DELAY DIFFERENTIAL EQUATIONS AND CONTINUATION

DELAY DIFFERENTIAL EQUATIONS AND CONTINUATION JEAN-PHILIPPE LESSARD Abstract. In these lecture notes, we demonstrate how rigorous numerics can help studying the dynamics of delay equations. We present a rigorous continuation method for solutions of finite and infinite dimensional parameter dependent problems, which is applied to compute branches of periodic solutions of a delayed Van der Pol equation and of Wright s equation. Contents. Introduction, Motivation and Examples. Rigorous Continuation of Solutions 6.. Predictor-Corrector Algorithms 7.. Background of Calculus in Banach Spaces 9.3. The Rigorous Continuation Method 0.4. A Finite Dimensional Example 6 3. Computing Branches of Periodic Solutions of Delay Equations 9 3.. A Delayed Van der Pol Equation 3.. Wright s Equation 36 References 36. Introduction, Motivation and Examples The main purpose of these lectures notes is to demonstrate how rigorous numerics can help gaining some understanding in the study of the dynamics of delay differential equations DDEs. One of the main motivating example we consider in these notes is Wright s equation, essentially because it is one of the simplest looking delay equation and it is arguably the most studied equation in the broad field of DDEs. Moreover, it has been the subject of active research for more than 60 years and has been studied by many different mathematicians e.g. see [,, 3, 4, 5]. As one will see later, the dynamics of this equation naturally leads to studying branches of periodic solutions parameterized by the parameter in the equation. This is why a large part of the notes is dedicated to the presentation of a rigorous continuation method for solutions of finite and infinite dimensional parameter dependent problems. This part will be independent from delay equations. While in this section, we focus on Wright s equation to introduce some concepts and ideas, the method introduced in these notes is quite general and can be applied to a large class of DDEs. Note however that the notes are not meant to provide a general introduction to the field of DDEs. The interested reader will find great introductory materials in the book of Hale and Verduyn Lunel [6], the book of Diekmann, van Gils, Verduyn Lunel and Walther [7], and in the recent survey paper of Walther [8]. In order to start the discussion, we begin by presenting a quote from R. Nussbaum taken from [9].

JEAN-PHILIPPE LESSARD An intriguing feature of the global study of nonlinear functional differential equations FDEs is that progress in understanding even the simplest-looking FDEs has been slow and has involved a combination of careful analysis of the equation and heavy machinery from functional analysis and algebraic topology. A partial list of tools which have been employed includes fixed point theory and the fixed point index, global bifurcation theorems, a global Hopf bifurcation theorem, the Fuller index, ideas related to the Conley index, and equivariant degree theory. Nevertheless, even for the so-called Wright s equation, y t = αyt [ + yt], α R which has been an object of serious study for more than forty-five years, many questions remain open. Roger Nussbaum, 00. This comment is still very true nowadays and is perhaps not surprising, as a large class of FDEs naturally give rise to infinite dimensional nonlinear dynamical systems. In order to understand this, let us consider an initial value problem associated to Wright s equation. More precisely, at a given time t 0 0, what kind of initial data guarantees the existence of a unique solution yt for all t > t 0? Since y t 0 is determined by yt 0 and yt 0, knowing the value of yt for all t > t 0 requires knowing the value of yt on the time interval [t 0, t 0 ]. In other words, the initial condition is a function y 0 : [t 0, t 0 ] R. Shifting time to 0, the initial data is given by y 0 : [, 0] R. Denote the space of continuous real-valued functions ined on [, 0] by C = C[, 0], R = {v : [, 0] R : v is continuous}. Given y 0 C, the initial value problem y t = αyt [ + yt], t 0 yt = y 0 t, t [, 0] has a unique solution e.g. see Theorem.3 of Chapter in [6], and this naturally leads to an infinite dimensional nonlinear dynamical system. Therefore a state space for the solutions of is the infinite dimensional function space C. This is the reason why Wright s equation falls into the class of functional differential equations. In Figure, find a cartoon phase portrait of Wright s equation visualized in the function space C. Denote by y t C the solution at time t. As time evolves, the solution y t of the initial value problem gain more and more regularity, somehow in a similar way that solutions of parabolic partial differential equations PDEs gain regularity. However, while the regularizing effect in parabolic PDEs can be instantaneous in time think for instance of the heat equation, the regularizing process in delay equations is much slower. In fact, this is a discrete regularizing process. For instance, if y 0 C = C[, 0], R and t 0 0,, then y t 0 = αyt 0 [ + yt 0 ], and so the solution y is differentiable at t 0. In other words, y t C for t 0, ]. Similarly, y t C for t, ], and more generally y t C k for t k, k]. This is why we call this a discrete regularizing process. At infinity, the solution of the initial value problem is C. As a consequence, this means that bounded solutions of Wright s equations are extremely regular. This a priori knowledge about the regularity of the bounded solutions will be crucial in designing the rigorous numerical methods. As a matter of fact, when studying periodic

DELAY DIFFERENTIAL EQUATIONS AND CONTINUATION 3 y t C y 0 Figure. A cartoon phase portrait of Wright s equation in the function space C = C[, 0], R. A point y t C in the phase portrait is a function. solutions of delay equations, we can get even analyticity of the solutions, as long as the delay equation itself is analytic [0]. The infinite dimensional nature of the problem comes directly from the presence of the delay in the equation. Suppose for the moment that the delay is absent from the equation, that is consider the scalar ordinary differential equation ODE y t = αyt[ + yt]. Then, the phase portrait of is simple and is portrayed in Figure. In particular, we get that the equilibrium solution 0 is asymptotically stable for all parameter values α > 0. 0 Figure. The phase portrait of for any α > 0. Adding a delay severely complicates the behaviour of the solutions of the equation. In fact, we see below that the effect of the delay in Wright s equation leads to a loss of stability of the zero equilibrium solution for all α > π/. This property is similar in some sense to Turing instability [], a phenomenon in which a stable equilibrium solution of an ODE becomes unstable after a diffusion term is added to the ODE. In other words, the steady state loses its stability after the finite dimensional ODE is transformed into an infinite dimensional reaction diffusion PDE. Let us discuss the history of Wright s equation, following closely the presentation of []. At the beginning of the 950s, the equation y t = log yt [ + yt] was brought to the attention of the number theorist Wright a former Ph.D. student of Hardy at Oxford because it arose in the application of probability methods to the theory of distribution of prime numbers. In 955, Wright considered the more general equation and studied the existence of bounded non trivial solutions for different values of α > 0 [3]. In 96, following the pioneer work of Wright, Jones demonstrated in [4] that non trivial periodic solutions of exist for α > π, and using numerical simulations, he remarked in [5]

