26. The Case of a Toroidal Manifold Filled with Trajectories of General Form

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In Sections 22–25, we have studied a dynamical system in the neighborhood

of an invariant manifold filled with a quasiperiodic trajectory of the system. In

what follows, we extend the results obtained in Sections 22–25 to the case where

the invariant toroidal manifold of the system is filled with trajectories of the general

form. In particular, we present a theorem on the reducibility of the dynamical

272 Investigation of a Dynamical System Chapter 4

system in the neighborhood of the invariant toroidal manifold M and a statement

on the exponential attraction of solutions from the neighborhood of the manifold

M to solutions on M and establish conditions for the invariance of the behavior

of trajectories of the dynamical system in the neighborhood of the manifold M

under small perturbations.

Consider the system

dx

dt

= X(x), (26.1)

where x = (x1, . . . , xn) is a point of the n-dimensional Euclidean space Rn

and X(x) ∈ Cr (Rn), r ≥ 1. Let f = f(ϕ) be a function from the space

Cs(Tm), s ≤ r, of 2π-periodic functions of ϕ = (ϕ1, . . . , ϕm) of smoothness

s ≥ 2 and with values in Rn. Let

M: x = f(ϕ), ϕ∈ Tm, (26.2)

be the invariant set of system (26.1) and let

rank ∂f(ϕ)

∂ϕ

= m, ϕ ∈ Tm. (26.3)

According to [Sam4], the first condition for the set M is satisfied if

_∂f(ϕ)

∂ϕ

Γ−1(ϕ)

_∂f(ϕ)

∂ϕ

_T

−E

_

X(f(ϕ)) = 0, ϕ∈ Tm,

where Γ(ϕ) =

_∂f(ϕ)

∂ϕ

_T ∂f(ϕ)

∂ϕ

, and the second condition means that M is a

toroidal manifold.

The system of equations (26.1) on M can be reduced to a dynamical system

on the torus Tm of the form

dϕ

dt

= a(ϕ), (26.4)

where, according to [Sam4], the function a(ϕ) has the form

a(ϕ) = Γ−1(ϕ)

_∂f(ϕ)

∂ϕ

_T

X(f(ϕ)), ϕ∈ Tm.

Assume that the m-frame

∂f(ϕ)

∂ϕ

can be complemented to a 2π-periodic basis

in Rn, and B(ϕ) is a complementing matrix from Cs(Tm). If we introduce

the local coordinates

x = f(ϕ) + B(ϕ)h (26.5)

Section 26 Toroidal Manifold Filled with Trajectories of General Form 273

in the neighborhood of the manifold M, then, taking into account the invariance

of M and Eq. (26.4) for the flow of trajectories on M, we can rewrite the system

of equations (26.1) in the neighborhood of M in the local coordinates ϕ, h as

follows:

dϕ

dt

= a(ϕ) + L1(ϕ, h)

_

X(f(ϕ) + B(ϕ)h) − X(f(ϕ)) − ∂B(ϕ)

∂ϕ

a(ϕ)h

_

,

dh

dt

= L2(ϕ, h)

_

X(f(ϕ) + B(ϕ)h) − X(f(ϕ)) − ∂B(ϕ)

∂ϕ

a(ϕ)h

_

. (26.6)

Here, L1(ϕ, h) and L2(ϕ, h) are blocks of the matrix inverse to the matrix

_∂f(ϕ)

∂ϕ

+ ∂B(ϕ)

∂ϕ

h,B(ϕ)

_

,

∂B(ϕ)

∂ϕ

a(ϕ) ≡

_m

ν=1

∂B(ϕ)

∂ϕν

aν(ϕ),

and ϕ and h are points from the domain

ϕ ∈ Tm, _h_ ≤ δ, (26.7)

where δ > 0 is sufficiently small.

Consider the variational equation for the manifold M

dϕ

dt

= a(ϕ),

dh

dt

= P(ϕ)h, (26.8)

where, by definition [Sam4],

P(ϕ) = L2(ϕ, 0)

_∂X(f(ϕ))

∂x

− ∂B(ϕ)

∂ϕ

a(ϕ)

_

. (26.9)

Let

ϕ = ψt(ϕ), ψ0(ϕ) = ϕ ∈ Tm, (26.10)

be a solution of the first equation of system (26.8) and let Ωt

0(P) be the fundamental

matrix of solutions of the second equation of system (26.8) for ϕ = ψt(ϕ).

