from delay differential equations to ordinary differential equations (through partial differential equations)

from delay differential equations to ordinary differential equations (through partial differential equations) dynamical systems and applications @ BCAM - Bilbao Dimitri Breda Department of Mathematics and Computer Science - University of Udine (I) December 10, 2013

outline basics [9,12] dynamical systems [9,10] three different views numerical analysis [1,5,11,15] from DDEs to ODEs (through PDEs) dynamical systems and applications @ BCAM 2/40

basics [9,12] dynamical systems [9,10] three different views numerical analysis [1,5,11,15]

delay: formal setting Given d 1 and τ > 0, let X := C([ τ, 0]; R d ) and F : X R d. A Retarded Functional Differential Equation (RFDE) is a relation where x t X is defined as Usual terminology: d is the dimension τ is the (maximum) delay [ τ, 0] is the delay interval x t is the state at time t X is the state space x (t) = F(x t ), x t (θ) := x(t + θ), θ [ τ, 0]. F is the Right-Hand Side (RHS, in general nonlinear, here autonomous). Common acronym: DDEs for Delay Differential Equations. from DDEs to ODEs (through PDEs) dynamical systems and applications @ BCAM 4/40

examples linear systems with discrete delay(s): F(ψ) = Aψ(0) + Bψ( τ) x (t) = Ax(t) + Bx(t τ) linear systems with distributed delay(s): F(ψ) = Aψ(0)+ 0 τ C(θ)ψ(θ)dθ x (t) = Ax(t)+ nonlinear delay logistic equation [13]: 0 τ C(θ)x(t+θ)dθ F(ψ) = rψ(0)[1 ψ( τ)] x (t) = rx(t)[1 x(t τ)] nonlinear Mackey-Glass equation [14]: F(ψ) = aψ( τ) 1 + [ψ( τ)] bψ(0) ax(t τ) c x (t) = 1 + [x(t τ)] bx(t) c from DDEs to ODEs (through PDEs) dynamical systems and applications @ BCAM 5/40

DDEs vs ODEs Contrary to ODEs for f : R d R d { x (t) = f(x(t)), t 0, x(0) = v R d, a function ϕ X is necessary for a specific solution of a DDE: { x (t) = F(x t ), t 0, x(θ) = ϕ(θ), θ [ τ, 0]. Theorem. F Lipschitz! solution and it is Lipschitz w.r.t. ϕ. A big difference: DDEs generate -dimensional dynamical systems. Other differences: no backward continuation if ϕ only continuous more smoothness of F more smoothness of x necessarily oscillations and chaos already for d = 1 (e.g., Mackey-Glass). from DDEs to ODEs (through PDEs) dynamical systems and applications @ BCAM 6/40

basics [9,12] dynamical systems [9,10] three different views numerical analysis [1,5,11,15]

local stability of equilibria Consider a linear(ized) continuous RHS L : X R d : x (t) = Lx t. (1) Looking for x(t) = e λt v, v R d \ {0}, leads to the characteristic equation Its solutions λ C are known as characteristic roots. det(λi d Le λ ) = 0. (2) Theorem. The zero solution of (1) is asymptotically stable R(λ) < 0 for all λ, unstable if R(λ) > 0 for some λ. Contrary to ODEs, (2) is a nonlinear eigenvalue problem: -many λ. Theorem. α R s.t. R(λ) < α for all λ and there are only finitely-many λ in any vertical strip of C. There exists an alternative characterization, which we follow for computation. from DDEs to ODEs (through PDEs) dynamical systems and applications @ BCAM 8/40

solution operator Well-posedness allows to define the solution operator T(t) : X X as T(t)ϕ = x t, t 0. It associates to the initial function ϕ in [ τ, 0] the piece of solution x in [t τ, t], shifted back to [ τ, 0]. Theorem. {T(t)} t 0 is a C 0 -semigroup of bounded linear operators: (a) T(t) is linear and bounded for all t 0 (b) T(0) = I X (c) T(t + s) = T(t)T(s) for all t, s 0 (d) {T(t)} t 0 is strongly continuous: lim t 0 T(t)ψ = ψ for all ψ X. What above is general for evolution maps of dynamical systems. What below is not always the case, but it holds for DDEs. Theorem. T(t) is compact for all t τ. from DDEs to ODEs (through PDEs) dynamical systems and applications @ BCAM 9/40

