0 ),

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Hence for t ∈ [0, t 1 ] and all (i) = (i 0 ) it holds that From (29), it follows that the function |x (i0 Observe that > 0, because the number of exit directions is finite. To finish the proof observe that Let us comment on why the assumptions of Theorem 4.2 can be quite easily satisfied. The condition d (i0) > 0 holds due to the following factors.
is small, because for sufficiently large m, the modes with high-wave numbers have a relatively weak influence on the modes with low frequency numbers. To discuss the condition a (i0)(i) > 0 observe first that In the above formula first term d (i0) is already positive. Observe that for the KS equation, is positive and goes to ∞ as (i) → ∞. It turns out that for the KS equation the last term tends to −∞ at a slower rate than the second one. Hence there are good chances to verify this condition (see Section 8 for more details).

4.2.
A criterion for the stability. A criterion for stability is basically contained in Theorem 3.7 in [26]. It is shown there that it is enough to check that if there exists l ∈ R such that for all blocks (i) (see also Remark 1), is stable (and attracting).

5.
The continuity of solutions of implicit equations through the selfconsistent bounds. Our goal in this section is to establish the regularity properties of the solution of the equation as a function of ν in the context of the self-consistent a priori bounds and to provide the mathematical basis for the rigorous numerical procedure for the computation of the derivatives of x(ν), which is required in the bifurcation analysis.
Let us assume additionally (just for simplicity, because this is not essential for the method) that for k = 1, 2, . . .
, all third partial derivatives vanish.
By taking the derivatives of (31) with respect to ν we obtain 5.1. The existence and uniqueness issue. In the computation of the derivatives of the steady states with respect to parameters one has to solve the following linear Equation (compare (32)-(34)): where z, x ∈ R d and D, N ∈ R d×d , D + N = ∂F ∂x at a fixed point for the KS equation (or its Galerkin projection), D is a diagonal matrix and D dominates N in a suitable sense. The dimension d can be infinite. We assume that we have a block decomposition of H.
We introduce the block-infinity norm given by Let H b,∞ be a completion of H in the above norm.
where z, x ∈ H b,∞ and D, N ∈ R d×d (we do not assume that D and N define maps on H b,∞ , but rather they are just collections of coefficients indexed by N 2 + ). Assume that there exists α < 1 such that inf Then Equation (37) has a unique solution, x ∈ H b,∞ , and Proof. Observe that from (39) it follows immediately that D (i)(i) is an isomorphism (see Def. 3.5). From Lemma 3.6 it follows that Consider the following map We will prove that P : First of all observe that dom (P ) = H b,∞ . Namely we have It remains to show that also D −1 N x ∈ H b,∞ and the linear map D −1 N is a contraction From the Banach contraction principle it follows that the map P has a unique fixed point, x * . This point is a unique solution to (37) and it is equal to a limit of the sequence P n (x) for any x ∈ H b,∞ .
To finish the proof observe the ball B = B 0, 5.2. Self-consistent bounds for the derivatives. We want to solve equation (35) in H (we have the existence and the uniqueness in the larger space H b,∞ ) over a range of ν, so that a solution for ν belongs to the set L, which forms the self-consistent a priori bounds for (35). We show now that such a solution exists under some mild assumptions, which are easily satisfied for the KS equations.
be two sets of linear mappings defined densely on H with values in H. We assume that X k ⊆ dom (D) and X k ⊆ dom (N ) for any k and any We assume for any z ∈ [Z], D ∈ [D] and N ∈ [N ] that the assumptions of Theorem 5.1 are satisfied and additionally the following inequalities hold. inf All these assumptions are satisfied for the KS equations as we will show later.
Under the above assumptions we are going to construct the set L given by which will form for M = m the topologically self-consistent a priori bounds containing the solution of (35) for all z ∈ We construct L using an algorithm for producing the self-consistent a priori bounds we used for the proof of the existence of fixed points for the KS equations in [23,26]. As a result of this algorithm we obtain a sequence L 0 ⊇ L 1 ⊇ L 2 ⊇ . . . , such that each set L i is of the form given by (51) and for i ≥ i 0 , L i forms the topologically self-consistent a priori bounds containing the solution of (35) for all We will use an additional index in the parameters defining L i , i.e., L i will be defined according to (51) by setting the values for x ± i,k . We will not change the set W L . Algorithm: Initialization: We define L 0 as follows. Let us fix > 0. We set One step. Assume that L i is defined. We define L i+1 as follows: for k > m, let b k be given by We define x ± i+1,k as follows.