Using the matrix P(ϕ), we define a function β(ϕ) as follows:

inf

S∈N

max

_h_=1

2_

S(ϕ)P(ϕ) +

1

2

∂S(ϕ)

∂ϕ

a(ϕ)

_

h, h

3

_S(ϕ)h, h_

≤ −β(ϕ), (26.11)

where N is the set of (n − m) × (n − m) positive-definite symmetric matrices

S = S(ϕ) ∈ C1(Tm) and _・, ・_ is the scalar product in Rn.

274 Investigation of a Dynamical System Chapter 4

Assume that

β0 = inf

ϕ∈Tm

β(ϕ) > 0. (26.12)

Condition (26.12) is sufficient [Sam4] for the following inequality to be satisfied

for all t ∈ R+ and ϕ ∈ Tm:

_Ωt

0(P)_ ≤ Le

−γt, (26.13)

where γ is an arbitrary positive number satisfying the inequality γ < β0 , and

L = L(γ) is a certain positive constant.

Further, we define a function α1(ϕ) by the inequality

inf

S1∈N1

max

_ψ_=1

2_

S1(ϕ)∂a(ϕ)

∂ϕ

+

1

2

∂S1(ϕ)

∂ϕ

a(ϕ)

_

ψ,ψ

3

_S1(ϕ)ψ,ψ_

≤ α1(ϕ), (26.14)

where N1 is the set of m-dimensional square positive-definite symmetric matrices

S1 = S1(ϕ) ∈ C1(Tm).

Using α1(ϕ), we can obtain the following estimate for the derivatives of the

function ψt(ϕ) with respect to ϕ [Sam4]:

___

∂lψt(ϕ)

∂ϕl1

1 . . . ∂ϕlm

m

___

≤ L1 exp

__t

0

la(ψτ (ϕ))dτ + μt

%

, t∈ R+, (26.15)

where l = l1 + . . . + lm, μ is an arbitrarily small positive number, and L1 =

L1(l, μ) is a certain positive constant.

Assume that

inf

ϕ∈Tm

[β(ϕ) − lα1(ϕ)] > 0 (26.16)

for a certain integer l ∈ [1, s − 1].

The following analog of Theorem 23.1 for the system of equations (26.6) is

true:

Theorem 26.1. Suppose that the right-hand side of system (26.6) satisfies the

smoothness conditions given above, and inequalities (26.12) and (26.16) are true.

Then one can find a constant μ > 0 and a matrix Ψ(ψ, h) ∈ Cl−1

Lip (Tm × Kμ)

such that the change of variables

ϕ = ψ +Ψ(ψ, h)h (26.17)

Section 26 Toroidal Manifold Filled with Trajectories of General Form 275

reduces the system of equations (26.6) to the form

dψ

dt

= a(ψ),

dh

dt

= P(ψ, h)h, (26.18)

where P(ψ, h) is a matrix that belongs to Cl−1

Lip (Tm × Kμ) and coincides with

P(ψ) for h = 0.

Theorem 26.1 is proved by analogy with Theorem 23.1 with the difference that

one should take into account estimate (26.15) for the derivatives of the function

ψt(ϕ) and inequality (26.16).

The verification of conditions (26.12) and (26.16) encounters certain difficulties.

These difficulties can be avoided if, for the fundamental matrices of solutions

Ωt

0(P) and Ωt

0

_ ∂a

∂ϕ

_

of the systems

dh

dt

= P(ψt(ϕ))h and

dg

dt

= ∂a(ψt(ϕ))

∂ϕ

g, (26.19)

respectively, estimates of the following form are known:

_Ωt

0(P)_ ≤ Le

−β0t, t∈ R+,

___

Ωt

0

_ ∂a

∂ϕ

____≤ L1e+α1t, t∈ R+,

(26.20)

where β0 and α1 are positive constants. In this case, for inequality (26.16) to be

satisfied, it is sufficient that

β0

α1

> l, (26.21)

where l ∈ [1, s − 1].

Remark 2. Condition (26.16) can be satisfied for a(ϕ) _≡ const only for a

finite value of l.

In this case, the change of variables (26.17) has finite smoothness. However,

if

a(ϕ) ≡ const, (26.22)

then the value of l is equal to s − 1, and, for s = ∞, the change of variables

(26.17) is infinitely differentiable. If condition (26.22) is satisfied, then Theorem

26.1 coincides with Theorem 23.1.