infinitesimal generator and -ODE C 0 -semigroups have infinitesimal generators. Here follows that associated to x (t) = Lx t. Theorem. The infinitesimal generator is the linear unbounded operator A : D(A) X X with action Aψ = ψ and domain D(A) = {ψ X : ψ X and ψ (0) = Lψ}. Theorem (abstract Cauchy problem). u(t) = T(t)ϕ = x t solves { u (t) = Au(t), t 0, u(0) = ϕ D(A). A linear DDE is an -dimensional linear ODE, governing the time-evolution of the state u(t) = x t in X. from DDEs to ODEs (through PDEs) dynamical systems and applications @ BCAM 10/40

spectra and stability The eigenvalues of the matrix A rule the dynamics of the linear ODE x (t) = Ax(t) (in R d ). The linear DDE x (t) = Lx t (in R d ) can be seen as u (t) = Au(t) (in X). Thanks to compactness, the dynamics depends on the spectrum σ(a) of A. Theorem. A and T(t) for t τ have only point spectrum (eigenvalues) and σ(t(t)) \ {0} = e tσ(a). The characteristic equation and dynamical systems standpoints are the same: Theorem. λ is a characteristic root λ σ(a). (but not numerically, like for ODEs: polynomial roots vs matrix eigenvalues). from DDEs to ODEs (through PDEs) dynamical systems and applications @ BCAM 11/40

basics [9,12] dynamical systems [9,10] three different views numerical analysis [1,5,11,15]

DDE { x (t) = Lx t, t 0, x(θ) = ϕ(θ), θ [ τ, 0]. from DDEs to ODEs (through PDEs) dynamical systems and applications @ BCAM 13/40

DDE -ODE from DDEs to ODEs (through PDEs) dynamical systems and applications @ BCAM 13/40

-ODE { u (t) = Au(t), t 0, u(0) = ϕ D(A). from DDEs to ODEs (through PDEs) dynamical systems and applications @ BCAM 13/40

PDE formulation So far (1) x(t) R d solves the Cauchy problem for the DDE { x (t) = Lx t, t 0, x(θ) = ϕ(θ), θ [ τ, 0], and (2) u(t) = x t X solves the abstract Cauchy problem for the -ODE { u (t) = Au(t), t 0, Consider now Since x t (θ) = x(t + θ), we have u(0) = ϕ D(A). v(t, θ) := x t (θ) = [u(t)](θ). x t (θ) x(t + θ) x(t + θ) = = = x t(θ) t t θ θ. Hence (3) v(t, θ) R d solves the initial-boundary value problem for the PDE v(t, θ) v(t, θ) =, t 0, θ [ τ, 0], t θ v(t, θ) t = Lv(t, ), t 0, θ=0 v(0, θ) = ϕ(θ), θ [ τ, 0]. from DDEs to ODEs (through PDEs) dynamical systems and applications @ BCAM 14/40

-ODE { u (t) = Au(t), t 0, u(0) = ϕ D(A). from DDEs to ODEs (through PDEs) dynamical systems and applications @ BCAM 15/40

DDE { x (t) = Lx t, t 0, x(θ) = ϕ(θ), θ [ τ, 0]. from DDEs to ODEs (through PDEs) dynamical systems and applications @ BCAM 15/40

DDE PDE v(t, θ) t = v(t, θ) θ characteristic lines t + θ = constant. from DDEs to ODEs (through PDEs) dynamical systems and applications @ BCAM 15/40

DDE PDE from DDEs to ODEs (through PDEs) dynamical systems and applications @ BCAM 15/40

PDE v(t, θ) v(t, θ) =, t 0, θ [ τ, 0], t θ v(t, θ) t = Lv(t, ), t 0, θ=0 v(0, θ) = ϕ(θ), θ [ τ, 0]. from DDEs to ODEs (through PDEs) dynamical systems and applications @ BCAM 15/40

