Rapidly and slowly oscillating periodic solutions of a delayed van der Pol oscillator

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1 Rapidly and slowly oscillating periodic solutions of a delayed van der Pol oscillator Gabor Kiss Jean-Philippe Lessard Abstract In this paper, we introduce a method to prove existence of several rapidly and slowly oscillating periodic solutions of a delayed van der Pol oscillator. The proof is a combination of pen and paper analytic estimates, the contraction mapping theorem and a computer program using interval arithmetic. Using this approach we extend some existence results obtained by Nussbaum in [Ann. Mat. Pura Appl., 4, 6 6, 974]. Keywords Delay differential equations Rigorous numerics Rapid and slow periodic solutions Contraction mapping theorem van der Pol equation Introduction In 9, to model oscillations of some electric circuits, van der Pol proposed one of the most influential system of nonlinear differential equations, see []. 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 [,, 4, 5]. In [7], Grafton establishes existence of periodic solutions to ÿt εẏt y t yt τ, ε, τ >, a van der Pol equation with a retarded position variable. His results are based on his periodicity results developed in [8]. In [9], using slightly different notations, Nussbaum considers the more general class of equations ÿt εẏt y t yt τ κyt, ε, τ >, κ R, and establishes the following result. Theorem. Nussbaum, 974. If κ <, κτ < ε and κτ equation has a nonzero periodic solution of period greater than τ., then University of Szeged, Bolyai Institute, Aradi vértanú tere., 67 Szeged, Hungary McGill University, Department of Mathematics and Statistics, 85 Sherbrooe St W, Montreal, QC HA B9.

2 We refer to as Nussbaum s equation. The techniques that Nussbaum uses are sophisticated fixed point arguments, however, as he remars, he has to restrict the size of κ in order to guarantee that the zeros of y are at least a distance τ apart, p. 87 of [9]. Moreover, he mentions that numerical simulations suggest the existence of periodic solutions to for a large range of κ <. Thus, it seems natural to try to develop a computational tool in order to establish existence results for periodic solutions to for parameter values seemingly inaccessible using Nussbaum s methods. Before formulating our results, we need the following definition. Definition.. Let yt be a periodic solution to with the minimal period p. Assume that z < z <... < z n p are such that yz i, i,..., n, and that these are the only zeroes of y. If z i z i > τ for all i,..., n then y is a slowly oscillating periodic solution. Similarly, y is said to be a rapidly oscillating periodic solution if z i z i < τ for all i,..., n. Our aim is to prove the following statements, whose proofs are a combination of pen and paper analytic estimates and computer programming using interval arithmetic. Theorem.. Let ε.5 and τ. For 77 distinct values of κ [.85,.79], there exists a nontrivial periodic solution of. As for the range of periods p of these solutions we have that p [.69,.4]. The parameter values of Theorem. were obtained by performing a numerical continuation on κ with step size.5. Theorem.4. Let ε.5 and τ 5. For 8 distinct parameter values of κ [ 4.684,.476], there exists a nontrivial periodic solution of. As for the range of the periods p of these solutions we have that p [.5, 5.6]. The parameter values of Theorem.4 were also obtained by performing a numerical continuation on κ with varying step size. Remar.5. The norms of the computed solutions are shown in Figure while their periods are plotted in Figure. Also, some illustrative time profiles can be seen in Figure and 4; these figures indicate that, indeed, there are both slowly and rapidly oscillating solutions to. Furthermore, our results relax the periodicity condition in Nussbaum s theorem. The trade-off here clearly is that our method applies to a given specific equation. Also, unlie in the case of finite dimensional dynamical systems, see [6], we do not have tools to address stability properties of these periodic solutions. Nevertheless, since these periodic solutions are visible via numerical simulations suggests their stability. The proofs of Theorem. and Theorem.4 are based on adapting ideas from [] and [] to the context of second order delay equations with cubic nonlinearities. Moreover, we would lie to mention that Nussbaum considers in [9] a large family of functional differential equations of which is a particular member. Establishing existence results of periodic solutions to other members of the family of equations considered in [9] is an ongoing project.