4 JEAN-PHILIPPE LESSARD that a given periodic solution seemed to be globally attractive, that is seemed to attract all initial conditions. In Figure 3 and Figure 4, we reproduced some of the numerical simulations of Jones using the integrator for delay equations dde3 in MATLAB. The periodic form he referred to is in fact a slowly oscillating periodic solution. yt 4 3.5 3.5.5 0.5 0-0.5 - y0t =t y0t = t +0.8 4 y0t =t y0t = t 0.5 0 4 6 8 0 4 6 8 0 t Figure 3. Numerical integration of Wright s equation with α =.4 with different initial conditions y 0 ined on the interval [, 0]. yt 3.5 3.5.5 0.5 0-0.5 - α =.6 α = α =. α =.4 0 4 6 8 0 4 6 8 0 t Figure 4. Numerical integration of Wright s equation with the initial condition y 0 t = t + 0.8 4 for different parameter values of α. Definition.. A slowly oscillating periodic solution SOPS of is a periodic solution yt with the following property: there exist q > and p > q + such that, up to a time translation, yt > 0 on 0, q, yt < 0 on q, p, and yt + p = yt for all t so that p is the minimal period of yt. A geometric interpretation of a SOPS can be found in Figure 5. After Jones observation in [5], the question of the uniqueness of SOPS in became popular and is still under investigation after more than 50 years. The next conjecture is sometimes called Jones Conjecture. Conjecture. Jones, 96. For every α > π, has a unique SOPS. A result of Walther in [6] shows that if Jones Conjecture is true, then the unique SOPS attracts a dense and open subset of the phase space. A result from Chow and Mallet-Paret

DELAY DIFFERENTIAL EQUATIONS AND CONTINUATION 5 yt 3.5.5 0.5 0-0.5 - q> p q> 6 8 0 4 6 8 t Figure 5. A slowly oscillating periodic solution. from [7] shows that there is a supercritical Hopf bifurcation of SOPS from the trivial solution at α = π/. This branch of SOPS which bifurcates forward in α from 0 is denoted by F 0. We refer to Figure 6 for a geometric interpretation of bifurcation. y F 0 π Figure 6. A supercritical Hopf bifurcation of SOPS from 0 at α = π/. α Regala then proved in his Ph.D. thesis [8] a result that implies that there cannot be any secondary bifurcations from F 0. Hence, F 0 is a regular curve in the α, y space. Later, Xie used asymptotic estimates for large α to prove that for α > 5.67, Wright s equation has a unique SOPS up to a time translation [9, 0]. Denote by A 0 = π, 5.67] the parameter range not covered by the work of Xie. In [], it was demonstrated, using the techniques that we introduce in the present notes, that the branch F 0 does not have a fold over the parameter range [π/ + ε,.3], with ε = 7.365 0 4. Considering the work that has been done in the last 50 years, Jones Conjecture can be reformulated as follows. Conjecture.3 Jones Conjecture reformulated. The branch of SOPS F 0 does not have any fold over A 0 \[π/+ε,.3] and there are no connected components isolas of SOPS disjoint from F 0 over A 0. Two different scenarios would therefore violate Jones Conjecture. The first scenario is the existence of a fold on F 0 over A 0 \ [π/ + ε,.3] which would provide the existence of α A 0 at which more than one SOPS could co-exist. The second scenario is the existence of an isola F over A 0 which could again force the existence of more than one SOPS. These two scenarios are simultaneously portrayed in Figure 7.

6 JEAN-PHILIPPE LESSARD y F 0 π α Figure 7. Two scenarios which would violate Jones Conjecture: the existence of a fold on F 0 or the existence of an isola in the parameter range α π, 5.67]. * F 5.67 Conjecture.3 naturally leads to studying branches of periodic solutions of DDEs. This is the main topic of these lectures notes. More precisely, we introduce a general continuation method to compute global branches of periodic solutions of DDEs using Fourier series and the ideas from rigorous computing e.g. see []. Note that the study of periodic solutions in DDEs is rich [, 3, 4, 5, 6, 7, 8, 9]. Rather than focussing only on continuation of periodic solutions in DDEs, we present in Section a more general approach to prove existence of branches of solutions for operator equations F x, λ = 0 posed on Banach spaces. In Section 3, we apply the general method to the context of periodic solutions of DDEs.. Rigorous Continuation of Solutions Throughout this section, let X, X and Y, Y denote Banach spaces. The vectors spaces X and Y are general and can be either finite or infinite dimensional. Let F : X R Y a C mapping see Definition., and consider the general problem of looking for solutions of 3 F x, λ = 0, where λ R is a parameter. The unknown variable x could represent various types of dynamical objects, e.g. a steady state of a PDE, a periodic solution of a DDE, a connecting orbit of an ODE, a minimizer of an action functional, etc. It is important to realize that the solution set S = {x, λ X R F x, λ = 0} X R may contain different types of bifurcations and may be complicated e.g. see Figure 8. There exists a vast literature on numerical continuation methods to compute solutions of 3. Methods to compute periodic orbits [3, 33], connecting orbits [34, 35, 36] and more generally coherent structures [37] are by now standard, and softwares like AUTO [38] and MATCONT [39] are accessible and well documented. We refer to [40, 4] for more general references on continuation methods. Next we briefly introduce two main algorithms to compute solutions of 3, namely the parameter continuation and the pseudo-arclength continuation. These methods fall into the class of predictor-corrector algorithms.

DELAY DIFFERENTIAL EQUATIONS AND CONTINUATION 7 kxk S Figure 8. Global branches of steady states of a system of reaction-diffusion PDEs introduced in [30] and studied with rigorous numerics in [3]... Predictor-Corrector Algorithms. In this section, we assume that the Banach spaces are finite dimensional and given by X = Y = R n X = Y = C n is also an option, we consider a map F : R n R R n and we study numerically the problem F x, λ = 0. At this point, considering X and Y finite dimensional is not a strong restriction, as any computer algorithm needs to be applied to a problem with a finite resolution. The mapping F could be a finitedimensional projection of an infinite dimensional operator, e.g. a Galerkin approximation or a discretization scheme. The first predictor-corrector algorithm we introduce is parameter continuation.... Parameter Continuation. This method involves a predictor and a corrector step: given, within a prescribed tolerance, a solution x 0 at parameter value λ 0, the predictor step produces an approximate solution ˆx 0 at nearby parameter value λ = λ 0 + λ for some λ 0, and the corrector step, takes ˆx as its input and produces with Newton s method, once again within the prescribed tolerance, a solution x at λ. The predictor is obtained by assuming that at the solution x 0, λ 0, the jacobian matrix D x F x 0, λ 0 is invertible, which in turns implies by the implicit function theorem that the solution curve is locally parametrized by λ. In this case, close to x 0, λ 0, we have λ F x, λ = 0 D xf x, λ dx F dx λ+ x, λ = 0 dλ λ dλ λ = D F xf x, λ x, λ. λ At x 0, λ 0, a tangent vector to the curve is ẋ 0 = dx dλ λ 0 and is obtained with the formula F ẋ 0 = D x F x 0, λ 0 λ x 0, λ 0. Once the tangent vector ẋ 0 is obtained, the predictor is ined by ˆx = x 0 + λ ẋ 0. Then, fixing λ = λ 0 + λ, we correct the predictor ˆx using Newton s method x 0 = ˆx, x n+ = x n D x F x n, λ n F x, λ, n 0,