For nonlinear F, the DDE {x (t) = F(x t ), t 0, nonlinear case x(θ) = ϕ(θ), θ [ τ, 0], is still equivalent to the PDE v(t, θ) v(t, θ) =, t 0, θ [ τ, 0], t θ v(t, θ) t = F(v(t, )), t 0, θ=0 v(0, θ) = ϕ(θ), θ [ τ, 0]. The abstract Cauchy problem reads { u (t) = A(u(t)), t 0, u(0) = ϕ D(A), with A the nonlinear differential operator Aψ = ψ and D(A) = {ψ X : ψ X and ψ (0) = F(ψ)}. From now on let d = 1 without loss of generality and remember that v(t, θ) = x t (θ) = [u(t)](θ). from DDEs to ODEs (through PDEs) dynamical systems and applications @ BCAM 16/40

basics [9,12] dynamical systems [9,10] three different views numerical analysis [1,5,11,15]

polynomial interpolation Let a function f : [ 1, 1] R be known on given nodes t i, i = 0, 1,..., N. Polynomial interpolation is a way to approximate f: find p N Π N s.t. p N (t i ) = f(t i ). Theorem.!p N Π N interpolating f for any choice of N + 1 distinct nodes. There are several ways to represent p N. We use the Lagrange form: N p N (t) = l j (t)f(t j ), where l j (t) = N k=0 k j j=0 t t k t j t k, t [ 1, 1]. Proof: p N (t i ) = f(t i ) since l j (t i ) = δ i,j, the Kronecker s delta. Interpolation is fine for smooth f and good nodes, as Chebyshev nodes are: ( ) iπ t i = cos. N from DDEs to ODEs (through PDEs) dynamical systems and applications @ BCAM 18/40

pseudospectral approximation PSA: whatever you do with f, do it instead with p N (dimension: finite). Example on differentiation. Assume we know y i = f(t i ) for i = 0, 1,..., N. Set y = (y 0, y 1,..., y N ) T. Let z = (z 0, z 1,..., z N ) T with z i = p N(t i ) approximating f (t i ). Differentiation and interpolation being linear, we write z = D N y, where D N R (N+1) (N+1) is the differentiation matrix d 0,0 d 0,1 d 0,N l 0(t 0 ) l 1(t 0 ) l N (t 0) d 1,0 d 1,1 d 1,N D N =........ = l 0(t 1 ) l 1(t 1 ) l N (t 1)......... d N,0 d N,1 d N,N l 0(t N ) l 1(t N ) l N (t N) from DDEs to ODEs (through PDEs) dynamical systems and applications @ BCAM 19/40

pseudospectral approximation PSA: whatever you do with f, do it instead with p N (dimension: finite). Example on differentiation. Assume we know y i = f(t i ) for i = 0, 1,..., N. Set y = (y 0, y 1,..., y N ) T. Let z = (z 0, z 1,..., z N ) T with z i = p N(t i ) approximating f (t i ). Differentiation and interpolation being linear, we write z = D N y, where D N R (N+1) (N+1) is the differentiation matrix d 0,0 d 0,1 d 0,N l 0(t 0 ) l 1(t 0 ) l N (t 0) d 1,0 d 1,1 d 1,N D N =........ = l 0(t 1 ) l 1(t 1 ) l N (t 1)......... d N,0 d N,1 d N,N l 0(t N ) l 1(t N ) l N (t N) d N D N from DDEs to ODEs (through PDEs) dynamical systems and applications @ BCAM 19/40

spectral accuracy Compare PSA on Chebyshev nodes (red) with finite differences (blue) f (t) to approximate f for f(t) = e t2. f(t + h) f(t), h = 2/N, h 10 2 10 4 O(N 1 ) 10 6 10 8 10 10 O(N N ) 10 0 10 1 10 2 10 3 10 4 10 5 10 6 N PSA exploits all the smoothness of f: vs finite convergence order. For Chebyshev nodes, moreover, D N is known explicitly. from DDEs to ODEs (through PDEs) dynamical systems and applications @ BCAM 20/40

