27. Discrete Dynamical System in the Neighborhood of a Quasiperiodic Trajectory

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Let x = x(n, x0), x0 ∈ Rq, n = 0,Ѓ}1,Ѓ}2, . . . , denote a solution of the

system of difference equations

x(n + 1) − x(n) = X(x(n)), x(0) = x0, (27.1)

Section 27 Discrete Dynamical System 283

where n = 0,Ѓ}1, . . . is discrete time, x ∈ Rq, and X(x) ∈ Cr(Rq). Assume

that system (27.1) has an invariant surface

M: x = f(ϕ), ϕ∈ Tm, (27.2)

where f(ϕ) ∈ Cr(Tm), filled with quasiperiodic trajectories

x(n, f(ϕ)) = f(ωn + ϕ), n= 0,Ѓ}1, . . . , ϕ ∈ Tm. (27.3)

Here, ω = (ω1, . . . , ωm) is a frequency basis of the quasiperiodic function

f(ωt), and ϕ is an arbitrary point in Tm.

Assume that

rank ∂f(ϕ)

∂ϕ

= m, ϕ ∈ Tm, (27.4)

and the matrix

∂f(ϕ)

∂ϕ

can be complemented to a 2π-periodic basis, i.e., there

exists a matrix B(ϕ) ∈ Cr(Tm) such that

det

_∂f(ϕ)

∂ϕ

,B(ϕ)

_

_= 0, ϕ∈ Tm. (27.5)

Under the assumptions made above, we investigate the behavior of trajectories

of system (27.1) originating in a small neighborhood of the manifold M. First,

note that the invariance of the manifold M and the quasiperiodicity of trajectories

on it require that the following identity be satisfied:

f(ϕ + ω) = f(ϕ) + X(f(ϕ)), ϕ∈ Tm. (27.6)

Assumptions (27.4) and (27.5) guarantee the introduction of local coordinates

(ϕ, h) = (ϕ1, . . . , ϕm, h1, . . . , hq−m) in the neighborhood of M according to

the formula

x = f(ϕ) + B(ϕ)h (27.7)

and representation (27.1) in the neighborhood of M in the form

ϕ(n + 1) − ϕ(n) = ω + A(ϕ(n), h(n))h(n),

h(n + 1) − h(n) = P(ϕ(n), h(n))h(n), (27.8)

where the matrices A(ϕ, h) and P(ϕ, h) of the corresponding dimensions are

2π-periodic in ϕ and sufficiently smooth with respect to ϕ and h in the domain

_h_ ≤ δ, ϕ ∈ Tm. (27.9)

284 Investigation of a Dynamical System Chapter 4

Here, n = 0,Ѓ}1,Ѓ}2, . . . and δ is a sufficiently small positive number. To establish

this fact, we perform the change of variables (27.7) in (27.1). As a result,

we obtain the following system of equations for the determination of the matrices

A = A(ϕ, h) and P = P(ϕ, h):

f(ϕ + ω + A(ϕ, h)h) + B(ϕ + ω + A(ϕ, h)h)[h + P(ϕ, h)h]

− [f(ϕ) + B(ϕ)h]

= X(f(ϕ) + B(ϕ)h),

or, with regard for identity (27.6),

f(ϕ + ω + Ah) − f(ϕ + ω) + B(ϕ + ω + Ah)(h + Ph) − B(ϕ)h

= X(f(ϕ) + B(ϕ)h) − X(f(ϕ)).

We represent the last equation in the form

∂f(ϕ + ω)

∂ϕ

Ah + B(ϕ + ω + Ah)Ph

= X(f(ϕ) + B(ϕ)h) − X(f(ϕ))

−

__

f(ϕ + ω + Ah) − f(ϕ + ω) − ∂f(ϕ + ω)

∂ϕ

Ah

_

+ [B(ϕ + ω + Ah) − B(ϕ + ω)]h + [B(ϕ + ω) − B(ϕ)]h

_

.