We set . We show first that for any block (i) and Indeed Hence This proves condition C4. Observe that (59) implies that C5 is satisfied. It remains to prove (57). Observe that from (58) it follows that for any λ ∈ [0, 1] and x ∈ L 0 such that |x (i) | = R, Let us define a homotopy H : From (60) it follows that From the continuation property of the degree it follows that The following Lemma follows directly from the rule of construction of L i+1 starting from L i .
In view of the above lemma it is clear that to show that the algorithm produces the topologically self-consistent a priori bounds for (35) it is enough to show that for i large enough conditions C2 and C3 are satisfied.
As the first step in this direction we have the following Lemma 5.4. There exists a constant E 1 , such that for all k, it is true that Proof. We have for k > m We will repeatedly use the following lemmas.
Then there exists a constant S(s, γ) such that for all k ∈ N + , it holds that Lemma 5.6. There exist i and a constant E i , such that Proof. By induction. i = 1 is treated in Lemma 5.4. As an induction assumption, we assume that for i ≥ 1, it is true that Using Lemma 5.5, we obtain and so we have proven that if γ i ≤ s, then γ i+1 = min(γ i + 3, t).
Lemma 5.7. If |x ± i,k | ≤ Ei k γ i and γ i > 5, then conditions C2 and C3 are satisfied on L i for F (x) = z + (D + N )x.
Proof. Condition C2 is manifestly satisfied. For condition C3 it is enough to prove the convergence statement for f k .
We have
In both cases we assume that the domain of interest is a compact and convex set, Λ, Λ ⊆ R for (71) and Λ ⊆ R 2 for (72). We assume in both cases that we have constructed V = W ⊕ k>m [x − k , x + k ] the topologically self-consistent a priori bounds, where we were able to verify the uniqueness. Hence we have a function x(ν) (x(ν, x 1 )) for ν ∈ Λ ((ν, x 1 ) ∈ Λ).
Throughout this section we assume: • Bounds on V : |x i | ≤ C i s , for x ∈ V . • We have a block decomposition.
• F1 and F2 for F and all its partial derivatives are satisfied on Λ × V .
Proof. We provide the proof essentially for the derivative with respect to ν only. The proof for (72) is the same and hence will be omitted; only the place where there is some difference in estimates will be discussed. Let us fix ν ∈ Λ, and let h = 0 be such that ν + h ∈ Λ. Then from Lemma 3.1 we obtain for all i, Hence the difference ratio r(h) = x(ν+h)−x(ν) h satisfies the following system of equations.
The above equation was considered in Section 5.2. To apply Theorem 5.8 to obtain bounds for r(h) we need to provide bounds for c iν for i large enough.
Since c iν ∈ ∂Fi ∂ν (Λ, V ) I , then from (79) we have Since Hence (46) is satisfied with t = s andC = CΘ/β. When considering (72) and the partial derivative with respect of x 1 , instead of From (78) it follows that for someG and so Hence (46) is satisfied with t = s + 3 andC =G/β. From Theorem 5.8 we obtain topologically self-consistent a priori bounds, L (independent of h), r(h) ∈ L and |r i (h)| ≤ C1 i s . We will prove now that r(h) → r * , where r * is a solution of the following equation.
First of all, this is an equation studied in the previous section, L are also the topologically self-consistent a priori bounds for it and r * ∈ L.
Since the set L is compact, it is enough to prove that for any sequence h n → 0 for n → ∞, such that ratios r(h n ) →r, we have thatr satisfies (87).
To prove this observe that from Lemma 3.1 it follows that c iν → ∂Fi ∂ν (ν, x(ν)) and c ij → ∂Fi ∂xj (ν, x(ν)), and so by passing to the limit we see that indeedr satisfies (87); hencer = r * . Uniqueness of solutions of (87) and continuity of coefficients in this equation with respect to ν, x, and r imply continuity of x (ν).
Proceeding inductively it is now easy to prove the following.
Theorem 5.10. Take the same assumptions as in Theorem 5.9 as well as the assumption that d 3 F = 0. Then the function x(ν) (x(ν, x 1 )) is C ∞ on Λ, and for any l ≥ 1 there exists a constant C 1 such that |x Moreover, for Equation (71) for any ν 1 , ν 2 ∈ Λ and any i, In the case of Equation (72) for any (ν 1 , y), (ν 2 , y) ∈ Λ and (ν, y 1 ), (ν, y 2 ) ∈ Λ, any operator, D (l−1) , of the partial derivatives of order l − 1, and any i, the following inequalities are satisfied.