3 y l y l κ 5 4 κ Figure : Left: The norm of the computed 67 solutions of Theorem. with respect to the parameter κ for τ and ε.5. Right: The norm of the computed 8 solutions of Theorem.4 with respect to the parameter κ for τ 5 and ε p p κ 5 4 κ Figure : Left: The period of the computed 67 solutions of Theorem. with respect to the parameter κ for τ and ε.5. Right: The period of the computed 8 solutions of Theorem.4 with respect to the parameter κ for τ 5 and ε.5. Setting up a fixed point problem If yt is a periodic solution of with a period p >, then yt c e ilt, where L π p and the c are complex numbers satisfying c c, since y R. Denoting the real and the imaginary part of c respectively by a and b, one gets that a a and b b. As a result, b Imc and so it is not a variable. An equivalent expansion for is given by yt a [a cos Lt b sin Lt]. 4 As the frequency L of is not nown a-priori, it is left as a variable. Since periodic solutions of analytic delay differential equations are analytic e.g. see [], their Fourier coefficients decay faster then any algebraic decay. This will motivate the

4 . a 4 b. y y.. y t c t y t d 5 t Figure : Some of the computed 67 solutions with τ and ε.5. On panel a, b, c and d, κ.85,.57,.8 and.79, respectively..5 a 4 b..5 y y.5. t c 4 4 t d..5 y t.5 y t Figure 4: Some of the computed 8 solutions with τ 5 and ε.5. On panel a, b, c and d, κ 4.8,.64,.486 and.476, respectively. choice of Banach space in which we will embed the Fourier coefficients. Let { def L, a, x a, b, > 5 and x def x, x,..., x,..., and denote the first and the second component of x by x, and x,, respectively. Given a growth rate s >, consider the weight functions { ω s, s 6, 4

5 which are used to define the norm where x max{ x,, x, }. def x s sup x ω, s 7 Lemma.. For a given s >, consider the space of sequences with algebraically decaying tails Ω s def {x x, x, x.... : x s < } is a Banach space. Moreover, assume that yt given by 4 is a periodic solution of Nussbaum equation and consider the associated x given by 5. Then for any fixed s >, the space Ω s contains x. Proof. The fact that Ω s is a Banach space is standard and the proof is omitted. Let yt given by 4 a periodic solution of and let x the associated vector given component-wise by 5. Since yt is a periodic solution of the analytic delay differential equation, it follows from [] that it is analytic. Therefore the Fourier coefficients of yt decay exponentially fast to zero. Hence, the sequences {a } and {b } converge to zero faster than any algebraic decay. This implies that for any s >, x Ω s. Formally, using, we get ẏt c ile ilt, and Thus becomes ÿt c L e ilt yt τ c e ilτ e ilt. [ L εil κ e ilτ ] c e ilt ε c e ilt c e ilt c i Le ilt. 8 To obtain the Fourier coefficients in, one taes the inner product on both sides of 8 with e ilt, Z, yielding def g [ L εil κ e ilτ ] c iεl c c c. 5

6 Observing that we get that def S j Z j Z j Z c c c c c c c c c S, g [ L εil κ e ilτ ] c iεl j Z c c c, which is a countable system of nonlinear complex-valued equations. Using 5, one gets that Reg x L κ cos Lτa εl sin Lτb εl a a b a b a b b b j Z Img x L κ cos Lτb εl sin Lτa εl a a a a b b b b a. j Z For a periodic function yt with expansion, let us define the map hx y a a. Hence, the map f whose zeros correspond to periodic solutions oscillating around of is given component-wise by hx, ; Reg f x def x 9 Reg x, >. Img x As in the case of x, denote the first and the second component of f by f, and f,, respectively. For the sae of simplicity of the presentation, let us introduce R L def L κ cos Lτ εl sin Lτ εl sin Lτ L. κ cos Lτ Also, for given three bi-infinite vectors a a Z, b b Z and c c Z, we define the discrete cubic convolution term component-wise by def a b c a b c, Z j Z 6