This yields

∂f(ϕ + ω)

∂ϕ

A + B(ϕ + ω + Ah)P

=

_1

0

∂X(f(ϕ) + tB(ϕ)h)

∂x

dtB(ϕ)

−

__1

0

_∂f(ϕ + ω + tAh)

∂ϕ

− ∂f(ϕ + ω)

∂ϕ

_

dtA

+ [B(ϕ + ω + Ah) − B(ϕ + ω)] + [B(ϕ + ω) − B(ϕ)]

%

. (27.10)

Section 27 Discrete Dynamical System 285

According to assumption (27.5), for fixed M > 0 there exists δ = δ(M) > 0

such that

det

_∂f(ϕ + ω)

∂ϕ

,B(ϕ + ω + Ah)

_

_= 0 (27.11)

for all ϕ and h from domain (27.9) and an arbitrary matrix A satisfying the

condition

_A_ ≤ M, (27.12)

where the norm of the matrix A is consistent with the norm of the vector h.

Therefore, Eq. (27.10) has a solution of the form

A = L1(ϕ, Ah)Q(ϕ, h,A), P = L2(ϕ, Ah)Q(ϕ, h,A), (27.13)

for all ϕ and h from domain (27.9) and A from domain (27.12). Here, Q =

Q(ϕ, h,A) is the matrix function defined by the right-hand side of Eq. (27.10),

and L1(ϕ, Ah) and L2(ϕ, Ah) are the blocks of the matrix inverse to the matrix _∂f(ϕ + ω)

∂ϕ

,B(ϕ + ω + Ah)

_

.

The matrix Q admits the following representation:

Q = B(ϕ + ω) − B(ϕ) + ∂X(f(ϕ))

∂x

B(ϕ) + Q1(ϕ, h,A), (27.14)

where Q1 = Q1(ϕ, h,A) is a matrix defined in the domain _h_ ≤ δ, ϕ ∈ Tm,

_A_ ≤ M, r − 1 times continuously differentiable with respect to its variables

and such that

Q1(ϕ, 0,A) = 0. (27.15)

The matrices L1(ϕ, Ah) and L2(ϕ, Ah) possess properties analogous to properties

of the matrix Q; moreover, L1(ϕ, 0) and L2(ϕ, 0) are the blocks of the

matrix inverse to the matrix

_∂f(ϕ + ω)

∂ϕ

,B(ϕ + ω)

_

.

In the space CLip(Tm×Kμ), we define the subset C(M,K) of matrix functions

A = A(ϕ, μ) that satisfy the conditions

_A(ϕ, h)_ ≤ M, _A(ϕ

_

, h

_) − A(ϕ, h)_ ≤ K(_ϕ

_ − ϕ_ + _h

_ − h_)

for any (ϕ_, h_) and (ϕ, h) from Tm × Kμ.

Let us prove that the first equation in (27.13) has a solution in C(M,K) for

the properly chosen constants M, K, and μ. To do this, we define an operator

S : A → SA = L1(ϕ, Ah) × Q(ϕ, h,A) on the set C(M,K). For r ≥ 2, this

286 Investigation of a Dynamical System Chapter 4

operator maps C(M,K) into a subset of the space CLip(Tm × Kμ). Furthermore,

SA = (SA)(ϕ, h) satisfies the following estimates for arbitrary (ϕ, h)

and (ϕ_, h_) from Tm × Kμ :

_(SA)(ϕ, h)_ ≤ c1(1 + μ + μM2),

_(SA)(ϕ

_

, h

_) − (SA)(ϕ, h)_

≤ c2(1 + μMK +M2)(_ϕ

_ − ϕ_ + _h

_ − h_), (27.16)

where c1 and c2 are positive constants independent of M, K, and μ ≤ δ

K

. By

a proper choice of sufficiently large M and K and sufficiently small μ, one can

guarantee that inequality (27.16) implies that SA belongs to the set C(M,K).

For a pair of matrix functions A = A(ϕ, h) and A1 = A1(ϕ, h) from the set

C(M,K), we have

_SA − SA1_ ≤ c3μ(1 +M)_A − A1_,

where c3 is a constant independent of M, K, and μ. For sufficiently small μ,

this estimate implies that S is a contraction operator on C(M,K). According

to the principle of contracting mappings, the equation A = SA has a unique

solution on the set C(M,K).