Proof. The proof is by induction. We will illustrate the induction step for l = 2 for equation (72) for the computation of ∂x ∂ν∂x1 . This example contains all essential ingredients for the whole proof, (due to the fact that d 3 F = 0).
Hence similarly as in the proof of Theorem 5.9 we see that Thus to conclude the proof it is enough to show that z i satisfies (46) with t ≥ s. We will do it term by term. For i large enough we have 6. Bifurcations.

Symmetries and invariant subspaces of the KS equation.
We have the following easy lemma. Lemma 6.1 . If u(t, x) is a solution of the KS equations (we ignore the boundary conditions) for some ν, then u(t, x) = ku(k 2 t, kx) is the solution of the KS equations for ν = ν k 2 .
From the above lemma it follows that for odd and periodic boundary conditions in terms of Fourier coefficients we obtain the following fact: if a k (t) is a solution of (3) for some ν then for any k ∈ N we obtain a solution of (3) forν = ν k given bỹ Observe that the shift by π,ũ (t, x) = u(t, x + π), maps solutions of (1-2) into solutions of the same problem. In terms Fourier coefficients, this symmetry, denoted by R, is given by Here are a few simple consequences of the above symmetries.
They are called unimodal. If a 1 > 0, then it is called a positive unimodal point; otherwise it is called negative unimodal. The same terminology applies to the branches of the unimodal fixed points. gives rise to two k-modal fixed points. These points are mapped one onto another under the symmetry R, and hence their stability is the same.
Observe that for each k ∈ N + the space of k-modal functions given by the condition a s = 0 if s / ∈ kN + is invariant.

Bifurcations in the KS equation.
Most of the steady state bifurcations in the KS equations are the bifurcations of the invariant subspace, the k-modal subspace (see [15]). Some of them are intersections of two regular branches (for example an intersection of the trimodal branch with the bi-tri branch for ν ≈ 0.11039383), but most of them are the symmetry breaking pitchfork bifurcations. Namely, we have a branch of the steady states laying in a lower dimensional subspace, which bifurcates into three steady states, one being the continuation of the branch in the lower dimensional subspace and two new ones related to one another by the symmetry R. This is the case for the bifurcation which happens when two unimodal solutions collide with the negative bimodal branch for ν ≈ 0.247833. The negative bimodal branch belongs to the fixed point set for R and the unimodal solutions are mapped one onto another by R.
In an attempt to establish rigorously the existence of the pitchfork bifurcation, we use the approach presented in [4]. This means that we perform the Liapunov-Schmidt reduction and then the bifurcation problem is reduced to solving G(ν, x) = 0, where G : R 2 → R is a smooth function.
To be more specific: let A ∈ R m×m be a nonsingular matrix commuting with R defining a new coordinate system such that in the new coordinates the first direction is close to the apparent 'bifurcation direction' and is spanned by odd modes only. To perform the Liapunov-Schmidt reduction we solve the equatioñ as a function y(ν, x 1 ) = (x 2 , x 3 , . . . )(ν, x 1 ) = 0 for (ν, x) ∈ Λ × X 1 . y and its derivatives are computed rigorously using the self-consistent a priori bounds method outlined in Section 5. We define the bifurcation function, G, by G(ν, x) =F 1 (ν, x, y(ν, x 1 )). Hence to solve equationF (ν, x 1 , . . . ) = 0 in the set Λ × X 1 × V (V are the self-consistent a priori bounds obtained in the construction of y(ν, x 1 )), it is enough to solve the equation 6.3. The saddle-node and pitchfork bifurcation. Our goal is to establish a type of Implicit Function Theorem for a problem (91) with the explicit bounds for the domain of the existence of the solution. We begin first with a theorem describing the saddle-node bifurcation, which will be later applied in the analysis of the pitchfork bifurcation.
Then there exist 0 < x 0 ≤ x 1 and a function ν : such that all solutions of the equation g(ν, x) = 0 in Z belong to the graph of the function ν(x). Moreover, the following is true.