276 Investigation of a Dynamical System Chapter 4

The theorem below enables one to describe the behavior of solutions of system

(26.1) originating in the neighborhood of M.

Theorem 26.2. Suppose that X(x) ∈ Cr(Rn) and the system of equations

(26.1) has the invariant toroidal manifold (26.2), where f(ϕ) ∈ Cs(Tm) for

r ≥ s ≥ 2.

Also assume that the following conditions are satisfied:

(i) the matrix

∂f(ϕ)

∂ϕ

can be complemented to a 2π-periodic basis in Rn;

(ii) the variational equation for the manifold M satisfies the condition of exponential

stability (26.12);

(iii) inequality (26.16) is true.

Then one can indicate a sufficiently small δ > 0 such that, for every y0,

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

_x(t, y0) − f(ψt(ψ0))_ ≤ L2e

−γ1t_y0 − f(ϕ0)_ (26.23)

for all t ∈ R+ and certain L2 > 0 and γ1 > 0, where γ1 = γ1(δ) → γ and

_ϕ0 − ψ0_ → 0 as δ → 0.

Inequality (26.23) proves that a solution of system (26.1) originating in a

small neighborhood of the manifold M is exponentially attracted as t → +∞ to

the corresponding solution of this system originating on M. The proof of Theorem

26.2 repeats the proof of Theorem 24.2 word for word.

Parallel with the system of equations (26.1), we consider the perturbed system

of equations

dy

dt

= X(y) + εY (y), (26.24)

where Y ∈ Cr(Rn) and ε is a small positive parameter. Let us clarify the

behavior of solutions of this system originating in a small neighborhood of the

manifold M. In the local system of coordinates (ϕ, h), system (26.24) takes the

form

dϕ

dt

= a(ϕ) + A(ϕ, h)h + εL1(ϕ, h)Y (f(ϕ) + B(ϕ)h),

dh

dt

= P(ϕ, h)h + εL2(ϕ, h)Y (f(ϕ) + B(ϕ)h),

(26.25)

which coincides with (26.6) for ε = 0.

Section 26 Toroidal Manifold Filled with Trajectories of General Form 277

We define a function α(ϕ) by the inequality

sup

S1∈N1

min

_ψ_=1

2_

S1(ϕ)∂a(ϕ)

∂ϕ

+

1

2

∂S1(ϕ)

∂ϕ

a(ϕ)

_

ψ,ψ

3

_S1(ϕ)ψ,ψ_

≥ α(ϕ) (26.26)

and require that the following condition be satisfied for a certain integer p ∈

[1, s − 1]:

inf

ϕ∈Tm

[β(ϕ) + pα(ϕ)] > 0. (26.27)

According to the perturbation theory of invariant toroidal manifolds, the validity

of inequalities (26.12) and (26.27) is a sufficient condition [Sam4] for the

system of equations (26.25) to have the invariant torus

h = u(ϕ, ε), ϕ∈ Tm, (26.28)

for all ε ∈ [0, ε0], where ε0 > 0 is sufficiently small, u ∈ Cp−1

Lip (Tm), and

lim

ε→0

_u_p−1,Lip = 0. (26.29)

Let p ≥ 2. Then the change of variables

h = u(ϕ, ε) + z,

where u is the function from (26.28), reduces the system of equations (26.25) to

the form

dϕ

dt

= a(ϕ) + F(ϕ, ε) + A(ϕ, z, ε)z,

dz

dt

= P(ϕ, z, ε)z, (26.30)

where F ∈ Cp−1

Lip (Tm), (A, P) ∈ Cp−2

Lip (Tm × Kδ0) for every ε ∈ [0, ε0],

δ0 = δ0(ε0) → 0 as ε0 → 0, and

lim

ε→0

_F_p−1,Lip = 0, lim

ε→0

_A(ϕ, z, ε) − A(ϕ, z)_p−2,Lip = 0,

lim

ε→0

_P(ϕ, z, ε) − P(ϕ, z)_p−2,Lip = 0. (26.31)

For every ε ∈ [0, ε0], the system of equations (26.30) has the form (26.25).

