NONAUTONOMOUS DYNAMICAL

SYSTEMS

Russell Johnson

Universita di Firenze

Luca Zampogni

Universita di Perugia

Thanks for collaboration to R. Fabbri, M. Franca, S.

Novo, C. Nunez, R. Obaya, and many other colleagues.

Dynamics and Differential Equations

Dedicated to Prof. George R. Sell

Minneapolis, 22–25/06 /2016.

1

Introduction

A nonautonomous dynamical system (NDS)

can be of continuous or discrete type.

Usually a continuous NDS is generated by the

solutions of a nonautonomous differential equa-

tion

x′ = f(t, x), t ∈ R, x ∈ B (1)

where B is a Banach space. A discrete NDS is

usually obtained by solving

xn+1 = fn(xn), n ∈ N, fn : B → B.

Let us consider the case (1) when B = Rd.

Suppose that each initial condition (t0, x0) gives

rise to a unique global solution ϕ(t, t0, x0). For

t ∈ R, set

τt : R×Rd → R×Rd : τt(t0, x0) = (t0+t, ϕ(t, t0, x0)).

Then τt | t ∈ R is a one-parameter group

of homeomorphisms, or dynamical system, on

R× Rd. This is simple but already important.

2

If f satisfies recurrence conditions in t, one

can often “compactify” time (Bebutov, Sell).

That is, there exists a compact metric space

Ω, a dynamical system τt on Ω, and a func-

tion F : Ω × Rd → Rd such that the solutions

ϕ(t, t0, x0) of

x′ = F (τt(ω), x) (1ω)

define a dynamical system τt on Ω× Rd via

τt(ω, x0) = (τt(ω), ϕ(t, ω, x0)).

Moreover there is a point ω0 ∈ Ω such that

(1ω0) coincides with (1ω).

The upshot is that one can apply methods of

dynamical systems theory to study the solu-

tions of the family (1ω).

3

Topics of current interest include:

– Bifurcation Theory

– Oscillation Theory, Control Theory

– Spectral Theory of Ordinary Differential Equa-

tions

– Sturm-Liouville problems, Integrable sys-

tems

– Nonautonomous Functional Differential Equa-

tions

– Finite-Interval problems.

4

One needs a good linear theory to make in-

roads on these problems. In fact some are in-

herently linear, and one can study the others

by the method of linearization along a solution.

So, let us consider the linear nonautonomous

system

x′ = a(t)x, x ∈ Rd. (2)

Here a(·) takes values in the set Md of d × dreal matrices. Suppose that a(·) is bounded

and uniformly continuous (for most purposes

boundedness and measurability are enough).

We assume that a(·) has some recurrence prop-

erties (e.g., almost periodicity, Birkhoff recur-

rence, chain recurrence,...).

5

One studies a constant coefficient system x′ =ax by making systematic use of the eigenval-

ues and generalized eigenspaces of a. Let us

look for quantities which are related to the

nonautonomous equation (2) and which are

analogous to:

(α) real parts of eigenvalues

(β) generalized eigenspaces

(γ) imaginary parts of eigenvalues.

6

Introduce the Bebutov hull (Ω, τt) of a(·),

and let A : Ω→ Md be the corresponding func-

tion. Consider the family

x′ = A (τt(ω))x. (2ω)

Let Φω(t) be the fundamental matrix solution

of (2ω). One obtains a flow τt on Ω× Rd as

follows:

τt(ω, x) = (τt(ω),Φω(t)x) .

This is a so-called skew-product flow.

7

(α) The notion of real part of an eigenvalue has

two very useful generalizations to the nonau-

tonomous case:

– Lyapunov exponents

– Dynamical (Sacker-Sell) spectrum.

If ω ∈ Ω, the corresponding (maximal) Lya-

punov exponent is

β(ω) = lim supt→∞

1

tln ||Φω(t)||.

The existence of the limit and the properties

of β(·) are studied in the Oseledets theory, of

which more later.

To define the dynamical spectrum, we intro-

duce the concept of exponential dichotomy, or

ED for short (Perron, Coppel, Palmer,...).

8

Definition 1 The family (2ω) admits an EDover Ω if there are constants K > 0, k > 0 and acontinuous projection-valued function P : Ω→Md : ω 7→ Pω = P2

ω such that

||Φω(t)PωΦ−1ω (s)|| ≤ Ke−k(t−s), t ≥ s

||Φω(t)(I − Pω)Φ−1ω (s)|| ≤ Kek(t−s), t ≤ s.

Thus an ED exists when the “solution space”Ω× Rd admits a hyperbolic splitting.