The last equation coincides with the first equation in (27.13), and, for sufficiently

small μ > 0 , its solution determines in CLip(Tm×Kμ) the unique matrix

A = A(ϕ, h) of the right-hand side of system (27.8). The implicit-function theorem

guarantees that A(ϕ, h) belongs to the space Cr−1

Lip (Tm × Kμ).

With the use of the obtained matrix A(ϕ, h), the second equation in (27.13)

determines the matrix P = P(ϕ, h) of the right-hand side of system (27.8),

which also belongs to the space Cr−1

Lip (Tm × Kμ).

Thus, in the small neighborhood of the manifold M, the dynamical system

(27.1) reduces to the form (27.8) with the matrices A and P from Cr−1

Lip (Tm ×

Kμ). The problem is to find conditions under which there exists a change of

variables ϕ → ψ that transforms system (27.1) into the quasiperiodic system

ψ(n + 1) − ψ(n) = ω,

h(n + 1) − h(n) = R(ψ(n), h(n))h(n). (27.17)

Theorem 27.1. Suppose that the conditions presented above are satisfied and

the matrix P(ϕ, 0) satisfies the inequality

_E + P(ϕ, 0)_ ≤ d < 1, ϕ∈ Tm. (27.18)

Section 27 Discrete Dynamical System 287

Then one can find μ > 0 and a matrix U(ϕ, h) ∈ Cr−2

Lip (Tm × Kμ), 2 ≤

r < ∞, such that the change of variables

ϕ = ψ + U(ψ, h)h (27.19)

reduces system (27.8) to the form (27.17) with the matrix

R(ψ, h) = P(ψ + U(ψ, h)h, h). (27.20)

Proof. We define the required transformation ϕ → ψ by the formula

ψ = ϕ + V (ϕ, h)h, (27.21)

where V = V (ϕ, h) is a matrix function from C(Tm × Kμ). According to

Eqs. (27.8) and (28.17) and the change of variables (27.21), we obtain the following

relation for the determination of the matrix V :

V (ϕ(n) + ω + A(ϕ(n), h(n))h(n), P1(ϕ(n), h(n))h(n))P1(ϕ(n),

h(n))h(n) − V (ϕ(n), h(n))h(n) + A(ϕ(n), h(n))h(n) = 0.

Therefore, V satisfies the equation

V (ϕ, h) = V (ϕ + ω + A(ϕ, h)h, P1(ϕ, h)h)P1(ϕ, h) + A(ϕ, h), (27.22)

where P1(ϕ, h) = E + P(ϕ, h). We set

ϕ1(ϕ, h) = ϕ + ω + A(ϕ, h)h, h1(ϕ, h) = P1(ϕ, h)h (27.23)

and rewrite (27.21) in the form

V (ϕ, h) = V (ϕ1(ϕ, h), h1(ϕ, h))P1(ϕ, h) + A(ϕ, h). (27.24)

For sufficiently small μ > 0, condition (27.18) yields the following inequality

for all (ϕ, h) ∈ Tm × Kμ :

_P1(ϕ, h)_ ≤ d1 = const < 1. (27.25)

This reasoning leads to the successive approximations for a solution of Eq. (27.24)

V1(ϕ, h) = A(ϕ, h),

Vi+1(ϕ, h) = Vi(ϕ1(ϕ, h), h1(ϕ, h))P1(ϕ, h) + A(ϕ, h), i≥ 1, (27.26)

288 Investigation of a Dynamical System Chapter 4

and the estimates

_V1(ϕ, h)_ ≤ _A(ϕ, h)_ ≤ max

Tm×Kμ

_A(ϕ, h)_ = M1,

_Vi+1(ϕ, h)_ ≤ M1

_i

ν=0

dν

1, i≥ 1.

By virtue of the last estimates, sequence (27.26) converges uniformly in (ϕ, h) ∈

Tm × Kμ, and its limit function

V (ϕ, h) = lim

i→∞

Vi(ϕ, h)

is a solution of Eq. (27.22) that belongs to the space C(Tm × Kμ).