linear case: PSA of eigenvalues The idea is to reduce A to a matrix and use its eigenvalues as approximations. Recall that Aψ = ψ and D(A) = {ψ X : ψ X and ψ (0) = Lψ}. Let p N interpolate ψ X on the Chebyshev nodes in [ τ, 0] θ i = τ ( ) iπ 2 cos τ, i = 0, 1,..., N. N 2 Notice that θ 0 = 0 and θ N = τ. Then (Aψ)(θ i ) = ψ (θ i ) p N(θ i ) for i = 1,..., N while at θ 0 = 0 (i = 0) we impose to respect the condition in D(A). (Aψ)(0) = ψ (0) = Lψ Lp N from DDEs to ODEs (through PDEs) dynamical systems and applications @ BCAM 21/40

discretized infinitesimal generator Being L, differentiation and interpolation linear, the relation between the values ψ(θ i ) and the approximations of (Aψ)(θ i ) is given by a matrix. Indeed, if then for i = 0 while for i = 1,..., N ψ(θ) p N (θ) = (Aψ)(0) Lp N = N l j (θ)ψ(θ j ), j=0 (Aψ)(θ i ) p N(θ i ) = N Ll j ψ(θ j ) j=0 N d i,j ψ(θ j ). Therefore A is discretized by Ll 0 Ll 1 Ll N d 1,0 d 1,1 d 1,N A N =........ R(N+1) (N+1). d N,0 d N,1 d N,N j=0 from DDEs to ODEs (through PDEs) dynamical systems and applications @ BCAM 22/40

discretized infinitesimal generator Being L, differentiation and interpolation linear, the relation between the values ψ(θ i ) and the approximations of (Aψ)(θ i ) is given by a matrix. Indeed, if then for i = 0 while for i = 1,..., N ψ(θ) p N (θ) = (Aψ)(0) Lp N = N l j (θ)ψ(θ j ), j=0 (Aψ)(θ i ) p N(θ i ) = N Ll j ψ(θ j ) j=0 N d i,j ψ(θ j ). Therefore A is discretized by Ll 0 Ll 1 Ll N d 1,0 d 1,1 d 1,N A N =........ R(N+1) (N+1). d N,0 d N,1 d N,N d N D N from DDEs to ODEs (through PDEs) dynamical systems and applications @ BCAM 22/40 j=0

example Take Lψ = aψ(0) + bψ( τ) x (t) = ax(t) + bx(t τ). Since θ 0 = 0, θ N = τ and l j (θ i ) = δ i,j we get Ll 0 = a, Ll j = 0, j = 1,..., N 1, Ll N = b. Therefore a 0 0 b d 1,0 d 1,1 d 1,N 1 d 1,N A N =............ d N,0 d N,1 d N,N 1 d N,N Observe: only the first row of A N is influenced by the RHS of the DDE, the rest depends only on τ and on the Chebyshev nodes in [ τ, 0]. from DDEs to ODEs (through PDEs) dynamical systems and applications @ BCAM 23/40

convergence Theorem. Let λ be an eigenvalue of A with multiplicity m. For N sufficiently large, A N has exactly m eigenvalues λ i counted with multiplicities and ( ) N/m max λ λ C1 i C 0 i=1,...,m N with C 0, C 1 constants and C 1 proportional to λ. 30 20 λ 9 λ 7 10 0 λ 5 I(λ) 10 0 λ 3 λ 1 error 10 5 λ 5 λ 7 λ 9 10 10 10 20 λ 3 30 4 3 2 1 0 1 R(λ) λ 1 10 15 20 30 40 N from DDEs to ODEs (through PDEs) dynamical systems and applications @ BCAM 24/40

extension to the nonlinear case PSA works well for the local stability analysis of equilibria through the computation of the eigenvalues of A. It is just the first step in the dynamical analysis. Next steps are stability of periodic orbits (Floquet multipliers) and detection of chaotic behaviors (Lyapunov exponents). Other directions concern with bifurcation analysis or different classes of functional equations (neutral and state-dependent, retardedadvanced, retarded partial, differential-algebraic, integro-differential, etc.). For most of the above, PSA is well-suited [2,3,4,6]. What about the nonlinear case? There is more behind the nature of PSA. Especially related to the following question: if DDEs are -ODEs, what do we loose or preserve by considering just a finite number of the latter? Observe that the interest is not in approximating a specific solution, but rather the whole dynamics. from DDEs to ODEs (through PDEs) dynamical systems and applications @ BCAM 25/40