Definition 2 A real number λ belongs to thedichotomy spectrum Σ of (2ω) if the trans-lated family

x′ = [−λI +A(τt(ω))]x

does not admit an ED over Ω.

Clearly if A(·) reduces to a constant a, thenΣ = <µ | µ is an eigenvalue of a. Sacker andSell proved that, if (Ω, τt) is chain recurrent,then Σ is a union of finitely many compactintervals. It is known that each Lyapunov ex-ponent β(ω) lies in Σ.

9

(β) The concept of generalized eigenspace hastwo nonautonomous versions: the Oseledetsbundles and the Sacker-Sell bundles (also Mil-lionscikov and Selgrade).

Oseledets Let µ be a τt-ergodic measure onΩ (thus µ is indecomposable in a certain sense).Then

Ω× Rd = W1 ⊕W2 ⊕ · · · ⊕Ws

where W1, . . . ,Ws are µ-measurable, τt-invariantvector subbundles of Ω×Rd. Within each sub-bundle Wi, all trajectories (τt(ω),Φω(t)x0) giverise to the same Lyapunov exponent (if x0 6=0):

β(ω, x0) = βi = limt→±∞

1

tln ||Φω(t)x0||.

10

Sacker-Sell Let (Ω, τt) be chain recurrent,

and let Σ = [a1, b1] ∪ . . . ∪ [ar, br] be the spec-

trum. There is a decomposition

Ω× Rd = W1 ⊕ W2 ⊕ . . .⊕ Wr

where W1, . . . , Wr are continuous, τt-invariant

vector subbundles of Ω × Rd. For each i =

1,2, . . . , r, the dynamical spectrum of (Wi, τt)is [ai, bi]. An Oseledets bundles is always con-

tained in a Sacker-Sell bundle; the reverse in-

clusion need not hold.

11

(γ) In the case of a constant-coefficient sys-

tem, one views the imaginary part of an eigen-

value as the source of “rotation” of certain

solutions of the system. This notion can be

rendered concrete for certain types of families

(2ω) via the concept of rotation number. Let

us briefly discuss the rotation number in the

context of two-dimensional families (2ω). In-

troduce polar coordinates

r2 = x21 + x2

2, θ = arctanx2

x1

and write out the θ-equation

θ′ = g(τt(ω), θ) = . . .

Definition 3 Let µ be an ergodic measure on

Ω. The rotation number is

α = limt→∞

θ(t)

t

where θ(t) is the solution of the θ-equation.

12

It is known that the limit exists for µ-almost

all ω ∈ Ω (it is easy to see that it is insensi-

tive to the initial value θ(0)). One obtains a

quantity which is very useful in the study of

the spectral theory of ordinary differential op-

erators, for example the Schrodinger operator

(in one space dimension), Sturm-Liouville op-

erators, the AKNS operator, and “operators”

of Atkinson type. The initial work involved in

applying the rotation number to these opera-

tors was carried out by J.-Moser, Kotani, and

other scientists.

A higher-dimensional version on the rotation

number has been developed for linear Hamil-

tonian systems, and has been applied to spec-

tral problems of Atkinson type and to con-

trol problems (J., J.-Nerurkar, Novo-Nunez-

Obaya,. . . ). These matters have been studied

by Fabbri, J., Novo, Nunez, Obaya; see the

recent book of these authors.

13

Let us now consider an application of the basic

tools of Nonautonomous Dynamics to the 1-D

Schrodinger operator. We will discuss ques-

tions concerning the generalized reflectionless

Schrodinger potentials (Lundina, Marchenko,

Kotani).

Let q : R → R be a bounded continuous func-

tion. Introduce the differential expression

L = −d2

dx2+ q(x).

Then L determines an unbounded self-adjoint

operator (Schrodinger operator) on L2(R). It

also determines two half-line operators L± de-

fined on L2[0,±∞] respectively, via the Dirich-

let boundary condition in x = 0:

L±ϕ = λϕ

ϕ(0) = 0.

14

EXAMPLE. Consider the soliton potential

q(x)=-2d2

dx2ln det(I +A(x)) (3)

where A(x) = (Aij(x))ni,j=1 and

Aij(x) =

√lilj

ηi + ηje−(ηi+ηj)x.

Here l1, . . . , ln and η1, . . . , ηn are positive num-

bers. If q(x) is of the form (3), then the

spectrum of the operator L consists of the

half-line [0,∞) (where it is absolutely continu-

ous) together with the finitely many eigenval-

ues −η21, . . . ,−η

2n.