Let us study the problem of the smoothness of the function V (ϕ, h). For this

purpose, we consider the function W = W(ϕ, h, μ) = V (ϕ, μh) for (ϕ, h) ∈

Tm × Kμ and sufficiently small μ > 0. This function satisfies the equation

W(ϕ, h, μ) = W(ϕ1(ϕ, μh), h1(ϕ, μh), μ)P1(ϕ, μh) + A(ϕ, μh) (27.27)

and is the limit of the successive approximations

W1(ϕ, h, μ) = A(ϕ, μh), (27.28)

Wi+1(ϕ, h, μ) = Wi(ϕ1(ϕ, μh), h1(ϕ, μh), μ)P1(ϕ, μh) + A(ϕ, μh), i≥ 1.

Differentiating expressions (27.28), we obtain the following equalities for derivatives:

∂W1(ϕ, h, μ)

∂ϕν

= ∂A(ϕ, μh)

∂ϕν

,

∂W1

∂hν

= μ

∂A(ϕ, μh)

∂(μh)ν

,

∂Wi+1(ϕ, h, μ)

∂ϕν

=

__m

j=1

∂Wi(ϕ1(ϕ, μh), h1(ϕ, μh), μ)

∂ϕ1,j

∂ϕ1,j(ϕ, μh)

∂ϕν

+

q_−m

j=1

∂Wi(ϕ1(ϕ, μh), h1(ϕ, μh), μ)

∂h1,j

∂h1,j(ϕ, μh)

∂ϕν

_

P1(ϕ, μh)

+Wi(ϕ1(ϕ, μh), h1(ϕ, μh), μ)∂P1(ϕ, μh)

∂ϕν

+ ∂A(ϕ, μh)

∂ϕν

, (27.29)

Section 27 Discrete Dynamical System 289

∂Wi+1(ϕ, h, μ)

∂hν

= μ

__m

j=1

∂Wi(ϕ1(ϕ, μh), h1(ϕ, μh), μ)

∂ϕ1,j

∂ϕ1,j(ϕ, μh)

∂(μh)ν

+

q_−m

j=1

∂Wi(ϕ1(ϕ, μh), h1(ϕ, μh), μ)

∂h1,j

∂h1,j

∂(μh)ν

_

P1(ϕ, μh)

+ μWi(ϕ1(ϕ, μh), h1(ϕ, μh), μ)∂P1(ϕ, μh)

∂(μh)ν

+ μ

∂A(ϕ, μh)

∂(μh)ν

, i≥ 1,

where ϕ1,j and h1,j are the jth coordinates of the vectors ϕ1(ϕ, μh) and

h1(ϕ, μh). For μ = 0, equalities (27.29) take the form

∂W1(ϕ, h, 0)

∂ϕν

= ∂A(ϕ, 0)

∂ϕν

,

∂W1(ϕ, h, 0)

∂hν

= 0,

∂Wi+1(ϕ, h, 0)

∂ϕν

= ∂Wi(ϕ + ω, h, 0)

∂ϕν

P1(ϕ, 0) +W1(ϕ + ω, h, 0)∂P1(ϕ, 0)

∂ϕν

+ ∂A(ϕ, 0)

∂ϕν

,

∂Wi+1(ϕ, h, 0)

∂hν

= 0, i≥ 1, (27.30)

and yield the following estimate:

max

Tm×Kμ

_W

_

i+1

_ ≤ d1 max

Tm×Kμ

_W

_

i

_ +M1, i≥ 1, (27.31)

where W

_

i is the matrix of derivatives of the iteration Wi, and M1 is a certain

constant.

It follows from (27.31) that

max

Tm×Kμ

_W

_

i+1

_ ≤ M1

1 − d1

, i≥ 1. (27.32)

For μ _= 0, relations (27.29) have the form of the matrix equalities

W

_

i+1(ϕ, h, μ) = W

_

i (ϕ1(ϕ, μh), h1(ϕ, μh), μ)P2(ϕ, μ, h)

+Wi(ϕ1(ϕ, μh), h1(ϕ, μh), μ)P

_

1(ϕ, h, μ)

+ A

_

1(ϕ, h, μ), i= 0, 1, 2, . . . , (27.33)

290 Investigation of a Dynamical System Chapter 4

where W

_

0 and W0 are zero matrices, W

_

i is the matrix of derivatives of the

iteration Wi, P

_

1, A

_

1, and P2 are matrix functions of the variables ϕ, h, and

μ that continuously depend on these variables for ϕ ∈ Tm, h ∈ Kμ, and μ ∈

[0, μ0], and μ0 is a sufficiently small positive number.