8 JEAN-PHILIPPE LESSARD ˆx ẋ 0 x S kxk x 0 0 Figure 9. Parameter continuation. to obtain the solution x at λ within the prescribed tolerance. We repeat this procedure iteratively to produce numerically a branch of solutions. We refer to Figure 9 to visualize one step of the parameter continuation algorithm. Sometimes it may be more natural to parametrize the branches of solutions of 3 by arclength or pseudo-arclength, especially when the solution curve is not locally parametrized by λ, for instance at points where the jacobian matrix is singular. This is for instance what is happening when a saddle-node bifurcations folds occur. An example of such phenomenon is given by F x, λ = x λ = 0 at the point x 0, λ 0 = 0, 0. Pseudo-arclength continuation, as opposed to parameter continuation, allows continuing past folds.... Pseudo-Arclength Continuation. In the pseudo-arclength continuation algorithm e.g. see Keller [4], the parameter value λ is no longer fixed and instead is left as a variable. The unknown variable is now X = x, λ. Consider the problem F X = 0 with the map F : R n+ R n. As before, the process begins with a solution X 0 given within a prescribed tolerance. To produce a predictor, we compute first a unit tangent vector to the curve at X 0, that we denote Ẋ0, which can be computed using the formula [ D X F X 0 Ẋ0 = D x F x 0, λ 0 F ] λ x 0, λ 0 Ẋ 0 = 0 R n. We now fix a pseudo-arclength parameter s > 0, and set the predictor to be ˆX = X 0 + s Ẋ 0 R n+. Once the predictor is fixed, we correct toward the set S on the hyperplane perpendicular to the tangent vector Ẋ0 which contains the predictor ˆX. The equation of this plan is given by EX = X ˆX Ẋ 0 = 0. Then, we apply Newton s method to the new function EX 4 X F X

DELAY DIFFERENTIAL EQUATIONS AND CONTINUATION 9 with the initial condition ˆX in order to obtain a new solution X given again within a prescribed tolerance. See Figure 0 for a geometric interpretation of one step of the pseudoarclength continuation algorithm. At each step of the algorithm, the function ined in 4 changes since the plane EX = 0 changes. With this method, it is possible to continue past folds. Repeating this procedure iteratively produces a branch of solutions. X ˆX kxk S X 0 Ẋ 0 s Figure 0. Pseudo-arclength continuation. Remark.. The above mentioned algorithms do not cover the case of bifurcations of solutions e.g. symmetry-breaking pitchfork bifurcations, branch points, Hopf bifurcations, etc. We refer for instance to the work [40] for numerical continuation methods handling bifurcations. Now that we have briefly introduced two classical algorithms to numerically compute branches of solutions of the general problem 3, we present an approach that combines the strength of the numerical continuation methods with the ideas of rigorous computing e.g. see []. Before introducing the rigorous continuation method in Section.3, we need some background from calculus in general Banach spaces... Background of Calculus in Banach Spaces. The space of bounded linear operators is ined by BX, Y = { E : X Y E is linear, E BX,Y < }, where BX,Y denotes the operator norm E BX,Y = sup Ex Y. x X = Note that BX, Y, BX,Y is a Banach space. Definition.. A function F : X Y is Fréchet differentiable at x 0 X if there exists a bounded linear operator E : X Y satisfying F x 0 + h F x 0 Eh Y lim = 0. h X 0 h X The linear operator E is called the derivative of F at x 0 and denoted by E = D x F x 0. We say that F : X Y is a C mapping if for every x X, F is Fréchet differentiable at x.

0 JEAN-PHILIPPE LESSARD Given a point x 0 X and a radius r > 0, denote by B r x 0 X the closed ball of radius r centered at x 0, that is B r x 0 = {x X x x 0 X r}. The proof of the following version of the Mean Value Theorem can be found in [43]. Theorem.3 Mean Value Theorem. Let x 0 X and suppose that F : B r x 0 X Y is a C mapping. Let K = sup x B rx 0 D x F x BX,Y. Then for any x, y B r x 0 we have that F x F y Y K x y X. While the following concept could be introduced more generally in the context of metric spaces, we present it in the context of Banach spaces to best suit our needs. Definition.4. Suppose that Λ is a set of parameters. A function T : X Λ X is a uniform contraction if there exists κ [0, such that, for all x, y X and λ Λ, T x, λ T y, λ X κ x y X. By the Contraction Mapping Theorem if T : X Λ X is a uniform contraction, then for every λ Λ there exists a unique x λ such that T x λ, λ = x λ. Thus the function g : Λ X given by gλ = x λ is well ined. As the following theorem indicates this function inherits the same amount of differentiability than T. The proof can be found in [43]. Theorem.5 Uniform Contraction Theorem. Assume that the set of parameters Λ is a Banach space, and consider open sets U X and V Λ. Assume that T : U V U is a uniform contraction with contraction constant κ. Define g : V U by T gλ, λ = gλ. If T C k U V, X, then g C k V, X for any k {,,..., }..3. The Rigorous Continuation Method. Now that we have introduced some basic notions from calculus in Banach spaces, we are ready to present the general rigorous continuation method. The idea of the proposed approach is to prove the existence of true solution segments of F x, λ = 0 close to piecewise-linear segments of approximations by applying the Uniform Contraction Theorem Theorem.5 over intervals of parameters. This approach has the advantage of being quite general and can be readily generalized to problems depending of several parameters e.g. see Remark.9. However, the rigorous error bounds quickly deteriorate as the width of the interval of parameters on which the uniform contraction theorem is applied grows. This is due to the fact that piecewise-linear approximations are coarse approximations of the solution branches of nonlinear problems. Expanding the solutions using high order Taylor approximations in the parameter could for instance increase significantly the error bounds e.g. see [44, 45], at the cost of complicating the analysis. This being said, let us mention the existence of a growing literature on rigorous numerical methods to compute branches of parameterized families of solutions [3, 46, 47, 48, 49]. Assume that numerical approximations of 3 have been obtained at two different parameter values λ 0 and λ, namely there exists x 0, λ 0 and x, λ such that F x 0, λ 0 0 and F x, λ 0. In other words, x 0, λ 0 and x, λ are approximately in the solution set S e.g. see Figure. The approximations can be computed first by considering a finite dimensional projection of F and then by using one of the two predictor-correctors algorithms presented in Section.. We refer to Section 3..3 for an example in the context of periodic