PSA of nonlinear DDEs Consider { x (t) = F(x t ), t 0, x(θ) = ϕ(θ), θ [ τ, 0]. For t 0 and τ = θ N < < θ 0 = 0 the Chebyshev nodes in [ τ, 0], let ū i (t) x t (θ i ) be an approximation in θ i of the state at time t. Their interpolant v(t, θ) = N l j (θ)ū j (t) j=0 is a polynomial in θ [ τ, 0], yet a function of t 0 since so are the ū i s. Question: does v satisfy a differential equation and, if so, what is the relation with the original DDE? from DDEs to ODEs (through PDEs) dynamical systems and applications @ BCAM 26/40

approximated PDE It is not difficult to prove that N v(t, θ) = l j (θ)ū j (t) j=0 satisfies v(t, θ) v(t, θ) = + ε(t, θ, v), t 0, θ [ τ, 0], t θ v(t, θ) t = F( v(t, )), t 0, θ=0 N v(0, θ) = l j (θ)ϕ(θ j ), θ [ τ, 0]. j=0 It corresponds to the PDE formulation of the DDE, but for the interpolation of the initial function ϕ and the error term ε(t, θ, v), which we see in a moment. So v(t, θ) approximates v(t, θ)...and ū j (t)? from DDEs to ODEs (through PDEs) dynamical systems and applications @ BCAM 27/40

back to ODEs The error term is ε(t, θ, v) = l 0 (θ) ( F( v(t, )) v(t, θ) θ and it vanishes at θ i for i = 1,..., N since l j (θ i ) = δ i,j. ) θ=0 Therefore v(t, ) is the polynomial of collocation of the original PDE: v(t, θ) v(t, θ) t = θ=θi θ, i = 1,..., N, θ=θi v(t, θ) t = F( v(t, )), (i = 0), θ=0 v(0, θ i ) = ϕ(θ i ), i = 0, 1,..., N. But this is equivalent to the Cauchy problem ū 0(t) = F( v(t, )), (i = 0), t 0 ū i (t) = N d i,j ū j (t), i = 1,..., N, t 0 j=0 ū i (0) = ϕ(θ i ), i = 0, 1,..., N, for a system of N + 1 ODEs, nonlinear only in the first one. from DDEs to ODEs (through PDEs) dynamical systems and applications @ BCAM 28/40

again the linear case If F = L is linear, the first ODE becomes ū 0(t) = L v(t, ) = N Ll j ū j (t). j=0 Therefore ū(t) = (ū 0 (t), ū 1 (t),..., ū N (t)) T solves {ū (t) = A N ū(t), ū(0) = φ R N+1, for φ = (ϕ(θ 0 ), ϕ(θ 1 ),..., ϕ(θ N )) T. Exactly the discretization of the original abstract Cauchy problem { u (t) = Au(t), u(0) = ϕ X. from DDEs to ODEs (through PDEs) dynamical systems and applications @ BCAM 29/40

PDE v(t, θ) v(t, θ) =, t 0, θ [ τ, 0], t θ v(t, θ) t = F(v(t, )), t 0, θ=0 v(0, θ) = ϕ(θ), θ [ τ, 0]. from DDEs to ODEs (through PDEs) dynamical systems and applications @ BCAM 30/40

PDE N + 1 ODEs φ = (ϕ(θ 0 ), ϕ(θ 1 ),..., ϕ(θ N )) T from DDEs to ODEs (through PDEs) dynamical systems and applications @ BCAM 30/40

