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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]; Rd) and F : X → Rd. A Retarded
Functional Differential Equation (RFDE) is a relation
x ′(t) = F(xt),
where xt ∈ X is defined as
xt(θ) := x(t+ θ), θ ∈ [−τ, 0].
Usual terminology:
d is the dimension
τ is the (maximum) delay
[−τ, 0] is the delay interval
xt is the state at time t
X is the state space
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)+
0∫−τ
C(θ)x(t+θ)dθ
nonlinear delay logistic equation [13]:
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)]c− bψ(0) ⇒ x ′(t) =
ax(t− τ)
1 + [x(t− τ)]c− bx(t)
from DDEs to ODEs (through PDEs) dynamical systems and applications @ BCAM 5/40
DDEs vs ODEs
Contrary to ODEs for f : Rd → Rd{x ′(t) = f(x(t)), t > 0,
x(0) = v ∈ Rd,
a function ϕ ∈ X is necessary for a specific solution of a DDE:{x ′(t) = F(xt), 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→ Rd:
x ′(t) = Lxt. (1)
Looking for x(t) = eλtv, v ∈ Rd \ {0}, leads to the characteristic equation
det(λId − Leλ·) = 0. (2)
Its solutions λ ∈ C are known as characteristic roots.
Theorem. The zero solution of (1) is asymptotically stable ⇔ <(λ) < 0 for all
λ, unstable if <(λ) > 0 for some λ.
Contrary to ODEs, (2) is a nonlinear eigenvalue problem: ∞-many λ.
Theorem. ∃α ∈ R s.t. <(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 = xt, 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 C0-semigroup of bounded linear operators:
(a) T(t) is linear and bounded for all t > 0
(b) T(0) = IX
(c) T(t+ s) = T(t)T(s) for all t, s > 0
(d) {T(t)}t>0 is strongly continuous: limt↓0T(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
C0-semigroups have infinitesimal generators. Here follows that associated to
x ′(t) = Lxt.
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 = xt 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) = xt 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 Rd).
The linear DDE
x ′(t) = Lxt (in Rd)
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} = etσ(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) = Lxt, 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
DDE → ∞-ODE
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
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) ∈ Rd solves the Cauchy problem for the DDE{x ′(t) = Lxt, t > 0,
x(θ) = ϕ(θ), θ ∈ [−τ, 0],
and (2) u(t) = xt ∈ X solves the abstract Cauchy problem for the ∞-ODE{u ′(t) = Au(t), t > 0,
u(0) = ϕ ∈ D(A).
Consider now
v(t, θ) := xt(θ) = [u(t)](θ).
Since xt(θ) = x(t+ θ), we have
∂xt(θ)
∂t=∂x(t+ θ)
∂t=∂x(t+ θ)
∂θ=∂xt(θ)
∂θ.
Hence (3) v(t, θ) ∈ Rd solves the initial-boundary value problem for the PDE
∂v(t, θ)
∂t=∂v(t, θ)
∂θ, t > 0, θ ∈ [−τ, 0],
∂v(t, θ)
∂t
∣∣∣∣θ=0
= Lv(t, ·), t > 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) = Lxt, 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
DDE → PDE
from DDEs to ODEs (through PDEs) dynamical systems and applications @ BCAM 15/40
PDE
∂v(t, θ)
∂t=∂v(t, θ)
∂θ, t > 0, θ ∈ [−τ, 0],
∂v(t, θ)
∂t
∣∣∣∣θ=0
= Lv(t, ·), t > 0,
v(0, θ) = ϕ(θ), θ ∈ [−τ, 0].
from DDEs to ODEs (through PDEs) dynamical systems and applications @ BCAM 15/40
nonlinear case
For nonlinear F, the DDE{x ′(t) = F(xt), t > 0,
x(θ) = ϕ(θ), θ ∈ [−τ, 0],
is still equivalent to the PDE
∂v(t, θ)
∂t=∂v(t, θ)
∂θ, t > 0, θ ∈ [−τ, 0],
∂v(t, θ)
∂t
∣∣∣∣θ=0
= F(v(t, ·)), t > 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) = xt(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 ti, i = 0, 1, . . . ,N.