The very easy proof is left to the reader. The model for Theorem 6.2 is given by the map g 1 (ν, x) = x 2 − ν in the neighborhood of the point (0, 0). By changing signs of ν and g we obtain the model maps g 2 (ν, x) = ν + x 2 , g 3 (ν, x) = ν − x 2 , and g 4 (ν, x) = −ν − x 2 for which we can state analogous theorems. Now we turn to an application of the above theorem to the pitchfork bifurcation.
Since G(ν, 0) = 0, it is natural to define a function and then solve g(ν, x) = 0 using Theorem 6.2. To do this we need a representation for g other than formula (98), because we cannot handle the singularity in the denominator in computer computations. We have the following lemma. Proof.
Now we can formulate a theorem about solutions of (97), which will be used in our bifurcation computations.
Proof. We introduce a function g as in Lemma 6.3, and then we rewrite the assumptions of Theorem 6.2 in terms of G. For example to establish (92) and (93) we observe that The assumptions of Theorem 6.4 are well suited for the functions, which behave like G(ν, x) = ax(x 2 − bν), where a, b are positive numbers. To handle the case of other sign combinations, we proceed as follows.
We define a function, G We will describe now how to define 1 and ν , so that the function G satisfies the assumptions of Theorem 6.4.
If ν = +1, then assume that the following conditions are satisfied.
If ν = −1, then assume that the following conditions are satisfied.
Then there exist 0 < x 0 ≤ x 1 and a function ν : such the set of all solutions of the equation G(ν, x) = 0 in Z is a union of the graph of the function ν(x) and the line x = 0. Moreover, the following is true.
Proof. For the proof observe that the function G satisfies the assumptions of Theorem 6.4.
6.4. Intersection of regular branches. In this section we state the bifurcation theorem, which handles the case of the intersection of two regular branches, one of which is contained in a lower dimensional invariant subspace (for example an intersection of the trimodal branch with the bi-tri branch for ν ≈ 0.11039383). Just as in case of the pitchfork bifurcation we analyze the equation G(ν, x) = 0, where G(ν, 0) = 0 (which corresponds to a solution branch contained in the lower dimensional subspace).
We introduce the function g by (98) and we compute it using Lemma 6.3. The task is now reduced to checking if the solution curve for g(ν, 0) = 0 intersects the line x = 0 at a nonzero angle and does not make any fold.
Assume that the following conditions are satisfied.
Then the solution of equation G(ν, x) = 0 is a sum of line x = 0 and a curve x . For the proof it is enough to show that the solution of equation g(ν, x) = 0 can be parameterized by x, as a curve (ν(x), x) and ν (x) = 0.
Since by Lemma 6.3 hence from (107) it follows that Since Z is connected we see that ∂g ∂ν (ν, x) has constant sign on Z, and so for each x ∈ [−x 1 , x 1 ] the equation g(ν, x) = 0 has at most one solution.

Details for the KS equation -
The local uniqueness issue. The goal of this section is to derive the formulas necessary to verify the assumptions of Theorem 3.7 for the KS equation. We can write the KS equation (see Section 1.1) in the Fourier domain as follows.
N k (a) = −k k−1 n=1 a n a k−n + 2k ∞ n=1 a n a n+k .
The formal first derivatives of F are given by For the second derivatives, we have the following formulas.
All higher order derivatives vanish. We start with the verification of F1 and F2.
Theorem 7.1. Let V be the self-consistent a priori bounds for the KS equation, such that |x k | ≤ C k s for s > 5. Then conditions F1 and F2 are satisfied on V .
From the formulas for ∂Fi ∂xj given above and the assumed polynomial decay rate of x i on V we have Hence to prove F2 it is enough to show that the series Hence the series in F2 converges for s > 1 2 .
The following lemma does not require any proof.

Lemma 7.2. Let A : H → H be a linear coordinate change of the form
Consider now the KS equation. We assume that V is the self-consistent a priori bounds for a fixed point. Let the numbers m < M be as in conditions C1, C2, C3. We assume that a ± k = ± C k s for k > M (as in [23]). Let A ∈ R m×m be a coordinate change around an approximate fixed point in X m for m-dimensional Galerkin projection of (114). This matrix induces a coordinate change in H. It is optimal to choose A so that the m-dimensional Galerkin projection of F is very close to the diagonal matrix (or to the block diagonal one when the complex eigenvalues are present).