A soliton potential has a very important prop-

erty. Let g(x, y, λ) be the Green’s function (in-

tegral kernel) of (L−λ)−1 for λ in the resolvent

of L. Then

<g(x, x, λ+ i0) = 0 (λ > 0, x ∈ R), (4)

where ”λ+ i0” means the limit of g(x, x, λ+ iε)

as ε decreases to zero. One says that q (or L)

is reflectionless.

15

To summarize: a soliton potential is reflection-

less in (0,∞), and has finitely many eigenvalues

in (−∞,0). Let c < 0, and set

Rc = cls

q | q is a soliton potential,

and the eigenvalues of

L = −d2

dx2+ q(x) lie in (c,0)

.

Here the closure is taken in the topology of

uniform convergence on compact subsets of

R. One says that Rc consists of generalized

reflectionless Schrodinger potentials.

Let GR =⋃c<0

Rc.

Theorem (Lundina, 1985)

The set Rc is compact

(in the compact-open topology).

16

There is a well-known connection between the

Schrodinger operator L and the Korteweg-de

Vries equation

∂u

∂t= 3u

∂u

∂x−

1

2

∂3u

∂x3, u(0, x) = q(x).

It was proved by Gardner-Green-Kruskal-Miura

that a regular solution u(t, x) of the K-dV equa-

tion has the property that the family of Schrodinger

operators

Lt = −d2

dx2+ u(t, x)

is isospectral (in L2(R)). This fact can be

used to determine explicit solutions of the K-

dV equation in certain cases (including that of

soliton potentials).

17

Let q be a generalized reflectionless Schrodinger

potential, that is q ∈⋃c<0

Rc = GR. Marchenko

tried to prove in 1991 that each such q gives

rise to a regular solution of the K-dV equa-

tion. Along the way he worked out a nice

parametrization of the elements in GR. About

ten years later Kotani succeeded in proving

that indeed each q ∈ GR gives rise to a solution

u(t, x) of the K-dV equation, which in fact is

meromorphic in the entire complex (t, x)−space.

He used the theory of the infinite-dimensional

Grassmann-type space Gr2, which is due to

Sato-Segal-Wilson.

18

From now on we set c = −1 and unify R =

Rc = R−1. We will study certain compact sub-

sets of R which are invariant under the natural

translation flow, defined by

τx(q)(·) = q(x+ ·) (x ∈ R, q ∈ R).

Let −∞ = a0 < b0 < aj < bj ≤ 0 (j ≥ 1),

where b0 ≥ −1 and (aj, bj) | j ≥ 1 are pairwise

disjoint nonempty open intervals. Set

E = R \∞⋃j=0

(aj, bj),

then set

QE = q ∈ R | Lq = −d2

dx2+ q(x)

has spectrum E and is reflectionless

By reflectionless we mean that the appropriate

generalization of condition (2) holds, namely

<gq(x, x, λ+ i0) = 0

(a.a λ ∈ E, all x ∈ R).(2bis)

19

Note: condition (2bis) holds on 0 < λ < ∞for every q ∈ R. However if q ∈ R and Lq has

spectrum E, it need not be the case that (2bis)

holds on the set E ∩ [−1,0].

Proposition

Let E and QE be as above.Suppose also that E has locally positive measure.

Then QE is compact and translation invariant.

At this point the methods of Nonautonomous

Dynamics become relevant.

Introduce the set of divisors

DE = (yj, εj) | yj ∈ [aj, bj], εj = ±1, 1 ≤ j <∞.

The points (aj,±1) and (bj,±1) are identified,

so DE is a product of countably many circles.

20

Now we want to define a map π : QE → DEwhich will provide information about QE and

about the flow (QE, τx). To do this we need

to carry out a preliminary discussion. Let q ∈QE.

Consider the map gq : λ 7→ gg(0,0, λ), which

is defined and meromorphic on ΩE = C \ E.

The map gq(·) is strictly monotone increasing

on each interval (aj, bj). There are three cases

to consider.

(i) The map gq has a (unique) zero µj ∈ (aj, bj).

(ii) The map gq is positive in (aj, bj); in this

case set µj = aj.

(iii) The map gq is negative in (aj, bj); in this

case set µj = bj.

21

We define a divisor

dq = (µ1, ε1), . . . , (µj, εj), . . . ∈ DE

where the signs εj are determined using theWeyl m-functions m±(λ). Recall that thesefunctions are defined as follows: if λ ∈ C \ E,there are nonzero solutions ϕ±(x, λ) ∈ L2[0,±∞)of Lϕ = λϕ. Set

m±(λ) =ϕ′±(0, λ)

ϕ±(0, λ).

