22. Statement and General Description of the Problem

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Let Cr(Tm) be the space of 2π-periodic functions f = (f1, . . . , fn) of a

variable ϕ = (ϕ1, . . . , ϕm) of smoothness r ≥ 0, and let λ = (λ1, . . . , λm)

be the frequency vector, i.e., a collection of m positive numbers that satisfy the

condition of linear independence over the field of integer numbers Zm, namely

(k, λ) =

_m

ν=1

kνλν _= 0 ∀k ∈ Zm\{0}.

A function

F(t) = f(λt), t∈ R, (22.1)

where f(ϕ) ∈ C(Tm) and C(Tm) = C0(Tm), is called a quasiperiodic function,

λ is called its frequency basis, and m is the dimension of the frequency basis.

By Cr(λ), we denote the collection of all quasiperiodic functions (22.1) with

frequency basis λ for which f ∈ Cr(Tm).

The true dimension of a frequency basis of the quasiperiodic function (22.1)

is defined as the number m such that F(t) ∈ C(λ) for a certain basis λ =

(λ1, . . . , λm) and F(t) _∈ C(ω) for an arbitrary basis ω = (ω1, . . . , ωs) for

which s < m. Denote by x(t, x0) a solution of the system of equations

dx

dt

= X(x) (22.2)

such that x(0, x0) = x0, where x = (x1, . . . , xn) and x0 = (x01

, . . . , x0

n)

are points of the n-dimensional Euclidean space Rn. Assume that the function

X = X(x) is r times continuously differentiable in Rn.

243

244 Investigation of a Dynamical System Chapter 4

For a set M ⊂ Rn, we denote by x(t,M) solutions x(t, x0) for an arbitrary

fixed x0 ∈ M.

Assume that system (22.1) has a quasiperiodic solution x = x(t, x0). By

definition, we have

x(t, x0) = f(λt + ψ0) (22.3)

for a certain function f ∈ C(Tm) and a certain basis λ = (λ1, . . . , λm). Assume

that m is the true dimension of the frequency basis. The closure of the

trajectory that passes through the point x0 consists of points M ∈ R defined by

the equation

x = f(ϕ), ϕ∈ Tm, (22.4)

where Tm denotes an m-dimensional torus. The set M is invariant because it is

the closure of the trajectory of a dynamical system.

According to the equation of motion (22.2), we have

f(λt + ψ0) = f(ψ0) +

_t

0

X(f(λτ + ψ0))dτ, t ∈ R.

Hence, passing to the limit in the sequence of functions f(λt+λtn), lim

n→∞λtn =

ϕ mod 2π, we obtain the identity

f(λt + ϕ) = f(ϕ) +

_t

0

X(f(λτ + ϕ))dτ,

which proves that

x(t, f(ϕ)) = f(ωt + ϕ) ∀t ∈ R, ϕ ∈ Tm.

Thus, the set M is filled with quasiperiodic trajectories of the dynamical

system (22.2) with the same frequency basis. The geometric structure of the set

M in the space Rn is described by the following statement [Sam4]:

Theorem 22.1. If x(t, x0) ∈ Cs(λ), where s ≤ r, then the set M is Cshomeomorphic

to an m-dimensional torus.

In what follows, a set M ⊂ Rn Cs-homeomorphic to an m-dimensional

torus is called an m-dimensional toroidal manifold of smoothness s, or, briefly,

an m-dimensional torus of smoothness s.

Section 22 Statement and General Description of the Problem 245

It follows from the arguments presented above that the system of equations

(22.2) on M reduces to a dynamical system on Tm of the form

dϕ

dt

= λ. (22.5)

We pose the problem of the investigation of the behavior of solutions of system

(22.2) that originate in a small neighborhood of manifold (22.4). For this

purpose, it is natural to represent the neighborhood of the manifold M in the

form of a product Tm × Kδ, where Kδ is the (n − m)-dimensional cube with

side δ, and introduce the local coordinates ϕ = (ϕ1, . . . , ϕm) on Tm and h =

(h1, . . . , hn−m) in Kδ instead of the Euclidean coordinates x = (x1, . . . , xn).

