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23. Theorem on Reducibility

Back

In what follows, we preserve the same notation as in Section 22. The statement

below is the main result of the present section.

Theorem 23.1. Suppose that the matrices A(ϕ, h) and P(ϕ, h) belong to

the space Cp(Tm × Kδ) for p ≥ 1, and the fundamental matrix of solutions

Section 23 Theorem on Reducibility 249

Ωt

0(ϕ) of system (22.16) satisfies inequality (22.19). Then one can find μ > 0

and a matrix Φ(ψ, h) belonging to the space Cp−1

Lip (Tm × Kμ) such that the

change of variables

ϕ = ψ +Φ(ψ, h)h (23.1)

reduces the system of equations (22.20) to the form

dψ

= λ,

= P(ψ +Φ(ψ, h)h, h) (23.2)

for (ψ, h) ∈ Tm × Kμ.

Proof. Let us prove the theorem for p = 1. We write the following equation

for the determination of the matrix Φ = Φ(ψ, h) :

∂Φ

∂ψ

λ + ∂Φ

∂h

P(ψ +Φh, h)h+ΦP(ψ +Φh, h) = A(ψ +Φh, h), (23.3)

where

∂Φ

∂h

Ph =

n_−m

ν=1

∂Φ

∂hν

(Ph)ν

and (Phν) is the νth coordinate of the vector Ph. To solve Eq. (23.3), we use

the method of passing from (23.3) to an operator equation, which is based on the

ideas of the method of integral manifolds [Bog, BoM1, MiLy]. Let C(M,K)

denote the set of matrix functions F = F(ψ, h) defined for all ψ ∈ Tm and

h ∈ Rn−m and satisfying the inequalities

_F(ψ, h)_ ≤ M, _F(ψ

, h

_) − F(ψ, h)_ ≤ K(_ψ

_ − ψ_ + _h

_ − h_)

for any (ψ, h) ∈ Tm × Rn−m and (ψ_, h_) ∈ Tm × Rn−m.

We define a scalar function z(τ ) of a scalar variable τ as follows:

z(τ) =

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

0 for |τ| ≥ 2μ,

−(τ + 2μ) for τ ∈ (−2μ,−μ),

τ for |τ| ≤ μ,

2μ − τ for τ ∈ (μ, 2μ).

We set g(h) = (z(h1), . . . , z(hn−m)). It is clear that, for all h_ and h from

Rn−m, we have

_g(h)_ ≤ μ(n − m) 1

2 , _g(h

_) − g(h)_ ≤ _h

_ − h_.

250 Investigation of a Dynamical System Chapter 4

Let

ψt = λt + ψ, ψ ∈ Tm.

Denote by XF

t = Ωt

0(ψ, g,F) a solution of the equation

= P(ψt + F(ψt,Xg)Xg,Xg)X (23.4)

that takes the value of the identity matrix E for t = 0. Here, g = g(h) and

h ∈ Rn−m. Since

t = Ωt

0(ψ) +

Ωt

s(ψ)

P(ψs + F(ψs,XF

s g)XF

s g,XF

s g) − P(ψs, 0)

s ds,

we conclude that the following estimate holds for XF

s :

_XF

_ ≤ Le

−γt

1 + μa

eγs_XF

_2ds

, t∈ R+, (23.5)

where a = K1(1 +M)(n − m) 1

2 and K1 is a constant independent of the first

derivatives of the matrix P(ψ, h) for ψ ∈ Tm and h ∈ Kδ.

Assume that _XF

_ ≤ L(1+μ) for t ∈ [0, T), where [0, T) is the maximum

half-interval on which XF

t satisfies the above inequality. It follows from (23.5)

that

_XF

_ ≤ Le

−γt

1 + μaL(1 + μ)

eγs______________XF

_ds

for t ∈ [0, T). Therefore, according to the Gronwall–Bellman inequality, we

have

_XF

_ ≤ Le

−γ1t ∀t ∈ [0, T), (23.6)

where

γ1 = γ − μaL2(1 + μ) ≥ γ

(23.7)

for sufficiently small μ. It follows from (23.6) that _XF

_ ≤ L(1+μ) ∀t ∈ R+,

and, hence,

_XF

_ ≤ Le

−γ1t, t∈ R+. (23.8)

Section 23 Theorem on Reducibility 251

On the set of functions C(M,K), we define an operator S : F → SF = W

according to the formula

W(ψ, h) = SF

= −

∞ _

ψs + F(ψs,XF

s g(h))XF

s g(h),XF

s g(h))

s ds. (23.9)

Let us prove that, for properly chosen M, K, and μ, the operator S maps the

set C(M,K) into itself.