7 to write f x R L a εl a a b a b a b b b b a a a a b b b b a for. In setting up our fixed point problem, the following is crucial. Lemma.. The operator f {f }, defined component-wise in 9, has the property that f : Ω s Ω s. Proof. It is not difficult to see that and sup L κ cos Lτ a εl sin Lτ b ω s < sup εl sin Lτ a L κ cos Lτ b ω s < for x Ω s. To proceed, for vectors u, v R m n, we introduce u v to denote the fact that u i,j v i,j, i,..., m, j,..., n. Since x Ω s, by definition, a x s ω s and b x s ω s, Z. Now, from Lemma. in [], there are C, and C, positive constants uniformly bounded so that a a b a b a b b b C, a a a a b b b b a ω s,. C, This implies that that is, fx Ω s. fx s sup f ω s <, Remar.. We want to emphasize that the idea of deriving a set of analytic estimates to bound the truncation error term are originally used in [, 5, 6] to develop constructive proofs of existence of equilibria to partial differential equations. However, in the present paper, we decided to write and prove all estimates explicitly to eep this wor self-contained. The following result establishes the correspondence between the periodic solutions of and the zeros of f. Its proof is omitted and a similar statement with a proof can be found for example in []. Lemma.4. For a sequence x x, x, x,... Ω s given in 5, fx if and only if yt in 4 is a solution to such that y a a. Now, we rewrite fx as a fixed-point equation T x x in Ω s where T is a Newton-lie operator. In order to define T, first consider f m : R m R m, a finite dimensional projection f, whose -th component is given by f m x,..., x m def f x,..., x m,,,..., m, 7

8 where j. Hereafter, we identify x L, ā, ā, b,..., ā m, b m with x, if x R m such that f m x. Now we define the finite part of T using numerics while its tail will be defined analytically. To define the finite part, consider the later described computational parameter M m e.g. see Lemma., and let A M R M M be a numerical approximation of the inverse of D x f M x. Assume that I M A M D x f M x <, which implies that A M is an invertible matrix. Furthermore, let def Λ f ϱ δ x, x δ ϱ where and def ϱ L κ cos Lτ def δ ε L sin Lτ ε L ā m ā b. Since κ <, a sufficient condition for the invertibility of Λ for all > M is m > κ. 4 L Indeed, 4 and M imply ϱ < and thus ϱ δ > for all M. Define the linear operator A : Ω s Ω s by { A Ax def M x F,,..., M Λ x 5, M. In this definition, and Λ ϱ δ ϱ δ, δ ϱ x F x, x,..., x M R M is the finite dimensional projection of x Ω s. The conditions and 4 imply the invertibility of the operator A defined in 5. Notice that m in 4 is an explicit lower bound on the dimension of the finite dimensional projection for A to be an invertible linear operator. Now, we complete the verification of the definition of A, that is, we show that A : Ω s Ω s. Given a matrix M denote by M the matrix whose components are the component-wise absolute values of M. Lemma.5. Let L > and ā R, and define and Ξ def ρ def ρ ρ ε L S M M M L κ > M ρ ε L S M ρ M, 8

9 where Then for all M, Thus A : Ω s Ω s. Proof. From 4, Thus S def ā m ā b. Λ cw Ξ. ϱ cos Lτ L κ L κ ϱ ρ, and then ϱ ϱ δ ϱ ϱ ϱ ρ. Finally, since ε >, δ ε L S, we get that δ ϱ δ ε L S ϱ δ ε L S ϱ ρ ε L S 4 ρ ε L S M M. L κ M ρ. That is, for M, Λ ρ ρ ε L S M M ρ ε L S M ρ M. Thus, for an x Ω s, { Ax s max { max max,...,m max,...,m { AM x F ω s } {, sup Λ x ω s } } M } { AM x F ω s }, Ξ sup { x ω} s M The conclusion of Lemma.5 allows us to define the Newton-lie operator T : Ω s Ω s by T x x Afx. 6 By injectivity of the operator A, the fixed points of T are in one-to-one correspondence with the zeros of f. Therefore, we now focus on the computation of the fixed points of T. To achieve this new tas, we use the radii polynomial approach.. 9