DELAY DIFFERENTIAL EQUATIONS AND CONTINUATION solutions of DDEs. Define the set of predictors between the approximations x 0, λ 0 and x, λ by 5 { x s, λ s x s = s x 0 + s x and λ s = sλ 0 + sλ, s [0, ]}. x s x S kxk r x 0 0 Figure. The set of predictors { x s, λ s s [0, ]}, approximating a segment of the solution set S. The radii polynomial approach, when successful, provides a tube of with r > 0 the shaded region in X R, where the true segment of solution curve is guaranteed to exist. Consider bounded linear operators A BX, Y and A BY, X. In practice, the operator A is chosen to be an approximation of D x F x 0, λ 0 while A is chosen to be an approximate inverse of D x F x 0, λ 0. Assume that A is injective and that 6 AF : X R X. The following theorem, often called the radii polynomial approach, is a twist of the standard Newton-Kantorovich theorem e.g. see [50]. Theorem.6 Radii Polynomial Approach. Assume that F C k X R, Y with k {,,..., }, and let Y 0, Z 0, Z, Z 0 satisfying 7 8 9 0 AF x s, λ s X Y 0, s [0, ] I AA BX,X Z 0 A[D x F x 0, λ 0 A ] BX,X Z, A[D x F x s + b, λ s D x F x 0, λ 0 ] BX,X Z r, b B r 0 and s [0, ]. Define the radii polynomial pr = Z rr + Z + Z 0 r + Y 0. If there exists r 0 > 0 such that pr 0 < 0,

JEAN-PHILIPPE LESSARD then there exists a C k function such that x : [0, ] s [0,] B r0 x s F xs, λ s = 0, s [0, ]. Furthermore, these are the only solutions in the tube s [0,] B r 0 x s. Proof. Recalling 6, ine the operator T : X [0, ] X by T x, s = x AF x, λ s. We begin by showing that for each s [0, ], the operator T, s is a contraction mapping from B r0 x s into itself. Now, given y B r0 x s and applying the bounds 7, 8, 9, and 0, we obtain D x T y, s BX,X = I AD x F y, λ s BX,X I AA BX,X + A[D x F x 0, λ 0 A ] BX,X + A[D x F y, λ s D x F x 0, λ 0 ] BX,X Z 0 + Z + Z r 0. We now show that for each s [0, ] the operator T, s maps B r0 x s into itself. y B r0 x s and apply the Mean Value Theorem Theorem.3 to obtain Let T y, s x s X T y, s T x s, s X + T x s, s x s X sup b B r0 x s D x T b, s BX,X y x s X + AF x s, λ s X Z 0 + Z + Z r 0 r 0 + Y 0 where the last inequality follows from. Recalling and using the assumption that pr 0 < 0 implies that T y, s x s X < r 0 for all s [0, ], the desired result. Letting a, b B r0 x s, apply the Mean Value Theorem and to obtain 3 T a, s T b, s X sup D x T b, s BX,X a b X b B r0 x s Z 0 + Z + Z r 0 a b X. Again, from the assumption that pr 0 < 0, it follows from Y 0 0 that 4 κ = Z 0 + Z + Z r 0 < Y 0 r 0, Define the operator T : B r0 0 [0, ] B r0 0 y, s T y, s = T y + x s, s x s. Consider now x, y B r0 0 and s [0, ]. Then, since x + x s, y + x s B r0 x s, we can use 3 and 4 to get T x, s T y, s X = T x + x s, s T y + x s, s X κ x y X.

DELAY DIFFERENTIAL EQUATIONS AND CONTINUATION 3 Since κ <, we conclude that T : B r0 0 [0, ] B r0 0 is a uniform contraction. By the Uniform Contraction Theorem Theorem.5, there exists g : [0, ] B r0 0 by T gs, s = gs. Since F C k X R, Y, then T C k B r0 0 [0, ], B r0 0, and therefore g C k [0, ], B r0 0. Let xs = gs + x s so that for all s [0, ] T xs, s = T gs + x s, s = T gs, s + x s = gs + x s = xs. Since T x, s = x AF x, λ s, we get that By assumption that A is injective, It follows from g C k [0, ], B r0 0 that T xs, s = xs AF xs, λ s = xs. F xs, λ s = 0, s [0, ]. x : [0, ] s [0,] B r0 x s is a C k function. Furthermore, it follows from the contraction mapping theorem that these are the only solutions in the tube s [0,] B r 0 x s [λ 0, λ ]. Theorem.6 provides a recipe to compute a local segment of solution curve and to obtain a uniform rigorous error bound r along the set of predictors connecting two numerical approximations x 0 and x. See Figure for a representation of the region shaded where the true segment of solution curve is guaranteed to exist. Assume now that this argument has been repeated iteratively over the set { x 0,..., x j } of approximations at the parameter values {λ 0,..., λ j } respectively. For each i = 0,..., j, this yields the existence of a unique portion of smooth solution curve S i in a small tube centered at the segment { s x i + s x i+ s [0, ]}. As the following results demonstrates, the set S = j is a global smooth solution curve of F x, λ = 0. Lemma.7 Globalizing the C k solution branch. Assume that the radii polynomial approach was successfully applied via Theorem.6 to show the existence of two C k segments S 0 and S of solution curves parameterized by the parameter λ over the respective parameter intervals [λ 0, λ ] and [λ, λ ]. Assume that the sets of predictors are ined by the three points x 0, x and x with x i C m for some fixed dimension m. Then the new segment of solution curve S 0 S is a C k function of λ. Proof. The continuity of S 0 S follows from the fact that at the parameter value λ = λ, the solution segment S 0 must connect continuously with the solution segment S by the existence and uniqueness result guaranteed by the Contraction Mapping Theorem. Let p 0 r the radii polynomial built with the predictors generated by x 0 and x, and ined by the bounds Y 0, i=0 S i