Polynomial interpolation is a way to approximate f: find pN ∈ ΠN s.t.
pN(ti) = f(ti).
Theorem. ∃!pN ∈ ΠN interpolating f for any choice of N+ 1 distinct nodes.
There are several ways to represent pN. We use the Lagrange form:
pN(t) =
N∑j=0
`j(t)f(tj),
where
`j(t) =
N∏k=0k6=j
t− tk
tj − tk, t ∈ [−1, 1].
Proof: pN(ti) = f(ti) since `j(ti) = δi,j, the Kronecker’s delta.
Interpolation is fine for smooth f and “good” nodes, as Chebyshev nodes are:
ti = cos
(iπ
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 pN (dimension: ∞ → finite).
Example on differentiation. Assume we know
yi = f(ti)
for i = 0, 1, . . . ,N. Set y = (y0,y1, . . . ,yN)T . Let z = (z0, z1, . . . , zN)T with
zi = p ′N(ti)
approximating f ′(ti). Differentiation and interpolation being linear, we write
z = DNy,
where DN ∈ R(N+1)×(N+1) is the differentiation matrix
DN =
d0,0 d0,1 · · · d0,Nd1,0 d1,1 · · · d1,N
......
. . ....
dN,0 dN,1 · · · dN,N
=
` ′0(t0) ` ′1(t0) · · · ` ′N(t0)
` ′0(t1) ` ′1(t1) · · · ` ′N(t1)...
.... . .
...
` ′0(tN) ` ′1(tN) · · · ` ′N(tN)
.
dN DN
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 pN (dimension: ∞ → finite).
Example on differentiation. Assume we know
yi = f(ti)
for i = 0, 1, . . . ,N. Set y = (y0,y1, . . . ,yN)T . Let z = (z0, z1, . . . , zN)T with
zi = p ′N(ti)
approximating f ′(ti). Differentiation and interpolation being linear, we write
z = DNy,
where DN ∈ R(N+1)×(N+1) is the differentiation matrix
DN =
d0,0 d0,1 · · · d0,Nd1,0 d1,1 · · · d1,N
......
. . ....
dN,0 dN,1 · · · dN,N
=
` ′0(t0) ` ′1(t0) · · · ` ′N(t0)
` ′0(t1) ` ′1(t1) · · · ` ′N(t1)...
.... . .
...
` ′0(tN) ` ′1(tN) · · · ` ′N(tN)
.
dN DN
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) ≈ f(t+ h) − f(t)
h, h = 2/N,
to approximate f ′ for f(t) = e−t2 .
100 101 102 103 104 105 106
10−10
10−8
10−6
10−4
10−2
N
O(N!N)
O(N!1)
PSA exploits all the smoothness of f: ∞ vs finite convergence order.
For Chebyshev nodes, moreover, DN 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 pN interpolate ψ ∈ X on the Chebyshev nodes in [−τ, 0]
θi =τ
2cos
(iπ
N
)−τ
2, i = 0, 1, . . . ,N.
Notice that θ0 = 0 and θN = −τ.
Then
(Aψ)(θi) = ψ ′(θi) ≈ p ′N(θi)
for i = 1, . . . ,N while at θ0 = 0 (i = 0) we impose
(Aψ)(0) = ψ ′(0) = Lψ ≈ LpN
to respect the condition in D(A).
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(θi) and the approximations of (Aψ)(θi) is given by a matrix. Indeed, if
ψ(θ) ≈ pN(θ) =
N∑j=0
`j(θ)ψ(θj),
then for i = 0
(Aψ)(0) ≈ LpN =
N∑j=0
L`jψ(θj)
while for i = 1, . . . ,N
(Aψ)(θi) ≈ p ′N(θi) =
N∑j=0
di,jψ(θj).
Therefore A is discretized by
AN =
L`0 L`1 · · · L`Nd1,0 d1,1 · · · d1,N
......