10 The radii polynomial approach Let Br def [ r ω s, be the ball of radius r > in Ω s centered at the origin, and be the ball centered at x. component-wise bounds Y r ω s ] 7 B x r def x Br 8 In order to show that T is a contraction, we derive Y, Z,, Z Y R, Z for such that, [T x x] [ Af x] Y 9 and sup [DT x bc] sup [DT x rurv] Z r. b,c Br u,v B Lemma.. For all M m, [Af x], T. Proof. If,, Z satisfy M m, then there exists j {,, } such that j m. Since, ā l b l for all l m, then f, x L κ cos Lτā ε L sin Lτ b ε L ā ā b ā b ā b b b, M. j Z A similar argument shows that f, x. This implies that f x, T, for all M. Finally, recalling 5, we get that [Af x] Λ f x, T, M. A consequence of the previous result is that for M, we let Y, T. Moreover, for M, assume that one may choose M s Z r ẐM r, where ẐM r R is independent of. We justify this assumption in Section.. Definition.. The M radii polynomials {p, p,..., p M } are defined by Y Z r r ω,,..., M s p r def Ẑ M r r ω s M ω s, M. Once these polynomials are constructed, the existence of periodic solutions to can be established by using the following result, whose proof can be found in [].

11 Lemma.. If there exists r > such that p r < for,..., M, then there exists a unique x B x r such that T x x, or, equivalently, such that f x. Therefore, the proofs of Theorem. and Theorem.4 are obtained by constructing the radii polynomials of Definition., by verifying the hypotheses of Lemma. and finally by applying Lemma.4. The tas is now to construct the radii polynomials associated to. We use the components of Y F def A M f F x to compute Y for,..., M. To derive the bound Z, for b, c Br, let b ur and c vr where u, v B, and consider A : Ω s Ω s defined by { A Df M x x def F,,... M Λ x, M, an almost inverse of the operator A. Now, we have DT x rurv [I AA ]rv A[Df x ru A ]rv. Since the first term is very small, for and i {, }, to derive the bound Z, we consider the expansion [Df x ru A ]rv 4 l. The bounds Z r, {,..., M } c l,i rl. 4 For {,..., M }, we generated coefficients c l,i using Maple. Bounds C l such that c l C l can be found in Table. Before going any further, we mae,i some comments on these bounds. First of all, since c,, we can define Thus we get that C l F C l C l C,,...,M. [DT x rurv] F [I M A M Df M x]v F r 4 l A M C l F rl Also, notice that except the infinite sum ω s ω s ω, all other convolutions involve the finite vector x and thus are finite sums. These finite discrete s convolutions are computed by using discrete fast Fourier transformation, for details on this see [4]. Finally, to bound the infinite discrete convolution in Table, one can use the following result.

12 Lemma.4 Quadratic estimates. For µ M and ν Z, let and α µ,ν φ µ,ν M φ µ,ν µν ω s ω s µν s µ ν s ω s Mµν M s s ωµν s. Then µν ω s ω s α µ,ν. 5 Proof. µν ω s ω s M M M M ω s ωµν s µν ωµν s ω s ω s µν ω s ω s µν ω s ω s µν ω s ω s µν M ω s ω s µν ω s ω s µν ω s µν ωmµν s M ω s Mµν ω s Mµν M ω s ω s µν x s dx ω s µν φ µ,ν M s s ωµν s φ µ,ν φ µ,ν φ µ,ν. Remar.5. To simplify our notations, whenever one of the indices is zero in a two-index notation, for instance in the case of φ,, we simply write φ. Before using 5 to formulate an explicit bound of the infinite convolution, we note that generic estimates are developed in [] to bound higher order convolutions.