4 JEAN-PHILIPPE LESSARD Z 0, Z and Z. Let r 0 > 0 such that p 0 r 0 < 0. By continuity of the radii polynomial p 0, there exists δ 0 > 0 and there exist bounds Ỹ0δ 0 and Z r, δ 0 such that AF x s, λ s X Ỹ0δ 0, s [ δ 0, + δ 0 ] A[D x F x s + b, λ s D x F x 0, λ 0 ] BX,X Z r, δ 0, b B r 0, s [ δ 0, + δ 0 ], and such that p 0 r 0 = Z r 0, δ 0 r 0 + Z + Z 0 r 0 + Ỹ0δ 0 < 0. Then, there exists of a C k branch of solution curve parameterized by λ over the range { sλ 0 + sλ s [ δ 0, + δ 0 ], extending smoothly in fact in a C k way the segment S 0 on both sides. Similarly, there exists δ such that the segment S can be extended smoothly over the parameter range { sλ + sλ s [ δ 0, + δ 0 ]. This implies that there is a C k overlap between S 0 and S. kxk x x 0 x S 0 S 0 Figure. Assume that S 0 and S are computed with the radii polynomial approach with predictors ined by three points x 0, x and x with x i C m for some fixed dimension m. Then the following situation is not possible: a piecewise smooth but not globally smooth piece of solution curve. Note that argument of using the continuity of the radii polynomial in the proof of Lemma.7 is not new and has been used in the previous works [3, 48, 49]. Repeating iteratively the argument of Lemma.7 leads to the existence of a smooth solution curve S of F = 0 near the piecewise linear curve of approximations, as portrayed in Figure 3. Remark.8 Bifurcations. In the present lecture notes we do not discuss how to handle some type bifurcations. Instead, we refer to the lecture notes of Thomas Wanner, where a rigorous computational method to prove existence of saddle-node bifurcations and symmetrybreaking pitchfork bifurcations is presented. Remark.9 Number of Parameters and Multi-parameter Continuation. In these lecture notes, we present the ideas in the context of equations depending on a single parameter λ R. However, the radii polynomial approach as presented below in Theorem.6 works also for problems depending on p > parameters. In fact, the method can be trivially extended to prove existence of solution manifolds within solutions sets of the form {x, Λ X R p F x, Λ = 0}. The only difference is that the bounds which need to be computed to apply the

DELAY DIFFERENTIAL EQUATIONS AND CONTINUATION 5 kxk x j x 0 S x x 4 x x j x 3 x j 0 3 4 j j j Figure 3. Computing rigorously a global branch of solutions. uniform contraction theorem have to be obtained uniformly over a compact set of parameters in R p instead of in R. A more advanced approach based on a rigorous multi-parameter continuation method, generalizing the concept of pseudo-arclength continuation, is introduced in [49] to compute solutions manifolds and to handle higher dimensional folds. Remark.0 Parameter Continuation vs Pseudo-Arclength Continuation. The method of Theorem.6 is based on parameter continuation: we compute branches of solutions parametrized by the parameter λ. The method can be extended to pseudo-arclength continuation where solutions are parametrized by pseudo-arclength e.g. see [48, 3]. Remark. Computing the bound Y 0. To compute the Y 0 bound satisfying 7, denote and consider the expansion x = x x 0 and λ = λ λ 0, [ F x s, λ s = F x 0, λ 0 + D x F x 0, λ 0 + s F x s, λ s s=0 s + h.o.t. F λ x 0, λ 0 ] x λ s

6 JEAN-PHILIPPE LESSARD Denote 5 6 Hence, y = [ D x F x 0, λ 0 y = s F x s, λ s s=0 ] F x λ x 0, λ 0 λ. AF x s, λ s X AF x 0, λ 0 X + Ay X + Ay X + δ where the extra term δ 0 can be obtained using Taylor remainder s theorem..4. A Finite Dimensional Example. In this section, we apply the radii polynomial approach Theorem.6 to prove the existence of branches of solutions of the problem 3 with F a mapping between finite dimensional Banach spaces. The example we consider is the problem of computing branches of steady states for the atmospheric circulation model introduced by Edward N. Lorenz in [5] 7 x = αx x x 3 + αλ x = x + x x βx x 3 + γ x 3 = x 3 + βx x + x x 3. Let us fix α = 0.5, β = 4 and γ = 0.5, and leave λ as a parameter. At these parameter values, equilibria of 7 are solutions of 8 F x, λ = 4 x x x 3 + λ 4 x + x x 4x x 3 + = 0. x 3 + 4x x + x x 3 In this case, the Banach spaces are X = Y = R 3 endowed with the sup-norm x = max x, x, x 3. At λ 0 = 0.8 and λ = 0.85, we used Newton s method to compute respectively x 0 = 0.0565585983890 0.4549579654079 and x = 0.0435054809 0.46688935398. 0.0968790048534 0.077744769808933 We wish to use Theorem.6 to prove the existence of a segment of solutions in the solution set S = {x, λ R 4 F x, λ = 0}. For s [0, ], recall the set of predictors 5 given by x s = s x 0 + s x and let λ s = sλ 0 + sλ. Denote by x = x x 0 and λ = λ λ 0. Recalling the Y 0 bound satisfying 7. Since the vector field is quadratic, recalling 5 and 6, we get the following expansion where F x s, λ s = F x 0, λ 0 + y s + y s, x y = x 3 x x 4 x x 3. 4 x x + x x 3

DELAY DIFFERENTIAL EQUATIONS AND CONTINUATION 7 The matrix A DF x 0, λ 0 is computed using MATLAB and is given by.034443070077 0.8834855388585 0 A =.6473748855484 0.0598974595854 0.93758400497068..43639056383.4966946078974 0.9049945930857 Using the inition of A, y and y, we compute Y 0 = 0.0048845505. For this finite dimensional example, we set A = DF x 0, λ 0, so that Z = 0 in 9. Recalling 8, we set Z 0 = I ADF x 0, λ 0. In this case, we computed Z 0 =.7 0 6. To facilitate the computation of Z satisfying 0, consider c B 0 R 3, that is c, and consider b B r 0 R 3, that is b r. Then, b c b 3 c 3 [D x F x s + b, λ s D x F x 0, λ 0 ]c = b c 4b c 3 + b c 4b 3 c 4b c + b c 3 + 4b c + b 3 c c x c 3 x 3 + s c x 4c x 3 + c x 4c 3 x, 4c x + c x 3 + 4c x + c 3 x and since s, we get the component-wise inequalities [D x F x s + b, λ s D x F x 0, λ 0 ]c 4 x + x 3 0 r + x + 4 x 3 + 5 x. 0 4 x + x 3 + 5 x Hence, letting Z = A 4 0 and Z 0 0 we can set = x + x 3 A x + 4 x 3 + 5 x 4 x + x 3 + 5 x Z r = Z r + Z0. Numerically, we obtain Z = 8.974747666638 and Z 0 = 0.4405944794. Recalling and that Z = 0, the radii polynomial is given by Note that pr = Z rr + Z 0 r + Y 0 = Z r + Z 0 + Z 0 r + Y 0 = 8.974747666638r 0.55940755778076r + 0.0048845505. I = {r > 0 pr < 0} [0.00694976548045, 0.0359430566]. Choosing for instance r 0 = 0.007 I, then by Theorem.6, there exists a C function x : [0, ] B r0 x s s [0,] such that F xs, λ s = 0 for all s [0, ] with F given in 8, and these are the only solutions in the set s [0,] B r 0 x s [λ 0, λ ].