To do this, we estimate the difference

Rt = XF

t (ψ

, h

_) − XF

t (ψ, h), t∈ R+,

where (ψ_, h_) and (ψ, h) are arbitrary points from Tm × Kμ. It follows from

the equation for XF

t that

Rt =

Ωt

s(P

_)[P

− Ps]XF

s ds, (23.10)

where

Ωt

s(P

_) = XF

t (ψ

, h

s (ψ

, h

_−1, ψ

t = λt + ψ

− Pt = P

t + F(ψ

t,XF

t (ψ

, h

_)g(h

_))

× XF

t (ψ

, h

_)g(h

_),XF

t (ψ

, h

_)g(h

− P

ψt + F(ψt,XF

t (ψ, h)g(h))XF

t (ψ, h)g(h),XF

t (ψ, h)g(h)

s = XF

s (ψ, h).

According to (22.18), we have Ωt

s(ψ) = Ωt−s

0 (ψs). Therefore, for Ωt

s(P_), the

following estimate of the form (23.8) is true:

_Ωt

s(P

_)_ ≤ Le

−γ1(t−s), t≥ s ≥ 0. (23.11)

Let us estimate the quantity I = _P

− Ps_. Using obvious notation, we get

I ≤ K1

δ + _F

_ − FXg_ + _X

_ − Xg_

≤ K1

δ + _X

_ − Xg_M + _F − F

___Xg_ + _X

_ − Xg_

252 Investigation of a Dynamical System Chapter 4

≤ K1

(1 + μKL(n − m) 1

2 )δ + (1 +M + μKL(n − m) 1

2 )_X

_ − Xg_

≤ K1

(1 + μKL(n − m) 1

2 )δ + (1 +M + μKL(n − m) 1

2 )Lδ1

+ (1 +M + μKL(n − m) 1

2 )μ(n − m) 1

2 _X

_ − X_]

≤ LK1(1 +M + μKL(n − m) 1

2 )

__ψ_ − ψ_

L + _h

_ − h_

+ μ

(n − m) 1

_XF

t (ψ

, h

_) − XF

t (ψ, h)_

, t≥ 0. (23.12)

We set

b = LK1

1 +M + μKL(n − m) 1

and rewrite (23.12) in the form

I ≤ b

__ψ_ − ψ_

L + _h

_ − h_ + μ(n − m) 1

_Rt

L, t≥ 0. (23.13)

Taking into account inequality (23.13), we derive from (23.10) the following inequality:

_Rt_ ≤ Lbe

−γ1t

_ψ − ψ

__L−1 + _h

_ − h_

+ μ(n − m) 1

2L−1_Rs_

ds, t ≥ 0. (23.14)

Estimate (23.14) has the form of the inequality

yt ≤ be

−γ1t

α + μ(n − m) 1

2 ys

ds, t ≥ 0, (23.15)

where yt = _Rt_L−1 and α = _ψ − ψ__L−1 + _h_ − h_. Since yt ≤ yt

for t ∈ R+, where yt is a solution of the equation obtained from (23.15) by

replacing the sign ≤ by =, we get

yt ≤ be

−γ1t_

α + μ(n − m) 1

2 c

t, t ∈ R+, c= sup

t∈R+

yt. (23.15_)

Section 23 Theorem on Reducibility 253

Let us determine c. It follows from the equation for yt that the value of c is

attained at the point τ ∈ R+, where

yτ = 0. Writing the last equality in more

detail, we get

−γ1c + be

−γ1τ (α + μ(n − m) 1

2 c) = 0.

This implies that, for μb(n − m) 1

2 < γ1, the number c can be estimated as

follows:

c = bαe−γ1τ

γ1 − μ(n − m) 1

2 be−γ1τ

≤ bα

γ1 − μb(n − m) 1

. (23.16)

If we set

μ ≤ γ1

2b(n − m)1/2

, (23.17)

then it follows from (23.15_ ) and (23.16) for all t ∈ R+ that

yt ≤ 2bαte

−γ1t. (23.18)

In view of the notation used, inequality (23.18) yields

_Rt_ ≤ 2Lbte

−γ1t(_ψ

_ − ψ_L−1 + _h

_ − h_), t∈ R+. (23.19)

Further, we consider the difference W(ψ_, h_) − W(ψ, h). Using (23.9), we

obtain the following estimate for this difference:

_W(ψ

, h

_) −W(ψ, h)_ ≤

∞ _

__Rs_ + _A

− As__XF

ds,

where A

t and At are the expressions obtained from the expressions for P

t and

Pt by replacing P by A. It is clear that the difference A

−At can be estimated

by analogy with the difference P

− Pt. As a result, we obtain the following

inequality of the form (23.13):