13 Lemma.6 Cubic estimates. For M, Proof. ω s ω s ω s M ω s α, α M M s M s s M φ ω s ω s M s M s s ω s [ M ω s φ, ω s ω s ] ωm s M s s M M ω s α ω s M s s ω s α M M s M s s. 6 ω s ω s ω s ω s ω s ω s ω s ω s ω s ω s ω s 7 ω s Now we bound each addend in 7. ω s ω s ω s ω s ω s

14 First, ω s ω s ω s Using Lemma.4, we get that ω s ω s φ M M ω s ω s ω s ω s M M M M M ω s α, α M ω s α, ω s α, ω s ω s ω s M α M M s s α M M s α M M s s ω s α, α M M s M s s ω s ω s M s M s s ω s Now taing µ and ν in Lemma.4, we get that [ M φ, ω ω ω ω ω ω ω M M s s ]. ω

15 Also, ω s ω s M M M ω s ω s ω s M M ω s x s dx ω s M s s. Finally, with µ and ν in Lemma.4, we get that ω s ω s ω s M ω s ω s M M M M M Clearly, 8- validate the statement. ω s ω s ω s α, α M ω s ω s M s α M ω s α M s s ω s α α M M s α M M s s ω s α α M M s M s s. Thus we can replace the infinite sums in Table by 6. Also, the first infinite sum in Table can be estimated by ω s m s m s. With these substitutions, we get new upper bounds C l F. Now, for,..., M, the th -component of Z r R can be defined as Z F r def [I F A M Df M x]v F r 5 4 l A M C l F rl.

16 . The analytic bound Ẑ M r In this section, assuming M m, we develop bounds Ĉli, l i {,,, 4}, i, satisfying c l,i s Ĉl. Notice that these bounds depend neither on nor i. To find these bounds, one can use Lemma.7. Let ν Z with ν M, and s, and define s [ ] [ def ν 4 ln ν γ,ν π 6 ν ν ν ] s and Then def α M,ν [ { M M γ M,ν ν ω s ω s ω s s α M,ν. }] M s s. Proof. Using Lemma A. from [5], we get φ,ν ν ω s s ν s s [ ] [ ν 4 ln ν π 6 ν ν ν ] s ω s γ,ν ω s γ M,ν Now, since ν M, we get that ω s ν ω s s γ M,ν ω s ω s ν M ν s M s s ν s s γ M,ν M s ω s M s s [ { M s M γ M,ν Finally, we get the bound ẐM r. ω s }] M s s. 6

17 Lemma.8. For M, define Then Ẑ M r def ε L S M s ρ ρ 4 M M c l,i rl. 4 l Proof. Using Lemma.5, we get [DT x rurv] ẐM r s M. 5 [DT x rurv] Λ [Df x rurv A rv] 4 Λ l c r l l 4 l ẐM r Ξ,i s rlĉl s. Now we are in the position of formulating the radii polynomials for. Namely, using Definition. and recalling the bounds Y F, Z F r and ẐM r in, and 4, respectively, we define A M f F x [I F A M Df M x]v F r p r def M s 4 l A M C l F r rl ω s M ε L S ρ ρ M M r l,,..., M ; 4 l 4 Proofs of Theorems. and.4 c l,i rl In order to prove Theorems. and.4, we use the following procedure. r ω s M, M. Procedure 4.. The following steps validates the assumptions of Lemma... Fix a decay rate s and a reduction dimension m.. Find x F an approximate solution of f m x F. This can be done by applying a predictor-corrector continuation algorithm based on Newton s method. See Section 4. for more details.. Compute the jabobian matrix D x f M x F. Compute an approximate inverse A M of D x f M x F. This is done by using the command A M invd x f M x F in MATLAB. Typically, I A M D x f M x F, and hence A M is an approximate inverse of D x f M x F. Chec conditions 4 and using interval arithmetic. This ensure that the linear operator A defined in 5, is invertible. 4. With interval arithmetic, compute the coefficients of Table and to complete the construction of the radii polynomials p,..., p M given in Definition.. 7