. . ....
dN,0 dN,1 · · · dN,N
∈ R(N+1)×(N+1).
dN DN
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(θi) and the approximations of (Aψ)(θi) is given by a matrix. Indeed, if
ψ(θ) ≈ pN(θ) =
N∑j=0
`j(θ)ψ(θj),
then for i = 0
(Aψ)(0) ≈ LpN =
N∑j=0
L`jψ(θj)
while for i = 1, . . . ,N
(Aψ)(θi) ≈ p ′N(θi) =
N∑j=0
di,jψ(θj).
Therefore A is discretized by
AN =
L`0 L`1 · · · L`Nd1,0 d1,1 · · · d1,N
......
. . ....
dN,0 dN,1 · · · dN,N
∈ R(N+1)×(N+1).
dN DNfrom DDEs to ODEs (through PDEs) dynamical systems and applications @ BCAM 22/40
example
Take
Lψ = aψ(0) + bψ(−τ) ⇒ x ′(t) = ax(t) + bx(t− τ).
Since θ0 = 0, θN = −τ and `j(θi) = δi,j we getL`0 = a,
L`j = 0, j = 1, . . . ,N− 1,
L`N = b.
Therefore
AN =
a 0 · · · 0 b
d1,0 d1,1 · · · d1,N−1 d1,N...
.... . .
......
dN,0 dN,1 · · · dN,N−1 dN,N
.
Observe: only the first row of AN 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, AN has exactly m eigenvalues λi counted with multiplicities and
maxi=1,...,m
|λ− λi| 6 C0
(C1
N
)N/mwith C0,C1 constants and C1 proportional to |λ|.
−4 −3 −2 −1 0 1−30
−20
−10
0
10
20
30
!(!)
"(!)
20 30 4010−15
10−10
10−5
100
N
erro
r
!1
!9
!7
!5
!3
!1
!9
!7
!5
!3
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 compu-
tation 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 (Lya-
punov exponents). Other directions concern with bifurcation analysis or dif-
ferent classes of functional equations (neutral and state-dependent, retarded-
advanced, 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
extension to the nonlinear case
PSA works well for the local stability analysis of equilibria through the compu-
tation 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 (Lya-
punov exponents). Other directions concern with bifurcation analysis or dif-
ferent classes of functional equations (neutral and state-dependent, retarded-
advanced, 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(xt), t > 0,
x(θ) = ϕ(θ), θ ∈ [−τ, 0].
For t > 0 and −τ = θN < · · · < θ0 = 0 the Chebyshev nodes in [−τ, 0], let
ui(t) ≈ xt(θi)
be an approximation in θi of the state at time t. Their interpolant
v(t, θ) =
N∑j=0
`j(θ)uj(t)
is a polynomial in θ ∈ [−τ, 0], yet a function of t > 0 since so are the ui’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
v(t, θ) =
N∑j=0
`j(θ)uj(t)
satisfies
∂v(t, θ)
∂t=∂v(t, θ)
∂θ+ ε(t, θ, v), t > 0, θ ∈ [−τ, 0],
∂v(t, θ)
∂t
∣∣∣∣θ=0
= F(v(t, ·)), t > 0,
v(0, θ) =
N∑j=0
`j(θ)ϕ(θj), θ ∈ [−τ, 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 uj(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) = `0(θ)
(F(v(t, ·)) −
∂v(t, θ)
∂θ
∣∣∣∣θ=0
)and it vanishes at θi for i = 1, . . . ,N since `j(θi) = δi,j.
Therefore v(t, ·) is the polynomial of collocation of the original PDE:
∂v(t, θ)
∂t
∣∣∣∣θ=θi
=∂v(t, θ)
∂θ
∣∣∣∣θ=θi
, i = 1, . . . ,N,
∂v(t, θ)
∂t
∣∣∣∣θ=0
= F(v(t, ·)), (i = 0),
v(0, θi) = ϕ(θi), i = 0, 1, . . . ,N.
But this is equivalent to the Cauchy problemu′0(t) = F(v(t, ·)), (i = 0), t > 0
u′i(t) =
N∑j=0
di,juj(t), i = 1, . . . ,N, t > 0
ui(0) = ϕ(θi), 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
u ′0(t) = Lv(t, ·) =
N∑j=0
L`juj(t).