It follows from inequalities (26.11) and (26.14), which determine the functions

β(ϕ) and α1(ϕ), that, for small changes in the values of a, P, and

∂a

∂ϕ

, the

changes in the values of β and α1 are also small. Therefore, the limit relations

278 Investigation of a Dynamical System Chapter 4

(26.31) imply that the functions β(ϕ, ε) and α1(ϕ, ε) defined with the use of

a(ϕ)+F(ϕ, ε) and P(ϕ, ε, 0) according to formulas (26.11) and (26.14) satisfy

the limit relations

lim

ε→0

[β(ϕ, ε) − β(ϕ)] = lim

ε→0

[α1(ϕ, ε) − α1(ϕ)] = 0 (26.32)

uniformly in ϕ ∈ Tm. Consequently, one can find ε0 = ε0(l) > 0 such that

inequalities (26.12) and (26.16) yield the following estimates for l ∈ [1, s − 1]:

inf

ε∈[0,ε0]

inf

ϕ∈Tm

β(ϕ, ε) > 0, (26.33)

inf

ε∈[0,ε0]

inf

ϕ∈Tm

[β(ϕ, ε) − lα1(ϕ, ε)] > 0. (26.34)

Since the right-hand side of the system of equations (26.30) belongs to the

space Cp−2(Tm × Kδ0) for every ε ∈ [0, ε0], we deduce from (26.34) the following

inequality for p ≥ 3:

inf

ε∈[0,ε0]

inf

ϕ∈Tm

[β(ϕ, ε) − l1α1(ϕ, ε)] > 0, (26.35)

where

l1 = min{l, p − 2} ≥ 1. (26.36)

For p ≥ 3, these arguments allow us to apply Theorem 26.1 to the system of

equations (26.30). As a result, we obtain the following statement:

Theorem 26.3. Suppose that the conditions of Theorem 26.1 are satisfied and

inequality (26.27) holds for p ≥ 3. Then there exist positive constants μ and

ε0 = ε0(p) and a matrix Ψ(ϕ, z, ε) that belongs to the space Cl1−1

Lip (Tm × Kμ)

for every ε ∈ [0, ε0] and satisfies the relations

lim

ε→0

_Ψ(ϕ, z, ε) − Ψ(ϕ, z, 0)_l1−1,Lip = 0 (26.37)

such that the change of variables

ϕ = ψ +Ψ(ψ, z, ε)z (26.38)

reduces the system of equations (26.30) to the form

dψ

dt

= a(ψ) + F(ψ, ε),

dz

dt

= P(ψ +Ψ(ψ, z, ε)z, z, ε)z. (26.39)

Section 26 Toroidal Manifold Filled with Trajectories of General Form 279

Condition (26.12) guarantees that

_z(t, ε)_ ≤ L3e

−βt_z(0, ε)_, t∈ R+, (26.40)

for solutions of system (26.39) such that _z(0, ε)_ ≤ μ for ε ∈ [0, ε0]. Here,

L3 = const and β = β(μ, ε) are certain positive quantities and β(μ, ε) → β0

as (μ, ε) → 0.

As noted above, the verification of inequalities (26.11), (26.14), and (26.26)

encounters certain difficulties. Therefore, it is natural to express the conditions of

Theorem 26.3 in terms of inequalities of the form (26.21). For this purpose, we

assume that the fundamental matrices of solutions Ωt

0(P) and Ωt

0

_ ∂a

∂ϕ

_

of the

corresponding equations of systems (26.19) satisfy the inequalities

_Ωt

0(P)_ ≤ Le

−β0t, t∈ R+,

___

Ωt

0

_ ∂a

∂ϕ

____≤ L1eα0|t|

, t∈ (−∞,∞),

(26.41)

where β0 and α0 are positive constants.

To express the conditions that guarantee the reducibility of the system of equations

(26.30) to the form (26.39) in terms of the parameters β0 and α0, we use

the following statement:

Lemma 26.1. If a ∈ C1(Tm), P ∈ C1(Tm), and inequalities (26.41) are

satisfied, then

inf

S∈N

max

_h_=1

2_

S(ϕ)P(ϕ) +

1

2

∂S(ϕ)

∂ϕ

a(ϕ)

_

h, h

3

_S(ϕ)h, h_

≤ −

_

β0 − α0

2

(1−L−2)

_

.

Proof. For μ > 0, the fundamental matrix Ωt

0(P +(β0 −μ)E) of solutions

of the equation

dx

dt

= [P(ψt(ϕ)) + (β0 − μ)E]x

satisfies the inequality

_Ωt

0(P + (β0 − μ)E)_ ≤ Le

−β0te(β0−μ)t = Le

−μt, t∈ R+.