18 5. Calculate numerically I r, r M {r > p r < }. This step is done by computing the roots of each quartic polynomial p r using the command roots in MATLAB. We then obtain an interval I {r > p r < }. Finally, we set I def M I. 6. If I then go to Step 8. If I then let r r r. With interval arithmetic, compute p r. If p r < for all,..., m then go to Step 7; else go to Step The proof succeeds. By Lemma. T defined in 6 has a unique fixed point x L, ã, ã, b ã, b, ã, b,... Ω s within the ball B x r. Now, the existence of a periodic solution ỹt ã [ ã cos Lt b cos Lt ] to with ỹ is guaranteed by invertibility of A and by Lemma The proof fails. Either increase M > m or increase m and go to Step. 4. Numerical solutions and proofs of Theorem. and.4 Consider κ as a free parameter. We isolate parameters where the trivial solution of loses its stability via Hopf bifurcation. This can be done analytically see [7], [8] and [9] or using numerical tools such as TRACE-DDE, see []. Then with those parameters, we find a numerical approximation of a nearby periodic solution of. This approximation of the periodic solution is an approximate zero of fx, κ, 6 where f is defined in 9. Using this approximation as an initial point, we run the pseudo arc-length continuation algorithm of [] on finite dimensional projection of 6 to generate other approximation of periodic solutions to. We then apply Procedure 4. to each of these numerical approximation of periodic solutions to. To prove Theorem., we used m 7 and s.5. Procedure 4. is then applied to each numerical approximations x F to establish the existence of r r x F > such that B xf r contains a unique zero of 6. Now, this zero corresponds to a nontrivial periodic solution ȳt of. The steps are carried out in the MATLAB code Theorem.m available at []. The code uses the interval arithmetic pacage INTLAB []. A sample approximate solution of Theorem. can be found in Table. The steps of the proofs are the same for Theorem.4, and are carried out in the MATLAB code Theorem.m available at []. A sample approximate solution of Theorem.4 can be found in Table 4. 8

19 5 Future projects As mentioned earlier, [9] considers a wide range of functional differential equations, including neutral equations. An ongoing project is to develop rigorous computational tools to study periodic solutions of equations of this type. Another project considers equations with non-polynomial nonlinearity i.e. trigonometric nonlinearities. We are also woring on developing computational techniques for state-dependent equations of threshold type relevant to modelling the dynamics of infectious diseases. 6 Acnowledgments G.K. was partially supported by European Research Council StG Nr and the grant MTM-4766 of MICINN Spain. G.K. also thans Gergely Röst for helpful comments and suggestions. J.-P. L. was partially supported by NSERC. 7 Appendix C, C C C C 4 ω s m ε L ε ω s ω s 4 τ ω s 4 ε L ε 4 ε 4 ε L 6 ε {,..., m }. ā b b b ā ā ω s ω s ω s L τ ā b ā b ω s ω s ω s ω s ω s ā b b b ā ā ω s ω s ω s ā b ω s ω s ω s ω s ω s Table : The bounds C j. References [] B Van der Pol. A theory of the amplitude of free and forced triode vibrations. Radio Review, 9:7 7, 9. [] Ovide Arino, Moulay Lhassan Hbid, and E Ait Dads. Delay differential equations and applications, volume 5. Springer, 6. 9

20 Ĉ Ĉ Ĉ Ĉ 4 ε L m m ā b m s m ā ā b b s m m s s m m ā ā b b s s m m ā ā b b s s m m ā ā m s ā ā m s m 4 L 4 τ m m ε m m 4 ā b s s s m m m ā ā b b s s s m m m m L ā 4 b m s s m s γ M, s m m s s m s γ M m 8 8 ā s m s 4γ M 8 s m ā ā 4 4 s 4 m m s m ε 4 s s s s m s m 4 s s m m s s m s s s s s m s m s s m s γ M m s γ M m s s m s γ M m s s m s m s s m s γ M s m s s m s m s m s γ M L 4 m s s m s γ M, s m m s s m s γ M ā b m s s m s 4γ M 48 ā m 6 m s m s s s s m s s m s m m s s s m s s s s s s m m s s m s γ M m s γ M m s s m s γ M m s s m s m s s m s γ M s m s s m s s m s γ M ε Table : The bounds Ĉj.

