from delay di erential equations to ordinary di erential ... · from delay di erential equations to...

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]




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.


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)


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).


basics [9,12]




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.


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.


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.


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).


basics [9,12]




DDE

{x ′(t) = Lxt, t > 0,

x(θ) = ϕ(θ), θ ∈ [−τ, 0].


DDE → ∞-ODE


∞-ODE

{u ′(t) = Au(t), t > 0,

u(0) = ϕ ∈ D(A).


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].


∞-ODE

{u ′(t) = Au(t), t > 0,

u(0) = ϕ ∈ D(A).


DDE

{x ′(t) = Lxt, t > 0,

x(θ) = ϕ(θ), θ ∈ [−τ, 0].


DDE → PDE

∂v(t, θ)

∂t=∂v(t, θ)

∂θ

⇓characteristic lines t+ θ = constant.


DDE → PDE


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].


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).


basics [9,12]




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

).


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


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.


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).


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


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].


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


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.


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?


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)?


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.


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.


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].


PDE → N+ 1 ODEs

φ = (ϕ(θ0),ϕ(θ1), . . . ,ϕ(θN))T


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.


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.


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→∞.


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.


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)


AUTO

MATCONT

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]).


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).


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!


bibliography

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Numer. Anal., 50:1456-1483, 2012

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Sacker-Sell spectrum for retarded functional differential equations,

Numer. Math., online, 2013


bibliography cont’d

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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


http://indy.cs.concordia.ca/auto/

http://matcont.sourceforge.net/

http://twr.cs.kuleuven.be/research/software/delay/ddebiftool.shtml

http://twr.cs.kuleuven.be/research/software/delay/ddebiftool.shtml

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