Then sign=m±(λ)

=λ= ±1 if =λ 6= 0, and also

signdm±(λ)

dλ= ±1 if λ ∈ R \ E. (5)

It turns out that

gq(λ) =1

m−(λ)−m+(λ).

So if gq(λ) = 0 and aj < λ < bj then eitherm−(λ) or m+(λ) has a pole at λ, and it followsfrom (5) that at most one of m±(λ) has a poleat λ.

22

Now return to the definition of the divisor dq(the pole divisor). If µj ∈ (aj, bj), set εj = 1 if

m+(·) has a pole at µj, and εj = −1 if m−(·)has a pole at µj. If µj = aj or µj = bj then

there is no need to worry about the sign εj,

since (aj,±1) and (bj,±1) are identified.

So we obtain a map π : QE → DE : q 7→ dq =

= (µj, εj) | j ≥ 1. We also obtain a ”pole

motion”: let τx(q) = q(x+·) be the translation,

and set

τx(dq) = (µj(x), εj(x)) | j ≥ 1 := dτx(q).

It is not a priori clear that we obtain a flow

on DE from this construction. This is because

trajectories could conceivably cross, and in any

case it is not clear that π is continuous.

23

The idea now is to produce examples via a

study of the map π. The starting point is the

following result.

Theorem

Let E ⊂ R be a closed set of the form

E = R \∞⋃j=0

(aj, bj),

where −∞ = a0 < b0 < aj < bj ≤ 0 (j ≥ 1),where b0 ≥ −1 and the closed intervals [aj, bj]

are pairwise disjoint. Suppose that E haslocally positive measure. Then

(a) The mapπ : QE → DE : q 7→ (µ1, ε1), . . . , (µj, εj), . . .

is continuous and surjective.

(b) If the half-line operators L± have purelyabsolutely continuous spectrum for all q ∈ QE,

then π is also injective,hence is a homeomorphism.

24

One can use this theorem to construct exam-ples of sets E for which QE has interestingstructure.

Example 1 There exists a closed set E ⊂ R,which satisfies the hypotheses of the Theo-rem, such that QE contains a minimal set Mwhich is almost automorphic in the sense ofBochner-Veech, but is not Bohr almost peri-odic. We explain the terminology. –A minimalflow M, τx) is Bohr almost periodic if thereis a metric d onM, which is compatible with itstopology, such that the flow τx is isometric:d(τx(q1), τx(q2)) = d(q1, q2) for all q1, q2 ∈ Mand all x ∈ R. A minimal flow (M, τx) isalmost automorphic if there exists an almostperiodic flow (M0, τx) and a flow homomor-phism h : M → M0 such that h−1(q0) is asingleton for some q0 ∈M0.–The almost periodic minimal set of the exam-ple turns out to be the character group JE ofthe infinitely connected domain ΩE = C \ E.The domain ΩE is of Parreau-Widom (PW)type.

25

This example contradicts the Kotani-Last con-

jecture, according to which if the operators L±

all have purely a.c. spectrum for all q ∈ QE,

then QE should consist entirely of Bohr almost

periodic potentials. For other examples: Avila,

Damanik-Yuditskii.

26

The Character Group of aParreau-Widom domain

The discussion is based on work by Sodin-

Yuditskii, Volberg-Yuditskii.

Let E = R \∞⋃j=0

(aj, bj) where a0 = −∞, −1 ≤

b0 < 0 and the nonempty open intervals (aj, bj)

are pairwise disjoint and contained in (b0,0).

We define the character group JE of the do-

main ΩE = C \E. For this, let cj (j = 1,2, . . . )

be a closed simple curve in ΩE which contains

λ0 = −2, passes through (aj, bj), and is orthog-

onal to R. Let cj be oriented clockwise. The

fundamental group ΓE is generated by these

closed curves. Let JE be the set of all charac-

ters on ΓE, that is, the set of all group homo-

morphisms from ΓE to the unit circle S1 ∼= R/Z.

One puts a group operation on JE via point-

wise addition of pairs α1, α2 ∈ JE:

(α1 + α2)(γ) = α1(γ) + α2(γ) ∈ R/Z.

Then JE is a compact Abelian topological group.

27

Suppose now that E has locally positive mea-

sure. We define a particular character δ ∈ JEas follows. Let ν be an ergodic measure on QE,

and let wν be the ν-Floquet exponent. Set

δ(cj) =ρj

π∈ R/Z

where ρj = ρν|(aj,bj) is the value of the ν-

rotation number on (aj, bj). This character de-

fines a translation flow τx on JE, as follows:

τx(α) = α+ δx (α ∈ JE, x ∈ R).