21 L ā. ā b ā b ā b ā b ā b ā b ā b ā b ā b ā b ā b ā b ā b ā b ā b9.9 ā b.976 ā b.77 ā b Table : Approximate zero ˆx F at τ, κ.695 and ε.5. L ā ā b ā b ā b ā b ā b ā b ā.8478 b.8478 ā b5.8 Table 4: Approximate zero ˆx F at τ 5, κ and ε.5. [] Thomas Erneux. Applied delay differential equations, volume of Surveys and Tutorials in the Applied Mathematical Sciences. Springer, New Yor, 9. [4] Jac K. Hale. Theory of functional differential equations. Springer-Verlag, New Yor, second edition, 977. Applied Mathematical Sciences, Vol.. [5] Roger D. Nussbaum. Functional differential equations. In Handboo of dynamical systems, Vol., pages North-Holland, Amsterdam,. [6] Roberto Castelli and Jean-Philippe Lessard. Rigorous numerics in floquet theory: computing stable and unstable bundles of periodic orbits. SIAM Journal on Applied Dynamical Systems, :4 45,. [7] RB Grafton. Periodic solutions of certain Liénard equations with delay. Journal of Differential Equations, :59 57, 97.

22 [8] RB Grafton. A periodicity theorem for autonomous functional differential equations. Journal of Differential Equations, 6:87 9, 969. [9] Roger D Nussbaum. Periodic solutions of some nonlinear autonomous functional differential equations. Annali di matematica pura ed applicata, 4:6 6, 974. [] Jean-Philippe Lessard. Recent advances about the uniqueness of the slowly oscillating periodic solutions of Wright s equation. J. Differential Equations, 485:99 6,. [] Gábor Kiss and Jean-Philippe Lessard. Computational fixed-point theory for differential delay equations with multiple time lags. J. Differential Equations, 54:9 5,. [] Roger D. Nussbaum. Periodic solutions of analytic functional differential equations are analytic. Michigan Math. J., :49 55, 97. [] Marcio Gameiro and Jean-Philippe Lessard. Analytic estimates and rigorous continuation for equilibria of higher-dimensional PDEs. J. Differential Equations, 499:7 68,. [4] Marcio Gameiro, Jean-Philippe Lessard, and Konstantin Mischaiow. Validated continuation over large parameter ranges for equilibria of PDEs. Math. Comput. Simulation, 794:68 8, 8. [5] Jan Bouwe van den Berg and Jean-Philippe Lessard. Chaotic braided solutions via rigorous numerics: chaos in the Swift-Hohenberg equation. SIAM J. Appl. Dyn. Syst., 7:988, 8. [6] Sarah Day, Jean-Philippe Lessard, and Konstantin Mischaiow. Validated continuation for equilibria of PDEs. SIAM J. Numer. Anal., 454:98 44, 7. [7] Gábor Kiss and Bernd Krausopf. Stabilizing effect of delay distribution for a class of second-order systems without instantaneous feedbac. Dynamical Systems: An International Journal, 6:85,. [8] Gábor Kiss and Bernd Krausopf. Stability implications of delay distribution for first-order and second-order systems. Discrete Contin. Dyn. Syst. Ser. B, :7 45,. [9] Odo Diemann and Stephan A van Gils and Sjoerd M. Verduyn Lunel and Hans- Otto Walther. Delay equations. Springer-Verlag, 995. [] Dimitri Breda, Stefano Maset, and Rossana Vermiglio. Trace-dde: a tool for robust analysis and characteristic equations for delay differential equations. In Topics in Time Delay Systems, pages Springer, 9. [] HB Keller. Lectures on numerical methods in bifurcation problems. Applied Mathematics, 7:5, 987. [] S.M. Rump. INTLAB - INTerval LABoratory. In Tibor Csendes, editor, Developments in Reliable Computing, pages Kluwer Academic Publishers, Dordrecht, 999.

23 [] G. Kiss and J.-P. Lessard, MATLAB codes available at