8 JEAN-PHILIPPE LESSARD Also, we applied the method on the intervals of parameters [λ, λ ] and [λ, λ 3 ] corresponding respectively to the segments { s x + s x s [0, ]} and { s x + s x 3 s [0, ]}, with λ = 0.89, λ 3 = 0.95, and x = 0.0374673 0.4764888735590 and x 3 = 0.0963874357 0.48486639859068. 0.060438037938 0.04353784636356 The MATLAB program script proof lorenz.m available at [67] performs the above computations. It uses the interval arithmetic package INTLAB developed by Siegfried M. Rump [5]. x 0.8 0.6 0.4 0. 0 0.5 0.6 0.7 0.8 0.9.. λ -0.0498-0.0499-0.05-0.050-0.050-0.0503-0.0504 0.848 0.848 0.849 0.849 0.850 0.85 0.85 0.85 Figure 4. Left A branch of equilibria for the model 7 computed using the pseudo-arclength continuation algorithm as presented in Section... The segments in red, green and purple were rigorously computed with the radii polynomial approach. The respective error bounds between the predictors and the actual solution segments are r 0 = 7 0 3 red, r 0 = 4. 0 3 green and r 0 = 3.9 0 3 purple. Right A zoom-in on the branch where the proof was performed at λ = 0.85. Remark. Proofs at fixed parameter values. It is important to recall that the purpose of the present section is to introduce a method to compute branches of solutions. If however we are interested in proving the existence of a solution at a fixed parameter value, then we can get dramatically better error bounds. Let us do this exercise for model 7 with the approximation x 0. In this case, x = 0 R 3, λ = 0 and Z 0 = 0, the radii polynomial is pr = 8.97474766665r +.9585794809 0 5 r+.4499947945 0 6, and I = {r > 0 pr < 0} [.45 0 6, 0.034567053563 ]. Therefore, there exists a unique x 0 B.45 0 6 x 0 such that F x 0, λ 0 = 0. In this case, the rigorous error bound is of the order of 0 6, as opposed to 0 3 for the branch of solutions. We are now ready to present the main application of the rigorous continuation method.

DELAY DIFFERENTIAL EQUATIONS AND CONTINUATION 9 0-3.5 Y 0.5 pr 0.5 0 0 0.005 0.0 0.05 Figure 5. The radii polynomial pr = Z r + Z 0 + Z 0 r + Y 0 associated to the numerical approximations x 0 and x as ined above. I r 0-3 0 I - - pr -3-4 -5-6 -7-8 0 0.005 0.0 0.05 0.0 0.05 0.03 Figure 6. The radii polynomial pr = Z r + Z 0 r + Y 0 associated to the single numerical approximation x 0. r 3. Computing Branches of Periodic Solutions of Delay Equations In this section, we show how the radii polynomial approach Theorem.6 can be used to compute rigorously branches of periodic solutions of DDEs. Rather than presenting the ideas for general classes of problems, we focus on presenting the ideas for specific examples, namely for a delayed Van der Pol equation and for Wright s equation. For the delayed Van der Pol equation, we present in details in Section 3. all steps, bounds, necessary estimates, choices of function spaces and the explicit coefficients of the radii polynomial, whereas in the case of Wright s equation, we only briefly discuss some results in Section 3. and refer to [] for more details. Note that the ideas presented here should be applicable to systems of N

0 JEAN-PHILIPPE LESSARD JOHN ~%/[ALLET-I~ - ROGER D. ~USS:BAUM: Global continuation, etv. 63 delay equations of the form then xt; ~, ~ is a solution of equation.~ with minimal period p satisfying 9 y n t = F yt, yt τ,..., yt τ d,..., y n t τ,..., y n t τ d,.o where y : R R N and F : R Nnd+ Im-~l < p < Im R N is89 a multivariate if m < o, polynomial. As.~ already mentioned in Section studying rigorously solutions of DDEs is a challenging problem, especially because they are naturally ined on infinite dimensional function p> if re=o, spaces. and On the other hand, as already mentioned in Section, for continuous dynamical systems like 9, individual solutions which exist globally in time are more regular than the.~ typical functions of the natural m-~ 89 phase <p<- space. m That if m>0. suggests that solving for the Fourier coefficients of the periodic solutions of 9 in a Banach space of fast decaying sequences is athe goodsets strategy. /,~ are pairwise In fact periodic disjoint. solutions I/ m>0 of and 9 ~ > are ~,,, analytic then equation since F.;. is analytic has at it is polynomial least m + [0]. distinct We therefore periodic solutions~ apriori know while that i/m the < Fourier 0 and A coefficients < ~, then it ofhas theat periodic least solutions Im[ periodic decay geometrically solutions. by the Paley-Wiener Theorem. Before presenting the rigorous numerical method, we briefly describe different methods used to study periodic solutions of 9, following PROOF. - closely The first the discussion part of the in theorem [53]. h~s already been proved. The bounds Fixed.0~ point.~ theory, ~nd the. fixedon point the index minimal and global period bifurcation p follow immediately theorems arefrom powerful the tools to study formula thep existence ~ ~ta, ~/Im~A, of solutions ~ of I infinite noted above, dimensional ~nd the dynamical fact thst systems. ~A~ ~ To > give for a few examples., ~se in/0. thethese context bounds of DDEs, also the imply ejective that fixed the sets point Z~ theorem are pairwise of Browder disjoint: [54] one and the fixedcan point easily index check canthgt be used the intervals to prove of existence values p ofgiven nontrivial in.0, periodic., solutions ~nd. [4, for 55, 56], andvarious the global integers bifurcation m are pairwise theorem disjoint. of Rabinowitz Finally, [57] the canconnectedness, be used to prove nnboundedncss the existence and characterize and disjointness the non of compactness the /~, the bounds of global.7, branches.8, of and periodic.9 for solutions X, ~e/~, [58, 59]. and This heavy the machinery ordering from functional analysis provides powerful existence results about solutions of DDEs, but its applicability may decrease if one asks more specific questions about the solutions of a given equation.... < For ~ ~< example, i_~< 0 it< appears Ao< A~< difficult ~<... in general to use the ejective fixed point theorem to quantify the number of periodic solutions or to use a global bifurcation theorem imply tothe conclude last sentence about existence in the statement of folds, of orthe more theorem. generally, [] of secondary bifurcations. The following Figure 7 taken from [9] shows that global branches of periodic solutions of DDEs may Figure be 6 complicated, depicts schematically as various the bifurcations branches may X~. occur on the branches. li~li Z 0 A ~ Fig. 6. The global Hopf branches :,~. Figure 7. Global branches of periodic solutions of delay differential equations. The picture is taken directly from [9].