N + 1 ODEs ū 0(t) = F( v(t, )), (i = 0), t 0 ū i (t) = N d i,j ū j (t), i = 1,..., N, t 0 j=0 ū i (0) = ϕ(θ i ), i = 0, 1,..., N. from DDEs to ODEs (through PDEs) dynamical systems and applications @ BCAM 30/40

role of D N Consider the linear part ū i(t) = N d i,j ū j (t), i = 1,..., N, j=0 and recall that d 1,0 d 1,1 d 1,N d N =., D N =........ d N,0 d N,1 d N,N Then ũ(t) = (ū 1 (t),..., ū N (t)) T satisfies the linear inhomogeneous ODE ũ (t) = D N ũ(t) + d N ū 0 (t), whose free dynamics depends on the eigenvalues of D N. D N is independent of the RHS of the DDE, so we study the easiest case F 0: { x (t) = 0, t 0, x(θ) = ϕ(θ), θ [ τ, 0]. It has constant solution x(t) = ϕ(0). Let ϕ(0) = 0 without loss of generality. from DDEs to ODEs (through PDEs) dynamical systems and applications @ BCAM 31/40

properties of σ( D N ) The first ODE gives ū 0 (t) = ϕ(0) = 0 and the linear part becomes ũ (t) = D N ũ(t). (1) Also ũ(t) for t τ must vanish as N to converge to the exact solution. Theorem. D N is nonsingular. Proof: e λt w solves (1) for λ σ( D N ) with eigenvector w. The interpolant N v(t, θ) = l 0 (θ)ū 0 (t) + l j (θ)e λt w j = q(θ)e λt (q Π N ) j=1 solves the approximated PDE, hence q (θ) = λq(θ) + l 0 (θ)q (0). Finally, l 0 Π N and q Π N 1 imply λ 0. Theorem. σ( D N ) C for all positive integers N. Conjecture. R(σ( D N )) as N. from DDEs to ODEs (through PDEs) dynamical systems and applications @ BCAM 32/40

equilibria and stability Theorem. A constant function of value c R is an equilibrium of the DDE x (t) = F(x t ) iff the constant vector (c,..., c) T R N+1 is an equilibrium of the ODE approximation {ū 0 (t) = F( v(t, )), ũ (t) = D N ũ(t) + d N ū 0 (t). Proof: based on two properties of Lagrange interpolation: N l j (θ) = 1 θ [ τ, 0] and j=0 N d i,j = 0 i = 0, 1,..., N. j=0 The second for i = 1,..., N reads D N (1,..., 1) T + d N = 0. Theorem. Linearization and approximation commute. Proof: the only nonlinear part of the ODE is the RHS F. Theorem. The linearized ODE approximation predicts accurately the local stability properties of the equilibria of the nonlinear DDE. Proof: it gives the discretized infinitesimal generator. from DDEs to ODEs (through PDEs) dynamical systems and applications @ BCAM 33/40

bifurcation of equilibria Concluding, the bifurcation analysis of equilibria of nonlinear DDEs can be tackled efficiently by standard tools for ODEs (e.g., AUTO [16], MATCONT [17]). A simple test. The nonlinear logistic DDE x (t) = rx(t)[1 x(t 1)] has equilibrium c = 1 for all r: asymptotically stable for r [0, π/2), unstable otherwise. At r = π/2 a Hopf bifurcation occurs and a limit cycle arises. 30 eigenvalues (N = 20) 10 0 r Hopf π/2 (MATCONT) 20 10 2 10 10 4 I(λ) 0 10 6 10 10 8 20 10 10 30 3 2 1 0 1 R(λ) 10 12 1 5 10 20 N from DDEs to ODEs (through PDEs) dynamical systems and applications @ BCAM 34/40

more on The same should apply to periodic orbits: those of the ODE approximation are spectrally accurate approximations of the exact ones; linearization and approximation always commute; PSA of Floquet multipliers is available [3]. Ongoing work to provide theoretical support: R(σ( D N )) as N : how fast? convergence of v(t, θ) to x t (θ): spectrally accurate for smooth x? The general idea (and hope) is that standard ODE tools as applied to the ODE approximation can easily provide information on the dynamics of the original DDE. Advantages: valid for any DDE bifurcation tools for ODEs complete and efficient; not so for DDEs [18] spectral accuracy (low N) theoretical insight (dynamical systems, functional analysis, [9]). from DDEs to ODEs (through PDEs) dynamical systems and applications @ BCAM 35/40