− At_ ≤ LK1(1 +M + μKL(n − m) 1

2 )

× (_ψ

_ − ψ_L−1 + _h

_ − h_ + μ(n − m)1/2L−1_Rt_), t≥ 0,

254 Investigation of a Dynamical System Chapter 4

where K1 is a positive constant that depends on the first derivatives of the matrix

A(ψ, h) for ψ ∈ Tm and h ∈ Kδ. This yields

_W(ψ

, h

_) −W(ψ, h)_

≤

∞ _

(M1 + μb(n − m) 1

2 )_Rs_ + b(_ψ

_ − ψ_ + L_h

_ − h_)e

−γ1s_

≤ (M1 + μb(n − m) 1

2 )

∞ _

_Rs_ds + b

γ1

L(_ψ

_ − ψ_ + _h

_ − h_), (23.20)

where

M1 = max

ψ,h

_A(ψ, h)_, b = LK1(1 +M + μKL(n − m) 1

2 ).

Inequality (23.19) yields

∞ _

_Rs_ds ≤ 2Lbγ

−2

1 (_ψ

_ − ψ_ + _h

_ − h_),

which, together with (23.20), guarantees the validity of the inequality

_W(ψ

, h

_) −W(ψ, h)_

≤ Lγ

−1

2bγ

−1

1 (M1 + μb(n − m) 1

2) + b

(_ψ

_ − ψ_ + _h

_ − h_).

It also follows from (23.9) that

_W(ψ, h)_ ≤ LM1γ

−1

1 .

Thus, W ∈ C(M,K), provided that M, K, and μ satisfy inequalities (23.7)

and (23.17) and the inequalities

LM1γ

−1

≤ M, Lγ

−1

2bγ

−1

1 (M1 + μb(n − m) 1

2) + b

≤ K.

For μ ≤ 1, the inequalities indicated are satisfied if

Section 23 Theorem on Reducibility 255

μ(n − m) 1

2L2K1(1 +M) ≤ γ

4 , 2LM1γ

−1 ≤ M,

μ(n − m) 1

2LK1

1 +M + μLK(n − m) 1

≤ γ

4 ,

2L2γ

−10

4K1γ

−1_

M1 + μK1(1 +M + μLK(n − m) 1

2 )(n − m) 1

1 +M + μLK(n − m) 1

≤ K. (23.21)

Inequalities (23.21) are satisfied due to the proper choice of M, K, and μ.

Namely, M and K are chosen from the conditions

2Lγ

−1M1 ≤ M,

2L2γ

−1(2 +M)

K1 + 4K1γ

−1[M1 + K1(2 +M)]

≤ K, (23.22)

and μ satisfies the inequalities

μ(n − m) 1

2L2K1(1 +M) ≤ γ

4, μ(n − m) 1

2LK1(2 +M) ≤ γ

4 ,

μ(n − m) 1

2LK ≤ 1. (23.23)

We fix M and K so large that conditions (23.22) are satisfied. Then, for these

values of M and K, one can choose μ0 > 0 so that inequalities (23.23) hold

for any μ ∈ (0, μ0].

Assume that M, K, and μ0 are chosen as indicated above. In this case,

W = SF ∈ C(M,K), i.e., the operator S maps the set C(M,K) into itself.

In C(M,K), we introduce a metric according to the formula

ρ(F1, F2) = sup _F1(ψ, h) − F2(ψ, h)_,

where the supremum is taken over (ψ, h) ∈ Tm × Rn−m. Thus, C(M,K) becomes

a complete metric space. Let us prove that the operator S, which acts from

C(M,K) into itself, is a contraction operator. For this purpose, we consider the

difference

WF −WF_ = SF − SF

where F and F_ are arbitrary functions from C(M,K). Equations (23.4) yield

− XF_

t =

Ωt

s(F)[Ps(F) − Ps(F

_)]XF_

s ds,

256 Investigation of a Dynamical System Chapter 4

where

Ωt

s(F) = XF

t (XF

s )−1, Ps(F) − Ps(F

= P

ψs + F(ψs,XF

s g)XF

s g,XF

s g

− P(ψs + F

_(ψs,XF_

s g)XF_

s g,XF_

s g)

, g= g(h). (23.24)