Therefore u(t) = (u0(t), u1(t), . . . , uN(t))T solves{u ′(t) = ANu(t),
u(0) = φ ∈ RN+1,
for φ = (ϕ(θ0),ϕ(θ1), . . . ,ϕ(θN))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)
∂t=∂v(t, θ)
∂θ, t > 0, θ ∈ [−τ, 0],
∂v(t, θ)
∂t
∣∣∣∣θ=0
= F(v(t, ·)), t > 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(θ0),ϕ(θ1), . . . ,ϕ(θN))T
from DDEs to ODEs (through PDEs) dynamical systems and applications @ BCAM 30/40
N+ 1 ODEs
u ′0(t) = F(v(t, ·)), (i = 0), t > 0
u ′i(t) =N∑j=0
di,juj(t), i = 1, . . . ,N, t > 0
ui(0) = ϕ(θi), i = 0, 1, . . . ,N.
from DDEs to ODEs (through PDEs) dynamical systems and applications @ BCAM 30/40
role of DNConsider the linear part
u ′i(t) =
N∑j=0
di,juj(t), i = 1, . . . ,N,
and recall that
dN =
d1,0...
dN,0
, DN =
d1,1 · · · d1,N...
. . ....
dN,1 · · · dN,N
.
Then u(t) = (u1(t), . . . , uN(t))T satisfies the linear inhomogeneous ODE
u ′(t) = DNu(t) + dNu0(t),
whose free dynamics depends on the eigenvalues of DN.
DN 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(DN)
The first ODE gives u0(t) = ϕ(0) = 0 and the linear part becomes
u ′(t) = DNu(t). (1)
Also u(t) for t > τ must vanish as N→∞ to converge to the exact solution.
Theorem. DN is nonsingular.
Proof: eλtw solves (1) for λ ∈ σ(DN) with eigenvector w. The interpolant
v(t, θ) = `0(θ)u0(t) +
N∑j=1
`j(θ)eλtwj = q(θ)eλt (q ∈ ΠN)
solves the approximated PDE, hence
q ′(θ) = λq(θ) + `0(θ)q′(0).
Finally, `0 ∈ ΠN and q ′ ∈ ΠN−1 imply λ 6= 0.
Theorem. σ(DN) ⊂ C− for all positive integers N.
Conjecture. <(σ(DN))→ −∞ 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(xt) iff the constant vector (c, . . . , c)T ∈ RN+1 is an equilibrium of the
ODE approximation {u ′0(t) = F(v(t, ·)),
u ′(t) = DNu(t) + dNu0(t).
Proof: based on two properties of Lagrange interpolation:
N∑j=0
`j(θ) = 1 ∀θ ∈ [−τ, 0] and
N∑j=0
di,j = 0 ∀i = 0, 1, . . . ,N.
The second for i = 1, . . . ,N reads DN(1, . . . , 1)T + dN = 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 sta-
bility 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.
−3 −2 −1 0 1−30
−20
−10
0
10
20
30
!(!)
"(!)
eigenvalues (N = 20)
1 5 10 2010−12
10−10
10−8
10−6
10−4
10−2
100
N
|rHopf# "/2| (MATCONT)
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 approx-
imation always commute; PSA of Floquet multipliers is available [3].
Ongoing work to provide theoretical support:
<(σ(DN))→ −∞ as N→∞: how fast?
convergence of v(t, θ) to xt(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
more on
The same should apply to periodic orbits: those of the ODE approximation are
spectrally accurate approximations of the exact ones; linearization and approx-
imation always commute; PSA of Floquet multipliers is available [3].
Ongoing work to provide theoretical support:
<(σ(DN))→ −∞ as N→∞: how fast?
convergence of v(t, θ) to xt(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]b(t) =
h∫0
β(X(a,St),S(t))F(a,St)b(t− a)da,
S ′(t) = f(S(t)) −
h∫0
γ(X(a,St),S(t))F(a,St)b(t− a)da,
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 eigen-
values, 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. Sanchez @ 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