DELAY DIFFERENTIAL EQUATIONS AND CONTINUATION 3.. A Delayed Van der Pol Equation. In the 90s, the Dutch electrical engineer and physicist Van der Pol proposed his famous Van der Pol equations to model oscillations of some electric circuits [60]. Since then, variants of the so-called van der Pol oscillator have been proposed as mathematical models of various real-world processes exhibiting limit cycles when the rate of change of the state variables depend only on their current states. However, there are many processes where this relation is also influenced by past values of the system in question. To model these processes, one may want to consider the use of functional differential equations, see [9, 6, 6]. In [63], Grafton establishes existence of periodic solutions to ÿt εẏt y t + yt τ = 0, ε, τ > 0, a Van der Pol equation with a retarded position variable. His results are based on his periodicity results developed in [64]. In [65], using slightly different notations, Roger Nussbaum considered the more general class of equations 0 ÿt εẏt y t + yt τ λyt = 0, ε, τ > 0, λ R, and establishes existence of periodic solution of period greater than τ, given that λ < 0, λτ < ε and λτ. We refer to 0 as Nussbaum s equation. The techniques that Nussbaum uses are sophisticated fixed point arguments, however, as he remarks, he has to restrict the size of λ in order to guarantee that the zeros of y are at least a distance τ apart e.g. see p. 87 of [65]. Moreover, he mentions that numerical simulations suggest the existence of periodic solutions to 0 for a large range of λ < 0. What we present now is an application of the rigorous continuation method introduced in Section, and we establish existence results for periodic solutions to 0 for parameter values outside of the range of parameters accessible with the above results of Nussbaum. The first step is to recast looking for periodic solutions of 0 as one of the form F x, λ = 0. 3... Setting up the F x, λ = 0 problem. Assume that yt is a periodic solution of 0 of period p > 0. Then yt = a k e ikωt, k= where ω = π p and the a k C are the complex Fourier coefficients. Given a complex number z C, denote by conjz the complex conjugate of z. During the rigorous continuation approach, we will verify a posteriori that the complex numbers satisfy a k = conja k in order to get that y R. As the frequency ω of is not known a-priori, it is left as a variable. Formally, using ẏt = a k ikωe ikωt, ÿt = a k k ω e ikωt and yt τ = a k e ikωτ e ikωt. k= Thus 0 becomes k= k= [ k ω εikω λ + e ikωτ ] a k e ikωt + ε k = a k e ik ωt k = a k e ik ωt k 3 = k= a k3 ik 3 ωe ik 3ωt = 0.

JEAN-PHILIPPE LESSARD To obtain the Fourier coefficients in, one takes the inner product on both sides of with e ikωt, yielding that for all k Z, [ ] g k = k ω εikω λ + e ikωτ c k + iεω a k a k a k3 k 3 = 0. Observing that S k = we get that k +k +k 3 =k k j Z and therefore a k a k a k3 k 3 = k +k +k 3 =k k j Z S k = k 3 k +k +k 3 =k a k a k a k3 k k k = k k +k +k 3 =k k j Z a k a k a k3 3 g k = µ k a k + iεkω 3 a3 k, where µ k = µ k ω, λ = k ω εikω λ + e ikωτ and a 3 k = k +k +k 3 =k k j Z k +k +k 3 =k k j Z a k a k a k3 S k a k a k a k3. Denote a = a k k Z and x = ω, a the infinite dimensional vector of unknowns. Denote gx, λ = g k x, λ k Z. In order to eliminate any arbitrary time shift, we append a phase condition given by 4 ηx = c k = 0, k <n for some n N, which ensures that y0 0. Combining 4 and 3, we let 5 F x, λ ηx =. gx, λ Let us now introduce the Banach space in which we look for solutions of F x, λ = 0 with F given in 5. Since periodic solutions of analytic delay differential equations are analytic e.g. see [0], their Fourier coefficients decay geometrically. This will motivate the choice of Banach space in which we will embed the Fourier coefficients. Given a weight ν >, ine the sequence space 6 l ν = {a = a k k Z a,ν < }, where 7 a,ν = a k ν k. k Z Define the Banach space endowed with the norm X = C l ν x X = max ω, a,ν.

DELAY DIFFERENTIAL EQUATIONS AND CONTINUATION 3 Note that F does not map X into itself. This is because a differential operator is in general unbounded on l ν. In order to overcome this problem we look for an injective linear smoothing operator A such that 6 holds, that AF x, λ X for all x X and λ R. The choice of the approximate inverse A is presented in Section 3..4. For now we take A as given and ine the Newton-like operator by 8 T x, s = x AF x, λ s, for s [0, ]. Given s [0, ], the injectivity of A implies that x is a solution of F x, λ s = 0 if and only if it is a fixed point of T, s. Moreover T, s now maps X back into itself. 3... Symmetry of the fixed points of T. We are interested in showing the existence of a real periodic solution y given by satisfying the symmetry property yt + p/ = yt, t R. In Fourier space, this means that the Fourier coefficients satisfy the relation 9 a j = 0, j Z. To do this, we design the method so that fixed points of T are in the symmetry space 30 X sym = R l ν, where 3 l ν = { a l ν a k = conja k k Z, and a j = 0 j Z }. Remark 3.. The condition a k = conja k is imposed in the function space l ν because we want y to be a real periodic solution, that is conjyt = yt. Lemma 3.. Assume that x s X sym and consider the closed ball B r x s X. Define T as in 8 and assume that the approximate inverse A satisfies 3 AF : X sym R X sym. Assume that for every s [0, ], T, s : B r x s B r x s is a contraction, and let x s X the unique fixed point of T, s in B r x s which exists by the Contraction Mapping Theorem. Then, x s X sym for all s [0, ]. Proof. By 3, T : X sym [0, ] X sym. Using that x s X sym B r x s, and that X sym is a closed subset of X for every s [0, ], we obtain that x s = lim n T n x s, s X sym. 3..3. Computation of the numerical approximations. We are ready to compute numerical approximations of F x, λ = 0 with F given in 5. Since the operator F is ined on an infinite dimensional space, this requires considering a finite dimensional projection. Given a = a k k Z l ν and a projection dimension m N, denote by a m = a k k <m C m a finite part of a of size m. Moreover, given x = ω, a X = C l ν, denote x m = ω, a m C m. Consider a finite dimensional projection F m : C m R C m of 5 given by 33 F m x m, λ = ηx m g m x m, λ,