Relations (27.30) are a particular case of relations (27.33) and coincide with

them for μ = 0. Therefore, in the case μ = 0, we obtain the following expressions

for the matrices P2, P

_

1, and A

_

1 :

P2(ϕ, h, 0) = diag {P1(ϕ, 0), . . . , P1(ϕ, 0), 0, . . . , 0},

P

_

1(ϕ, h, 0) =

_∂P1(ϕ, 0)

∂ϕ1

, . . . ,

∂P1(ϕ, 0)

∂ϕm

, 0, . . . , 0

_

,

A

_

1(ϕ, h, 0) =

_∂A(ϕ, 0)

∂ϕ1

, . . . ,

∂A(ϕ, 0)

∂ϕm

, 0, . . . , 0

_

.

The first of these expressions yields

_P2(ϕ, h, 0)_ = _P1(ϕ, 0)_ ≤ d1, (27.34)

which guarantees the following estimate for the norm of the matrix P2(ϕ, h, μ):

_P2(ϕ, h, μ)_ ≤ d1(μ), (27.35)

where d1(μ) → d1 as μ → 0.

Choosing μ0 > 0 sufficiently small, we get

d1(μ) ≤ d2 ≤ const < 1 (27.36)

for all μ ∈ [0, μ0]. Then relations (27.33), (27.35), and (27.36) yield

max

Tm×Kμ

_W

_

i+1(ϕ, h, μ)_ ≤ d2 max

Tm×Kμ

_W

_

i (ϕ, h, μ)_ +M2, i= 0, 1, 2, . . . .

Hence,

max

Tm×Kμ

_W

_

i+1(ϕ, h, μ)_ ≤ M2

1 − d2

, i= 0, 1, 2, . . . , (27.37)

where M2 is a certain positive constant.

Inequality (27.37) means that

max

ν,j

____ ∂Wi(ϕ, h, μ)

∂ϕν

___

;

___

∂Wi(ϕ, h, μ)

∂hj

___

%

≤ M2

1 − d2

Section 27 Discrete Dynamical System 291

for all i = 1, 2, . . . . Then

max

ν,j

____ ∂Vi(ϕ, h)

∂ϕν

___

;

___

∂Vi(ϕ, h)

∂hj

___

%

≤ M2

μ0(1 − d2) .

Thus, the sequence of the first derivatives of approximations (27.26) is uniformly

bounded. By analogy, one can establish the uniform boundedness of the sequence

of arbitrary derivatives of approximations (27.26) up to the order r−1 inclusive.

This is sufficient for V = V (ϕ, h) to belong to the space Cr−2

Lip (Tm × Kμ).

To complete the proof of the theorem, it remains to solve relations (27.21)

with respect to ϕ in the form

ϕ = ψ + U(ψ, h)h, (27.38)

where U = U(ψ, h) is a function from the space Cr−2

Lip (Tm ×Kμ). Substituting

(27.38) into (27.21), we obtain the following equation for the matrix U :

U = −V (ψ + Uh,h). (27.39)

Equation (27.39) has the form of the first equation in (27.13). Therefore,

the arguments used in the proof of the solvability of the first equation in (27.13)

remain true in the case of Eq. (27.39). This yields the solvability of Eq. (27.39) in

the space Cr−2

Lip (Tm × Kμ) and, hence, the solvability of Eq. (27.21) in the form

(27.38).

In order to obtain expression (27.20) for R, it remains to replace ϕ by its

value (27.38) in the matrix P(ϕ, h), which determines the right-hand side of the

second equation in system (27.8). Theorem 27.1 is proved.

The statement below characterizes the behavior of trajectories of the discrete

dynamical system (27.1) originating in a small neighborhood of M.

Theorem 27.2. Suppose that the conditions of Theorem 27.1 are satisfied.