We will use the new coordinates in H. We also change the norm so that the new coordinates are orthogonal. We define the splitting of P m H into blocks which are either 2-dimensional (the case of the complex eigenvalue) or one-dimensional (the real eigenvalue). For the instability consideration we may glue several 'unstable blocks' into one -see Section 8. For the KS equation there was no need to consider more complicated situations such as the nontrivial Jordan cells. For (i) > m all blocks are 1-dimensional (these coordinates are not affected by our coordinate change).
We would like to derive the formula for

PIOTR ZGLICZYŃSKI
We introduce S(l) and S N D ((i)) by Since many times in the estimates we need to estimate ∞ k=l 1 k s we elevate one such estimate to the status of a lemma.
We will estimate S(l) from the above using the following lemma.
Lemma 7.4. Assume that |a k (V )| ≤ C k s for k > M , and s > 1. Then Proof. It follows immediately from Lemma 7.3.
We set (131) Lemma 7.6. Let (i) = (i 1 , i 2 , . . . , i r ), (i) ≤ m. Then Proof. From Lemma 7.5 it follows that we can ignore the structure for all blocks different from (i) and use the following estimate.
Therefore we have To finish the proof it is enough to show that We have We have for i l ∈ (i) (observe that i l ≤ m) This finishes the proof.
7.1. Formulas for 1-dimensional blocks. Observe that if dim(i) = 1, then From the above observation and Lemma 7.6 we obtain the following lemma.
Observe that from our assumptions about the decomposition of H it follows that all blocks (i) such that (i) > m are one-dimensional.
Proof. Just as in the proof of Lemma 7.6 we can ignore the block structure here. It is easy to see that Therefore to finish the proof it is enough to show that We proceed as follows.
Proof. Just as in the proof of Lemma 7.6 we can ignore the block structure here. It is easy to see that Therefore to finish the proof, it is enough to show that To prove (139) observe that To prove (140) we proceed as follows.
The following lemma shows how to handle the case of large i.
Lemma 7.10. If for some n > m, ∆ n > 0, and 1 − νn 2 < 0, then Proof. From Lemma 7.9 it follows that where a = max k,l=1,...,m |A −1 kl |. Hence where f (i) is a positive decreasing function of i. It is easy to see that the function i → (νi 3 − i) is increasing and positive for i ≥ n.
In the computation of the derivatives of the steady states with respect to the parameters (see Theorem 5.1) we will be interested in the ratio We have the following.
Lemma 7.11. For i > m, the function r(i) is decreasing.
Observe that K(Q) has the following form.
Obviously K(K(Q)) = K(Q), the eigenvalues of K(Q) are λ 1,2 = α ± iβ, and The difference Q − K(Q), where Q is a 2-dimensional block obtained from linearization, measures how good this linearization really is. In our applications to fixed points for the KS equations it is usually very small, which is achieved by taking a small set V and m large enough. To measure this difference we will use the function δ(Q) given by Lemma 7.12. Let Q ∈ R n×n and |Q ij | ≤ . Then |Q| ≤ n .
Lemma 7.13. Let Q ∈ R 2×2 . Then Proof. Since then the assertion follows from Lemma 7.12.
Lemma 7.14. Let (i) = (i 1 , i 2 ), (i) ≤ m. Then Proof. From Lemma 7.13 it follows immediately that inf 8. Details for the KS equation -The instability. In this section we provide the formulas for the verification of the assumptions of Theorem 4.2.
We assume that we have the coordinates introduced in Section 7 and the same block decomposition. For the purpose of the proof of Theorem 4.2 we modify slightly this block decomposition as follows.
• Let (i 1 ), . . . , (i s ) be all the blocks such that all diagonal elements of Observe there are only finitely many such blocks (in fact, at most m).
• We create a new block (i 0 ) = (i 1 ) ∪ · · · ∪ (i s ). This means that in the block decomposition of H we have H (i0) = H (i1) ⊕ · · · ⊕ H (is) . Observe that the sum of non-diagonal elements appearing in Theorem 4.2 was already computed in Section 7, but the block decomposition is slightly different now. It reduces to the fact that on H (i0) the sum-norm was used, and now we have to use the Euclidean norm.
The next lemma addresses the computation of µ inf and µ sup .
Proof. We provide the proof for µ sup only. The other case is analogous. The formulas for the computation of (i) =(j) ∂F (j) ∂x (i) (Z) are given in Lemmas 7.7, 7.8, 7.9, and 7.6. Hence it remains to discuss how to verify in the finite computation that a (i0)(i) > 0 holds for all (i).