In view of the notation used, we get

_Ps(F) − Ps(F

_)_

≤ μ(n − m) 1

2K1

_F(ψs,XF

s g)XF

− F

_(ψs,XF_

s g)XF_

_ + _XF

− XF_

≤ μ(n − m) 1

2K1

Lρ(F,F

_)e

−γ1s

+ (1 +M + μ(n − m) 1

2KL)_XF

− XF_

for all s ≥ 0. The last inequality, together with inequality (23.11) for ψ_ = ψ

and h_ = h, yields

_XF

− XF_

_ ≤ μ(n − m) 1

2K1L2

−γ1(t−s)_

Lρ(F,F

_)e

−γ1s

+ (2 +M)_XF

− XF_

−γ1sds

≤ μK1L2(n − m) 1

Lρ(F,F

_)γ

−1

+ (2 +M)

_XF

− XF_

_ds

−γ1t, t∈ R+. (23.25)

Solving inequality (23.25), we obtain

_XF

− XF_

_ ≤ μ2L3K1(n − m) 1

2 γ

−1

1 e

−γ1tρ(F,F

_) (23.26)

for all t ∈ R+ and all μ that satisfy the inequality

μ(n − m) 1

2K1L2(2 +M)γ

−1

≤ ln 2. (23.27)

Section 23 Theorem on Reducibility 257

For the difference WF −WF_

, the following estimate is true:

_WF −WF__ ≤

∞ _

M1_XF

− XF_

_ + _As(F) − As(F

_)__XF_

ds,

where As(F)−As(F_) is the expression obtained from (23.24) by replacing the

matrix P by the matrix A.

Since

_As(F)−As(F

_)_ ≤ μ(n−m) 1

2K1

Lρ(F,F

_)e

−γ1s +(2+M)_XF

−XF_

for all s ≥ 0, we have

_WF −WF__ ≤

M1 + μ(2 +M)K1

L(n − m) 1

∞ _

_XF

− XF_

_ds + μ(n − m) 1

2K1

L(2γ1)−1ρ(F,F

≤ μ(n − m) 1

2L3K1

M1 + μ(2 +M)K1

L(n − m) 1

−2

+ K1

L(2γ1)−1

ρ(F,F

= μd1ρ(F,F

_), (23.28)

where d1 denotes the corresponding constant. For sufficiently small μ > 0,

inequality (23.28) yields

ρ(SF,SF

_) ≤ 1

2ρ(F,F

_), (23.29)

which proves that the operator S is contracting.

According to the principle of contracting mappings, the operator S has a

unique fixed point F(ψ, h) = Φ(ψ, h) in C(M,K). This means that

Φ(ψ, h) = −

∞ _

A[ψs +Φ(ψs,Xsg)Xsg,Xsg]Xsds, (23.30)

where Xt = XΦ

t is a solution of Eq. (23.4) for

F(ψ, h) = Φ(ψ, h), g= g(h).

258 Investigation of a Dynamical System Chapter 4

Let us establish a relationship between the matrix Φ(ψ, h) and solutions of the

system of equations (23.4). For this purpose, we substitute the following functions

for ψ and h in (23.30):

ψt = ψ + λt, ht = Xt(ψ, h)h,

where t ∈ R+ and h ∈ Kμ(L)−1 . As a result, we get

Φ(ψt, ht) = −

∞ _

ψs+t +Φ(ψs+t,Xs(ψt, ht)ht)Xs(ψt, ht)ht,

Xs(ψt, ht)ht

Xs(ψt, ht)ds.

It follows from the equation for Xs(ψ, h) that, for s ≥ t ≥ 0, the function

Xs(ψt, ht)ht is a solution of the equation

= P(ψs+t +Φ(ψs+t, y)y, y)y (23.31)

that takes the value

y0 = ht = Xt(ψ, h)h, t ∈ R+, (23.32)

for s = 0. According to Eq. (23.4), the function

y = Xs+t(ψ, h)h, s ≥ t ≥ 0,

is also a solution of the Cauchy problem (23.31), (23.32). It follows from the

uniqueness of a solution of the Cauchy problem (23.31), (23.32) that

Xs(ψt, ht)ht = Xs+t(ψ, h)h

for all s ≥ t ≥ 0, ψ ∈ Tm, and h ∈ Kμ(L)−1 . Taking this identity into account,

we conclude that

Φ(ψt, ht) = −

∞ _

A(ψτ +Φ(ψτ,Xτh)Xτ h,Xτh)XτX

−1

t dτ (23.33)

for all t ∈ R+.

We set

u(ψ, h) = Φ(ψ, h)h

Section 23 Theorem on Reducibility 259

for ψ ∈ Tm and h ∈ Kμ(L)−1 . Then it follows from (23.33) that

u(ψt, ht) = Φ(ψt, ht)ht

= −

∞ _

A(ψτ + u(ψτ, hτ ), hτ )hτ dτ, t ∈ R+. (23.34)

Differentiating (23.34) with respect to t, we establish that u(ψt, ht) satisfies the

equation

= A(ψt + u, ht)ht, t∈ R+.