Then there exists a sufficiently small δ > 0 such that, for every y0 satisfying the

inequality ρ(y0,M) ≤ δ, one can find values ϕ0 and ψ0 from Tm such that

_x(n, y0) − f(ωn + ψ0)_ ≤ K1dn3

_y0 − f(ϕ0)_ (27.40)

for all n = 0, 1, . . . and certain positive K1 and d3, where d3 = d3(δ) → d2

and _ϕ0 − ψ0_ → 0 as δ → 0.

292 Investigation of a Dynamical System Chapter 4

The proof of Theorem 27.2 is analogous to the proof of Theorem 24.2.

Corollary 4. If the conditions of Theorem 27.1 are satisfied, then a quasiperiodic

solution

x = x(n, f(ϕ)) = f(ωn + ϕ)

of system (27.1) is Lyapunov stable for any ϕ ∈ Tm.

The proof of this corollary is analogous to the proof of Corollary 1 in Section

24.

Corollary 5. Suppose that the conditions of Theorem 27.1 are satisfied and

(k, ω) _= 0 mod 2π for every integer-valued vector k = (k1, . . . , km) _= 0.

Then, for an arbitrary function F = F(x) continuous in the neighborhood of M

and an arbitrary solution x = x(n, y0) of system (27.1) for which ρ(y0,M) ≤ δ,

the following relation is true:

lim

n→∞

1

n

n_−1

ν=0

F(x(ν, y0)) = F0

= (2π)−m

_2π

0

. . .

_2π

0

F(f(ϕ))dϕ1 . . . dϕm. (27.41)

Proof. We represent the function F in the form

F(x) = P(x, ε) + R(x, ε),

where P(x, ε) is a polynomial that approximates F in the neighborhood of

M to within an arbitrary fixed ε > 0, i.e., |R(x, ε)| ≤ ε ∀x ∈ Uδ(M) ≡

{x: ρ(x,M) ≤ ε}. This yields

1

n

___n−1 _

ν=0

[F(x(ν, y0)) − P(x(ν, y0), ε)]

___

≤ ε (27.42)

for arbitrary n = 1, 2, . . . . Using inequality (27.40), we get

Section 27 Discrete Dynamical System 293

1

n

___

n_−1

ν=0

[P(x(ν, y0), ε) − P(f(ων + ψ0), ε)]

___

≤ K(ε)K1

n

n_−1

ν=0

dν

3

_y0 − f(ϕ0)_ ≤ 1

n

K2(ε)

1 − d3

, (27.43)

where K(ε) is the Lipschitz constant of the polynomial P for x ∈ Uδ. We

represent the function P(f(ϕ), ε) in the form

P(f(ϕ), ε) = Q(ϕ, ε) + R1(ϕ, ε),

where Q(ϕ, ε) is a trigonometric polynomial that approximates P(f(ϕ), ε) to

within ε, i.e., |R1(ϕ, ε)| ≤ ε ∀ϕ ∈ Tm. Therefore, the following estimate holds

for arbitrary n = 1, 2, . . . :

1

n

___

n_−1

ν=0

[P(f(ων + ψ0), ε) − Q(ων + ψ0, ε)]

___

≤ ε. (27.44)

By definition,

Q(ϕ, ε) =

_

_k_≤N

Qkei(k,ϕ),

where N = N(ε) is a sufficiently large integer and Qk = Qk(ε) are the Fourier

coefficients of the function Q(ϕ, ε). This yields the equalities

1

n

n_−1

ν=0

Q(ων + ψ0, ε) = Q0 +

1

n

n_−1

ν=0

_

1≤_k_≤N

Qkei(k,ω)νei(k,ψ0)

= Q0 +

1

n

_

1≤_k_≤N

Qk

_n_−1

ν=0

ei(k,ω)ν

_

ei(k,ψ0)

and estimates

___

1

n

n_−1

ν=0

Q(ων + ψ0, ε) − Q0

___

≤ 1

n

_

1≤_k_≤N

|Qk|

___

n_−1

ν=0

ei(k,ω)ν

___

≤ max

1≤_k_≤N

1

n

___

n_−1

ν=0

ei(k,ω)ν

___

_

1≤_k_≤N

|Qk|

= M(ε) max

1≤_k_≤N

1

n

|

n_−1

ν=0

ei(k,ω)ν|. (27.45)

294 Investigation of a Dynamical System Chapter 4

For the last sum in inequality (27.45), the following estimate is true:

___

n_−1

ν=0

ei(k,ω)ν

___

=

__n_−1

ν=0

cos ν(k, ω)

_2

+

_n_−1

ν=0

sin ν(k, ω)

_2_1

2

=

___

sin n(k, ω)

2

___

___

cosec

(k, ω)

2

___

≤

___

cosec

(k, ω)

2

___

,

(k, ω) _= 0 mod 2π.