It turns out that the same analysis as in Lemma 7.10 gives rise to the following.
Lemma 8.2. If for some n > m, a (n)(i0) > 0, and 1 − νn 2 < 0, then The file bifdata.txt [27] contains data from the proof of the uniqueness, the instability (plus the regularity computation) for the positive bimodal fixed point for ν = 0.127 + 10 −5 · [−1, 1]. For this steady state the unstable direction is two dimensional and corresponds to a pair of complex eigenvalues. 9. Algebra of polynomial bounds. In the computation of various derivatives of the implicit function defined as the z-term in Equation (35), we have sums of the following expressions (and since F is a second degree polynomial, in fact only terms of this type): where |y i | and |w i | satisfy some decay condition (arising from the self-consistent a priori bounds), whileF and its derivatives are also evaluated on the self-consistent a priori bounds. It turns out that the computation of these terms might be realized by an algebra, which we are going to develop in this section. Throughout this section we fix a positive integer M .
We will often use the triple (Y, E, β) to denote the polynomial bounds.
We introduce some arithmetic operations on bounds.
Definition 9.2. Assume that (X, E, β) and (Y, G, γ) are polynomial bounds. We define With the above definitions we can define the product of A · (Y, E, β), where A ∈ R m×m , m ≤ M . It also makes sense to apply projections P n and Q n = I − P n , where n ≤ M , to the polynomial bounds.
It turns out that we can compute all terms in (153) using the following functions: The goal of the next few lemmas is to define the operations QF and QI on the polynomial bounds. 9.1. Some estimates. Proof.
j>M +k Lemma 9.8. Assume identical assumptions as in Lemma 7.2. Then the following are true.
Let Y and W be polynomial bounds. We now turn to the computation of kj ∂ 2F i ∂xj ∂x k y j w k , where y ∈ Y and w ∈ W . We would like to stress here that ∂ 2F i ∂xj ∂x k are constants, and so there is no need to specify their arguments. Observe that where y i = m j=1 A −1 ij y j for i ≤ m and y i = y i otherwise. w is defined analogously. Hence it is enough to derive the formulas for kj ∂ 2 Fi ∂xj ∂x k y j w k . We have j,k y k+i w k = −2iQF i (y, w) + 2iQI i (y, w) + 2iQI i (w, y).
Hence it is enough to have an expression for ∂N k ∂xi w i . Using the formulas for the derivatives of the vector field of the KS equation we obtain, Summarizing we have shown that From the above formula it follows immediately that

10.
Details for the KS equation -The regularity issue. We assume that we have a coordinate change A as in Lemma 7.2, and as in Section 7, we assume that we have a block decomposition of H and V representing the topologically self-consistent bounds for F (ν, x) = 0 (161) where ν ∈ Λ = [ν 0 − δ, ν 0 + δ]. We assume that we have the uniqueness property of solutions (161) for ν ∈ Λ. This defines the function x(ν). In Section 5, it was shown that x(ν) is C ∞ .
As in the proof of the uniqueness in Section 7, we will perform all computations for the functionF = A • F • A −1 . Similarly we defineÑ as A • N • A −1 .
We have to solve where z is obtained from the implicit differentiation of F (ν, x) = 0 and depends upon the particular partial derivative we are willing to compute.
be obtained from the above iterative scheme. We define a candidate (an initial guess) for the self-consistent a priori bounds for (163) as follows.
Finally the candidate bounds are given by Let us comment about the bounds for y i when i > M . They are chosen so that we have max |z M +1 | = max |y M +1 | and so that the decay rate for |y i | is E i 4 . This procedure worked in most cases -i.e., it produced a candidate, which lead later to the self-consistent a priori bounds. It failed only when z ≈ 0 (this happened when considering the zero solution) and the coefficient E was very small. In this case when we did not get an isolation starting from the above guess, we produced a new guess by setting E = 0.01, and then it always worked.
Observe that the proof of Theorem 5.1 gives a guaranteed good candidate for the self-consistent a priori bounds, but it turns out that the refinement, as described in Section 10.3, of the bounds obtained in this way requires much larger M .
We have to compute the following partial derivatives of G for Theorem 6.4: For Theorem 6.6, we need They are given by the following formulas.
All these formulas can be computed using the algebra for polynomial bounds described in Section 9 and then extracting the first coordinate.