This implies that u(ψt, ht), ht is a solution of the system of equations

= A(ψt + u, h)h,

= P(ψt + u, h)h, t ∈ R+,

i.e., a solution ϕt, ht of system (22.20) whose initial value ϕ, h is chosen from

the conditions h ∈ Kμ(L)−1 ,

ϕ = ψ + u(ψ, h) (23.35)

is determined by the change of variables (23.1) with ϕ = ϕt and h = ht for all

t ∈ R+.

To complete the proof of the theorem for p = 1, it remains to establish that

the mapping (ψ, h) → (ϕ, h) determined by (23.35) is a Lipschitz homeomorphism

of the domain Tm×Kμ onto itself. Since the mapping indicated transforms

h identically, it remains to prove that, using (23.35), one can find

ψ = ϕ + v(ϕ, h), (23.36)

where the function v = v(ϕ, h) satisfies the Lipschitz condition with respect to

ϕ and h in the domain Tm × Kμ. Substituting (23.36) into (23.35), we obtain

the following equation for v :

v = −u(ϕ + v, h), (ϕ, h) ∈ Tm × Kμ.

The function u(ϕ, h) satisfies the inequalities

_u(ϕ + v, h)_ ≤ μM,

_u(ϕ

_ + v

, h

_) − u(ϕ + v, h)_

≤ μK(_ϕ

_ − ϕ_ + _v

_ − v_) + (M + μK)_h

_ − h_ (23.37)

260 Investigation of a Dynamical System Chapter 4

for arbitrary ϕ, h, v and ϕ_, h_, v_ from the domain Tm×Kμ×Rm. In the space

CLip(Tm × Kμ), we select the subspace of functions v = v(ϕ, h) for which

_v(ϕ, h)_ ≤ μM,

_v(ϕ

, h

_) − v(ϕ, h)_ ≤ (2M + 1)(_ϕ

_ − ϕ_ + _h

_ − h_). (23.38)

Here, (ϕ, h) and (ϕ_, h_) are arbitrary points of the domain Tm×Kμ. It follows

from (23.37) and (23.38) that the operator

S1 : v →−u(ϕ + v, h)

maps the subspace indicated into itself for μK ≤ 1

and is a contraction operator:

ρ(S1v, S1v

_) ≤ μKρ(v, v

_) ≤ 1

2ρ(v, v

_),

where

ρ(v, v

_) = max

ϕ,h

_v(ϕ, h) − v

_(ϕ, h)_.

This is sufficient for the equation for v to have a unique solution v = v(ϕ, h)

defined for (ϕ, h) ∈ Tm × Kμ and satisfying inequalities (23.38). For p = 1,

Theorem 23.1 is proved.

Remark 1. To prove this theorem for p ≥ 2, it remains to investigate the

smoothness of the change of variables (23.35) in the domain Tm ×Kμ. The case

p = 2 is principal for this investigation because, beginning with this case, the

function Φ(ψ, h) becomes continuously differentiable and turns into the classical

solution of Eq. (23.1) for ψ, h from the domain Tm×Kμ. It is clear that, for p ≥

2, it is necessary to take a p times continuously differentiable function g(h). To

study the smoothness of the function Φ(ψ, h), we use the fact that Φ(ψ, h) can

be obtained as the limit (as j →∞) of the successive approximations Φj(ψ, h),

j ≥ 1, defined by the equality

Φj(ψ, h)=−

∞ _

ψs+Φj−1(ψs,X(j−1)

s g)X(j−1)

s g,X(j−1)

s g

X(j−1)

s ds,

where Φ0(ψ, h) ≡ 0 and X(j−1)

t is a solution of the equation

= P[ψt +Φj−1(ψt,Xg)Xg,Xg]X, j = 1, 2, . . . ,

Section 24 Variational Equation and Theorem on Attraction 261

that takes the value E for t = 0. One can prove [Sam6] that the functions

Φj(ψ, h) and their partial derivatives with respect to ψ and h up to the order

p are continuous on the set Tm × Kμ and uniformly bounded for all j ≥ 0,

ψ ∈ Tm, and h ∈ Kμ by a constant c0. Then it follows from the Arzel`a theorem

[KoF] that the sequence Φj(ψ, h) is compact in Cp−1(Tm × Kμ) and the

(p − 1)th derivatives of the limit function satisfy the Lipschitz condition. These

arguments complete the proof of Theorem 23.1.

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