This estimate yields

___

1

n

n_−1

ν=0

Q(ων + ψ0, ε) − Q0

___

≤ 1

n

M(ε) max

1≤_k_≤N

___

cosec

(k, ω)

2

___

. (27.46)

We also have the inequality

|F0 − Q0| ≤ |F0 − P0| + |P0 − Q0| ≤ 2ε (27.47)

for the averages F0, Q0, and P0 of the functions F(f(ϕ)), Q(ϕ, ε), and

P(f(ϕ), ε). Combining inequalities (27.42)–(27.47), we get

___

1

n

n_−1

ν=0

F(x(ν, y0)) − F0

___

≤ 1

n

___

n_−1

ν=0

[F(x(ν, y0)) − P(x(ν, y0), ε)]

___

+

1

n

___

n_−1

ν=0

[P(x(ν, y0), ε) − P(f(ων + ψ0), ε)]

___

+

1

n

___

n_−1

ν=0

[P(f(ων + ψ0), ε) − Q(ων + ψ0), ε)]

___

+

1

n

___

n_−1

ν=0

Q(ων + ψ0), ε) − Q0

___

+|Q0 − F0|

≤ 4ε +

1

n

M1(ε), (27.48)

where

M1(ε) = K2(ε)

1 − d3

+M(ε) max

1≤_k_≤N

___

cosec

(k, ω)

2

___

.

Section 27 Discrete Dynamical System 295

We choose n0 = n0(ε) so large that the inequality

1

n

M1(ε) ≤ ε holds for

all n ≥ n0. Then relation (27.48) takes the form

___

1

n

n_−1

ν=0

F(x(ν, y0)) − F0

___

≤ 5ε ∀n ≥ n0,

which yields the limit relation (27.41). Corollary 2 is proved.

At the end of the section, we consider the perturbed system of equations

x(n + 1) − x(n) = X(x(n)) + εY (x(n)), (27.49)

where Y = Y (x) ∈ Cr(Rq) and ε is a small positive parameter. Upon the

change of variables (27.7), this system of equations takes the following form in

the neighborhood of the manifold M:

ϕ(n + 1) − ϕ(n) = ω + εa(ϕ(n)) + A(ϕ(n), h(n), ε)h(n),

h(n + 1) − h(n) = P(ϕ(n), h(n), ε)h(n) + εb(ϕ(n)),

(27.50)

where a = a(ϕ), A = A(ϕ, h, ε), P = P(ϕ, h, ε), and b = b(ϕ) are functions

from the space Cr−1

Lip (Tm × Kμ) for all sufficiently small ε > 0.

Applying to system (27.50) the perturbation theory of invariant toroidal manifolds

of discrete dynamical systems [MSM1, Nei] and the method of transformation

of system (27.8) to the form (27.17) presented above, we establish the

following statement:

Theorem 27.3. Suppose that the conditions of Theorem 27.1 are satisfied.

Then, for sufficiently small positive values of μ and ε0, there exists a change of

variables

ϕ = ψ + U(ψ, z, ε)h, h = u(ϕ, ε) + z

that reduces the system of equations (27.50) to the form

ψ(n + 1) − ψ(n) = ω + F(ψ(n), ε),

z(n + 1) − z(n) = R(ψ(n), z(n), ε)z(n),

where the functions u(ϕ, ε), U(ϕ, h, ε), F(ϕ, ε), and R(ϕ, h, ε) belong to the

space Cr−2

Lip (Tm × Kμ) for every ε ∈ [0, ε0] and

lim

ε→0

(_u(ϕ, ε)_r−2,Lip + _U(ϕ, h, ε) − U(ϕ, h)_r−2,Lip) = 0.

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