28

Next, introduce an Abel map from the set of

divisors DE to the character group JE, as fol-

lows. Let ω(λ, F ) be the harmonic measure of

a subset F ⊂ E with respect to the domain ΩE.

–This means that ω(·, F ) solves the Laplace

equation ∆ω = 0 in ΩE, and has boundary

value equal to the characteristic function of

F .– Let d = (y1, ε1), . . . , (yj, εj), . . . ∈ DE,

and define

A(d)(γk) =1

2

∞∑j=1

εj

∫ bjajω(dλ,Ek) ∈ R/Z (6)

where Ek is the part of E to the right of bk,

that is Ek = E ∩ [bk,∞).

There is no a priori reason to think that the

series on the right-hand side of (6) converges.

It does converge if ΩE is a Parreau-Widom

domain.

29

To explain what this means, we introduce the

Green’s function G(λ, λ0) of ΩE with logarith-

mic pole at λ0 = −2. We assume that ΩE is

regular for the Dirichlet problem, which means

that G assumes continuously the boundary value

zero at each point of E (which is the boundary

of ΩE). Further, let cj | j ≥ 1 be the points

in ΩE where the gradient ∇G(cj) = 0; there is

exactly one such point in each interval (aj, bj)

and there are no other such points in ΩE. One

now says that ΩE is of Parreau-Widom type if

∞∑j=1

G(cj) <∞.

30

Theorem (Gesztesy-Yuditskii).

Suppose that ΩE is of Parreau-Widom type.Then the Abel map A is well-defined, continuousand surjective, and maps τx-orbits in DE onto

τx-orbits in JE.

Moreover, there is a right inverse I : JE → DE,that is, a map such that A I(α) = α

for all α ∈ JE. This map I need not becontinuous (it is if E is homogeneous in the

sense on Carlesen).

31

To create sets E and corresponding domains

ΩE as in Example 1, we arrange that E satis-

fies all the hypotheses of the first theorem and

is also of Parreau-Widom type. We arrange

that the δ-translation flow τx(α) = α + δx is

minimal on JE. We further arrange that the

right inverse I : JE → DE satisfies the follow-

ing conditions

– I is not continuous;– I(α+ δx) = τx(I(α)) (α ∈ JE, x ∈ R)– I is of the first Baire class.

The last condition means that I(α) = limN→∞

IN(α)

(α ∈ JE) where each map IN : JE → DE is con-

tinuous. This implies that I has a residual set

of continuity points. This in turn implies that

M = clsI(α) | α ∈ JE ⊂ DE

is an almost automorphic, non-almost periodic

minimal subset of DE.

32

Example 2 This example will consist of a setE of the the type under discussion for whichthe divisor map is not injective. Let

E = [−1,∞) \∞⋃n=1

(an, bn),

where a = a0 = −3/4, b = b0 = −1/2, 0 > b1 >

a1 > a2 > b2 > . . . and where bj, aj → b = −1/2.

Note that the interval [−1,−3/4] ⊂ E. Let QEbe the corresponding set of potentials. Choosean ergodic measure ν on QE.

(i) One can choose the intervals (aj, bj) so thatb = −1/2 is irregular in the sense of potentialtheory. This implies that the ν-Lyapunov ex-ponent βν(λ) > 0.(ii) Use the Oseledets Theorem to see that forν-a.a. q ∈ QE the equation

Lqϕ =

(d2

dx2+ q(x)

)ϕ = bϕ

admits solutions ϕ±(x) which decay exponen-tially at x = ±∞.

33

Using the fact that Lqϕ = bϕ is oscillatory, we

can find q ∈ QE so that the Dirichlet problem

Lqϕ = bϕ

ϕ(0) = 0

admits a nonzero solution in L2[0,∞).

(iii) Introduce the spectral measures σ± of L±q :

σ±(dλ) = σ±,ac(dλ) + σ±,d(dλ) +r±b− λ

Choose q ∈ QE so that r+ > 0, this is pos-

sible by point (ii). If r+ ≥ 0, r− ≥ 0, and

r+ + r− = r+ + r−, then one can use the

Marchenko parametrization of GR (and bypass

the less informative Gel’fand-Levitan theory)

to show that there exists q ∈ QE with spectral

measures

σ±(dλ) = σ±,ac(dλ) + σ±,d(dλ) +r±b− λ

.

Since r+ > 0 one has that π is not injective.

34