In the coordinates ϕ, h, the equation of the manifold M takes the form

h = 0, ϕ∈ Tm, (22.6)

and the system of equations (22.2) takes the form (22.5).

Assume that x(t, x0) ∈ Cs(λ) and s ≥ 1. According to [Sam4], we have

rank ∂f(ϕ)

∂ϕ

= m ∀ϕ ∈ Tm,

where f(ϕ) is a function from (22.4).

The problem of the introduction of local coordinates (h, ϕ) reduces in this

case to the following algebraic problem: Find a matrix B(ϕ) whose columns

belong to the space Cs(Tm) and for which the n × n matrix

_∂f(ϕ)

∂ϕ

,B(ϕ)

_

is nondegenerate for all ϕ ∈ Tm; here,

_∂f(ϕ)

∂ϕ

,B(ϕ)

_

denotes the matrix m

columns of which are the columns of the matrix

∂f(ϕ)

∂ϕ

, and n − m columns

are the columns of the matrix B(ϕ). This problem is called the problem of the

complementation of the m-frame

∂f(ϕ)

∂ϕ

to a 2π-periodic basis in Rn, and it

has a solution [Sam4] for n = m + 1 or n ≥ 2m + 1. Assume that the matrix

∂f(ϕ)

∂ϕ

can be complemented to a 2π-periodic basis in Rn, and B(ϕ) is the

complementing matrix.

Under the assumptions imposed above on the manifold M, the local coordinates

(ϕ, h) of a point x from the neighborhood of M are determined by the

equality

x = f(ϕ) + B(ϕ)h. (22.7)

246 Investigation of a Dynamical System Chapter 4

Consider the matrix

Γ0(ϕ) = BT (ϕ)B(ϕ), ϕ∈ Tm, (22.8)

where BT (ϕ) is the transpose of B(ϕ). The matrix Γ0(ϕ) is the Gram matrix of

the linearly independent columns of the matrix B(ϕ); therefore, the eigenvalues

of this matrix are positive for all ϕ ∈ Tm. The periodicity of the matrix Γ0(ϕ)

in ϕ enables one to estimate these eigenvalues from below and from above by

positive constants γ0 and γ0 independent of ϕ. In this case, the quadratic form

(Γ0(ϕ)h, h) satisfies the inequalities

γ0_h_2 ≤ (Γ0(ϕ)h, h) ≤ γ0_h_2 ∀ϕ ∈ Tm, h ∈ Rn−m, (22.9)

where _h_2 = (h, h). Inequalities (22.9) yield

γ0_h_2 ≤ _x − f(ϕ)_2 ≤ γ0_h_2. (22.10)

In view of these estimates, under the transformation x → (ϕ, h) defined by

equality (22.7) a small neighborhood of the set M ⊂ Rn turns into the small

neighborhood

_h_ ≤ δ, ϕ ∈ Tm (22.11)

of the set h = 0, ϕ ∈ Tm in the space Rn−m × Tm. Using this fact, we rewrite

the equations of motion of system (22.2) (originating in a neighborhood of the

manifold M) in the local coordinates. To do this, we differentiate relation (22.7)

as a formula of a change of variables in system (22.2). As a result, instead of

(22.2), we obtain the system of equations

dϕ

dt

= L1(ϕ, h)X(f(ϕ) + B(ϕ)h),

dh

dt

= L2(ϕ, h)X(f(ϕ) + B(ϕ)h),

(22.12)

where L1(ϕ, h) and L2(ϕ, h) are blocks of the matrix inverse to the matrix _∂f(ϕ)

∂ϕ

+ ∂B(ϕ)h

∂ϕ

,B(ϕ)

_

, and (ϕ, h) are points of domain (22.11) with sufficiently

small positive δ. Note that expressions for the matrices L1(ϕ, h) and

L2(ϕ, h) in terms of the matrices

∂f

∂ϕ

and B can be obtained using the Frobenius

formula [Lan] for the construction of the inverse matrix for the matrix composed

of the following blocks:

Section 22 Statement and General Description of the Problem 247

L1(ϕ, h) =

__∂f

∂ϕ

+ ∂Bh

∂ϕ

_T

(E − BΓ−1

0 BT )

_∂f

∂ϕ

+ ∂Bh

∂ϕ

__−1

×

_∂f

∂ϕ

+ ∂Bh

∂ϕ

_T

(E − BΓ−1

0 BT ),

L2(ϕ, h) =

_

BT

_

E −

_∂f

∂ϕ

+ ∂Bh

∂ϕ

_

Γ−1

1

_∂f

∂ϕ

+ ∂Bh

∂ϕ

_T _

B

_−1

× BT

_

E −

_∂f

∂ϕ

+ ∂Bh

∂ϕ

_T

Γ−1

1

_∂f

∂ϕ

+ ∂Bh

∂ϕ

_T_

, (22.13)

where Γ0 = Γ0(ϕ) is matrix (22.8), E is the identity matrix, and

Γ1 = Γ1(ϕ, h) =

_∂f

∂ϕ

+ ∂Bh

∂ϕ

_T _∂f

∂ϕ

+ ∂Bh

∂ϕ

_

. (22.14)

Taking into account that manifold (22.6) is invariant for the system of equations

(22.12) with the flow of trajectories on it defined by system (22.5), we rewrite

(22.12) in the form

dϕ

dt

= λ + L1(ϕ, h)

_

X(f(ϕ) + B(ϕ)h) − X(f(ϕ)) − ∂B(ϕ)

∂ϕ

λh

_

,

dh

dt

= L2(ϕ, h)

_

X(f(ϕ) + B(ϕ)h) − X(f(ϕ)) − ∂B(ϕ)

∂ϕ

λh

_

, (22.15)

where

∂B

∂ϕ

λ =

_m

ν=1

∂B

∂ϕν

λν.

Parallel with system (22.15), we write the following auxiliary system of equations

obtained from (22.15) by omitting the terms of order _h_ for ϕ and of

order _h_2 for h on the right-hand sides of these equations:

dϕ

dt

= λ,

dh

dt

= P(ϕ)h, (22.16)

where

P(ϕ) = L2(ϕ, 0)

_∂X(f(ϕ))

∂x

− ∂B(ϕ)

∂ϕ

λ

_

. (22.17)

Denote by Ωt

0(ϕ), Ω00

(ϕ) = E, the normal fundamental matrix of solutions

of the second equation in (22.16). It is clear that Ωt

0(ϕ) ∈ Cs−1(Tm) for every

248 Investigation of a Dynamical System Chapter 4

t ∈ R. Moreover, for arbitrary t ∈ R, θ ∈ R, and ϕ ∈ Tm, the following

identity is true [Sam4]:

Ωt

0(ϕθ(ϕ)) = Ωt+θ

0 (ϕ), (22.18)

where ϕθ(ϕ) = λθ + ϕ.

Assume that the condition

_Ωt

0(ϕ)_ ≤ Le

−γt (22.19)

is satisfied for all t ∈ R+ = [0,∞) and ϕ ∈ Tm and certain positive constants

L and γ. We rewrite the system of equations (22.15) in the form

dϕ

dt

= λ + A(ϕ, h)h,

dh

dt

= P(ϕ, h)h, (22.20)

where

A(ϕ, h) = L1(ϕ, h)

__1

0

∂X(f(ϕ) + τB(ϕ)h)

∂x

dτB(ϕ) + ∂B(ϕ)

∂ϕ

λ

_

,

P(ϕ, h) = L2(ϕ, h)

__1

0

∂X(f(ϕ) + τB(ϕ)h)

∂x

dτB(ϕ) + ∂B(ϕ)

∂ϕ

λ

_

. (22.21)

It follows from relations (22.13), (22.14), and (22.21) that A and P are s − 1

times continuously differentiable functions of their variables in their domain of

definition (22.11).

Let Cp

Lip(Tm ×Kμ) denote the space of functions of (ϕ, h) that are defined

in the domain Tm × Kμ, Kμ = {h: _h_ ≤ μ}, have (in this domain) continuous

partial derivatives up to the order p inclusive, and are such that their pth

derivatives satisfy the Lipschitz condition with respect to (ϕ, h). The meaning of

the notation Cp(Tm × Kμ) is analogous for finite and infinite values of p.

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