4 JEAN-PHILIPPE LESSARD where g m x m, λ C m corresponds to the finite part of g of size m, that is g m = {g m k } k <m. More explicitly, given k < m, g m k x m, λ = µ k ω, λa k + iεkω 3 k +k +k 3 =k k i <m a k a k a k3. We can now apply the parameter continuation method as introduced in Section.. to the finite dimensional problem F m : C m R C m. Assume that at the parameter value λ = λ 0, x 0 = ω 0, ā 0 C C m is an approximate solution, that is F m x 0, λ 0 0. The next step is to introduce an approximate inverse operator A that satisfies 3 and the operator A, required to apply the radii polynomial approach Theorem.6. 3..4. Definition of the operators A and A. We now ine an approximate inverse A for D x F x 0, λ 0 so that 6 holds and the operator A which approximates D x F x 0, λ 0. Assume that x 0 = ω 0, ā 0 X sym. The Fréchet derivative D x F x 0, λ 0 can be visualized by block as 0 D D x F x 0, λ 0 = a η x 0, ω g x 0, λ 0 D a g x 0, λ 0 since ω F 0 x 0 = 0, and where ω g x 0, λ 0 : C l ν, D a η x 0 : l ν C is a linear functional D a g x 0, λ 0 : l ν l ν is a linear operator with ν < ν. We first approximate D x F x 0, λ 0 with the operator A 0 A a,0 = A ω, A, a, which acts on b = b 0, b X = C l ν component-wise as A b 0 = A a,0 b = D a mη x 0 b m A b = A ω, b 0 + A a, b l ν, where A ω, = ωg m x 0, λ 0 and A a, b l ν { A a, b = k is ined component-wise by k < m D a mg m x 0, λ 0 b m k, µ k ω 0, λ 0 b k, k m. Let A m a finite dimensional approximate inverse of D x F m x 0, λ 0 which is obtained numerically and which has the decomposition A m A m ω,0 A m a,0 = A m ω, A m C m m, a,

DELAY DIFFERENTIAL EQUATIONS AND CONTINUATION 5 where A m ω,0 C, Am a,0 C m, A m ω, Cm and A m a, Cm m. Assume moreover that A m satisfies the following symmetry assumptions: 34. A m a,0 j = conj. A m ω, k = conj 3. A m a, k, j = conj A m a,0 j A m ω, k, j = m +,..., m,, k = m +,..., m,, k, j = m +,..., m, A m a, k,j 4. A m ω, k = 0, k { m +,..., m }, 5. A m a, k,j+ = 0, k, j + { m +,..., m }. Assumption of 34 implies that A m a,0 0 R while assumption implies that A m ω, 0 R. We ine the approximate inverse A of the infinite dimensional operator D x F x 0, λ 0 by A = Aω,0 A a,0, A ω, A a, where A acts on b = b 0, b X = C l ν component-wise as Ab 0 = A m ω,0 b 0 + A m a,0 bm Ab = A m ω, b 0 + A a, b, where A m ω, Cm is understood to be an element of l ν by padding the tail with zeros, and A a, b l ν is ined component-wise by A a, b k = Let us now verify that 6 holds. A m a, bm, k < m k µ k ω 0, λ 0 b k, k m. Lemma 3.3. Let x X and λ R. Then AF x, λ X. Proof. Consider x = ω, a X and let F x, λ = ηx, gx, λ, with ηx given in 4 and g given component-wise in 3. For sake of simplicity of the presentation, we denote F 0 = ηx and F = gx, λ. We need to show that AF x ν <. Since 35 µ k ω, λ = k ω εikω λ + e ikωτ, lim k ± µ k ω, λ µ k ω 0, λ 0 = lim k ± k ω εikω λ + e ikωτ ω k ω 0 εik ω 0 λ 0 + e ik ω 0τ = <, ω 0 there exists C < such that µ k ω µ k ω 0, µ k ω 0 iεkω 3 < C, for all k m.

6 JEAN-PHILIPPE LESSARD Then, AF x ν = AF x k ν k = A m ω, F 0 + A a, F ν k k k Z k Z A m ω, F 0 ν k + A m a, F m ν k k <m k, k <m + µ k ω 0, λ 0 µ kω, λa k + iεkω 3 a3 k ν k k m A m ω, F 0 ν k + A m a, F m ν k k <m + C k m k <m k, a k ν k + C A m ω, k, k m + C a,ν + C a,ν 3 <, k <m a 3 k ν k F 0 ν k + A m a, F m k <m k k ν k where we used the fact that a 3,ν a,ν 3, because l ν is a Banach algebra. Let us now show that the operator A satisfies the symmetry assumption 3. Lemma 3.4. Let x X sym and λ R. Then AF x, λ X sym. Proof. Let x = ω, a X sym = R l ν, with l ν as ined in 3. This implies that a k = conja k and a k = 0 for all k Z. Again, denote F 0 = ηx and F = gx, λ. We begin the proof by showing that the operator F preserves the symmetry conditions, that is we show that F 0 R, F k = conjf k and F k = 0. Recalling the inition of the phase condition 4, F 0 = a k = a n + a n+ + + a n + a n = conja n + conja n + + a n + a n R. k n Also, from 35, we see that µ k ω, λ = conj µ k ω, λ. Then, F k = µ k ω, λa k iεkω a k a k a k3 3 Now, k +k +k 3 = k = conj µ k ω, λa k iεkω 3 iεkω = conj µ k ω, λa k + conj 3 = conj F k. k +k +k 3 =k a k a k a k3 k +k +k 3 =k 36 F k = µ k ω, λa k + iεkω a 3 k = µ k ω, λ0 + iεkω 3 3 k conja k conja k conja k3 k +k +k 3 =k since k + k + k 3 = k implies that there exists i {,, 3} such that k i is even. a k a k a k3 = 0,