the true (long-term) objective Recent sophisticated models of resource-consumer dynamics are based on delay integro-differential equations (DEs/DDEs) such as [8] h b(t) = β(x(a, S t ), S(t))F(a, S t )b(t a)da, 0 S (t) = f(s(t)) h γ(x(a, S t ), S(t))F(a, S t )b(t a)da, 0 with X and F solutions of external ODEs and discontinuities between juveniles and adults. Semigroup and stability theories are available [7], as well as the PSA of eigenvalues, but only for caricature models [2]. The ODE approximation presented for DDEs is adaptable to such models. Ongoing work: computation of eigenvalues (linear); bifurcation analysis (nonlinear). from DDEs to ODEs (through PDEs) dynamical systems and applications @ BCAM 36/40

research group J. Sánchez @ BCAM P. Getto @ TU Dresden (D) O. Diekmann @ Utrecht (NL) A. de Roos @ Amsterdam (NL) S. Maset @ Trieste (I) R. Vermiglio @ Udine (I)...thanks for your attention! from DDEs to ODEs (through PDEs) dynamical systems and applications @ BCAM 37/40

bibliography 1 Berrut and Trefethen, Barycentric Lagrange interpolation, SIAM Rev., 46:501-517, 2004 2 Breda, Diekmann, Maset and Vermiglio, A numerical approach for investigating the stability of equilibria for structured population models, J. Biol. Dyn., 7:4-20, 2013 3 Breda, Maset and Vermiglio, Approximation of eigenvalues of evolution operators for linear retarded functional differential equations, SIAM J. Numer. Anal., 50:1456-1483, 2012 4 Breda, Maset and Vermiglio, Pseudospectral approximation of eigenvalues of derivative operators with non-local boundary condition, Appl. Numer. Math., 56:318-331, 2006 5 Breda, Maset and Vermiglio, Pseudospectral differencing methods for characteristic roots of delay differential equations, SIAM J. Sci. Comput., 27:482-495, 2005 6 Breda and Van Vleck, Approximating Lyapunov exponents and Sacker-Sell spectrum for retarded functional differential equations, Numer. Math., online, 2013 from DDEs to ODEs (through PDEs) dynamical systems and applications @ BCAM 38/40

bibliography cont d 7 Diekmann, Getto and Gyllenberg, Stability and bifurcation analysis of Volterra functional equations in the light of suns and stars, SIAM J. Math. Anal., 39:1023-1069, 2007 8 Diekmann, Gyllenberg, Metz, Nakaoka and de Roos Daphnia revisited: local stability and bifurcation theory for physiologically structured population models explained by way of an example, J. Math. Biol., 61:277-318, 2010 9 Diekmann, van Gils, Verduyn Lunel and Walther, Delay Equations - Functional, Complex and Nonlinear Analysis, Springer AMS 110, 1995 10 Engel and Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer GTM 194, 1999 11 Gottlieb, Hussaini and Orszag, Theory and applications of spectral methods, Spectral methods for partial differential equations (Hampton 1982), SIAM, 1:54, 1984 12 Hale and Verduyn Lunel, Introduction to functional differential equations, Springer AMS 99, 1993 13 Hutchinson, Circular causal systems in ecology, Ann. N.Y. Acad. Sci., 50:221-246, 1948 from DDEs to ODEs (through PDEs) dynamical systems and applications @ BCAM 39/40

bibliography cont d 14 Mackey and Glass, Oscillations and chaos in physiological control systems, Science, 197:287-289, 1977 15 Trefethen, Spectral methods in MATLAB, SIAM SET, 2000 16 AUTO, http://indy.cs.concordia.ca/auto/ 17 MATCONT, http://matcont.sourceforge.net/ 18 DDE-BIFTOOL, http://twr.cs.kuleuven.be/research/software/ delay/ddebiftool.shtml from DDEs to ODEs (through PDEs) dynamical systems and applications @ BCAM 40/40