11.1. Short description of the procedure for the proof of the existence of bifurcations. Observe first that both bifurcations theorems (6.4 and 6.6) contain two types of assumptions: Global: a construction of the self-consistent bounds for (90) over Z = [ν 1 , ν 2 ] × [−a, a], on which we can evaluate ∂ 3 G ∂x 3 1 (Z) and ∂ 2 G ∂ν∂x1 (Z) in case of Theorem 6.4, or ∂ 2 G ∂ν∂x1 (Z) and ∂ 2 G ∂x 2 1 (Z) in case of Theorem 6.6. These partial derivatives of G should not contain zero, and this can be achieved by taking the set Z small enough.
Observe that if at this stage we are not able to construct Z on which the global conditions are satisfied, then the proof is inconclusive. We can neither claim nor exclude the existence of bifurcation. We can only hope that taking larger m, M and smaller (in diameter) Z will improve the situation. Local: Once we have the set Z over which the global conditions are satisfied we verify the remaining (local) conditions. Observe that each of them involves either some derivatives of G or the value of G at some point (ν 0 , x 0 ). The isolation algorithm presented in [23] applied to (90) and the algorithms for the computation of the partial derivatives y(ν, x) described in the previous section allow us to compute the desired values to an arbitrary accuracy (close to the round-off error) by taking M large enough. Hence we can check whether the local conditions are satisfied or violated. (In this case, we can rule out the existence of any bifurcation in Z). Hence this part of the proof is conclusive.
The computer procedure performing the proof of the existence of bifurcation works as follows. Input Parameters: • m -the dimension of the Galerkin projection, M = max(2m, 10).
The Procedure: Diagonalization: We define the coordinate change A as follows: we approximately diagonalize dP m F (ν 0 , x 0 ), the m-dimensional Galerkin projection of (3) for ν = ν 0 at x = x 0 . We choose an eigenvalue, λ 0 , which is the closest to zero (there should be such an eigenvalue; otherwise there is no bifurcation nearby). The eigenvector corresponding to λ 0 we choose as our 'bifurcation direction', and it will represent the 1 st coordinate in the new coordinate frame.
We have also to make sure that the subspace of k-modal functions is contained in the hyperplanex 1 = 0. In case of the bimodal branch (k = 2), the coordinate change must commute with the symmetry R, and from this we obtain the odd-symmetry property of the bifurcation function G.
Observe that it is easy to satisfy these properties during the diagonalization process, as the subspace of k-modal functions is invariant also for dP m F (ν 0 , x 0 ) if x 0 is k-modal itself. Global Conditions: Let X 1 = 10 −2 . We set Z = [ν 1 , ν 2 ] × [−X 1 , X 1 ], and we try to verify the global bifurcation conditions on Z. If they are not satisfied, then we set X 1 = X 1 /5 and try again, until X 1 < 10 −5 , when we decide that we fail. If we fail, then we increase M (hoping for an improvement in the diameters of the relevant partial derivatives of G on Z) and try again.
We do this until we succeed (and we jump to verify the local conditions), or else there is no improvement in values of the relevant partial derivatives G on Z (we use some ad-hoc stabilization criterion -for example: the diameter should shrink by a factor at least 0.9).
If we fail again, then we exit. We can only hope that either the increase of m or the shrinking of the diameter of [ν 1 , ν 2 ] will improve things. Local Conditions: We evaluate all local conditions (see the discussion at the beginning of this subsection). Consider for example the computation of ∂G ∂x (ν 1 , 0). We keep refining bounds for y(ν, x) by increasing M until ∂G ∂x (ν 1 , 0) does not contain 0 or its value stabilizes.
The k-modal fixed points were produced from the unimodal fixed point branch using the rescaling described in Lemma 6.1. The solutions on the unimodal branch are all attracting, and so can be easily found either by following the trajectory or by using the Newton Method starting at an approximate fixed point for the 2dimensional Galerkin projection. We picked up the bifurcation values from [12,15], where the bifurcation parameter was α = 4/ν. It turns out that these values were too crude for our purpose, and we refined them by an ad-hoc trial and error approach (this was not automated), until we can finally verify the global bifurcation condition.
The file bifdata.txt [27] contains the most relevant numerical data from the proofs of the bifurcations listed in Theorem 1.1.
The program was written in C++ (the GNU compiler was used). We used the CAPD package [2] to handle the interval arithmetic and graphics. All computations were performed on a Windows 98, Pentium III, 450 MHz computer. We tested the program also using Linux.