Let

S(ϕ) =

∞ _

0

(Ωτ

0[P + (β0 − μ)E])TΩτ

0[P + (β0 − μ)E]dτ. (26.42)

280 Investigation of a Dynamical System Chapter 4

It follows from the calculations carried out in [Sam4] (Chapter 3, Section 5) that

˙S

(ϕ) = −PT (ϕ)S(ϕ) − S(ϕ)P(ϕ) − 2(β0 − μ)S(ϕ) − E,

where

˙S

(ϕ) = d

dt

S(ψt(ϕ))|t=0.

For the matrix Yν(t) = ∂

∂ϕν

Ωt

0(P), we have

_Yν(t)_ =

___

_t

0

Ωt

τ (P)∂P(ψτ (ϕ))

∂ψ

∂ϕτ (ϕ)

∂ϕν

Ωτ

0(P)dτ

___

≤

_

L2L1K1

m

α0

_

e

−(β0−α0)t, t∈ R+,

where

max

ϕ∈Tm

___

∂P(ϕ)

∂ϕj

___

≤ K1, j= 1,m.

Taking this estimate into account, we get

___

∂

∂ϕν

{Ωt

0(P + (β0 − μ)E)TΩt

0(P + (β0 − μ)E)}

___ ≤

2L3L1K1

m

α0

e(α0−2μ)t, t∈ R+. (26.43)

Choosing μ from the condition

α0 < 2μ, (26.44)

one can verify that the integral

Iν =

∞ _

0

∂

∂ϕν

{(Ωτ

0(P + (β0 − μ)E))TΩ0(P + (β0 − μ)E)}dτ

is majorized by a convergent one and, furthermore,

_Iν_ ≤ 2L3L1K1

m

α0(2μ − α0) .

Section 26 Toroidal Manifold Filled with Trajectories of General Form 281

This is sufficient for the matrix S(ϕ) to belong to the space C1(Tm). Then

˙S

(ϕ) = dS(ϕ)

dϕ

a(ϕ)

and

Δ(ϕ) ≡ max

_h_=1

2_

S(ϕ)P(ϕ) +

1

2

∂S(ϕ)

∂ϕ

a(ϕ)

_

h, h

3

_S(ϕ)h, h_

= −

_

β0 − μ +

1

2 max

_h_=1

< S(ϕ)h, h >

_

. (26.45)

Since

max

_h_=1

_S(ϕ)h, h_ =

∞ _

0

max

_h_=1

_Ωτ

0(P + (β0 − μ)E)h,Ωτ

0(P + (β0 − μ)E)_ dτ

≤ L2

∞ _

0

e

−2μτ dτ =

L2

2μ

,

it follows from (26.45) that

Δ(ϕ) ≤ −[β0 − μ(1−L−2)].

Passing to the limit in the last estimate as μ → 1

2α0, we get

inf

S∈N

max

_h_=1

2_

S(ϕ)P(ϕ) +

1

2

∂S(ϕ)

∂ϕ

a(ϕ)

_

h, h

3

_S(ϕ)h, h_

≤ −

_

β0 − α0

2

(1−L−2)

_

.

The positive definiteness of the matrix S(ϕ) was proved in [Sam4] (Chapter 3,

Section 5).

According to Lemma 26.1, we can take the constant β0 − 1

2α0(1−L−2) as

the function β(ϕ) in inequalities (26.27) and (26.34). It is also obvious that the

values α1(ϕ) and α(ϕ) in these inequalities can be replaced by the constant α0.

282 Investigation of a Dynamical System Chapter 4

Thus, in order that the statement of Theorem 26.3 be true, it is sufficient to require

that the following inequalities be satisfied:

β0 − 1

2α0(1−L−2) − lα0 > 0 for l ∈ [1, s − 1],

β0 − 1

2α0(1−L−2) − pα0 > 0 for p ∈ [3, s − 1].

In order that both inequalities be satisfied, it suffices to set l = p and require that

β0

α0

> p+

1

2

(1−L−2) (26.46)

for integer p ∈ [3, s − 1].

Corollary 3. Suppose that the smoothness conditions presented in Theorem

26.3 are satisfied and inequalities (26.41) with constants satisfying condition

(26.46) are true. Then, for any ε ∈ [0, ε0], the change of variables (26.38) with

matrix Ψ(ψ, z, ε) ∈ Cp−3

Lip (Tm × Kμ) reduces the system of equations (26.30) to

the form (26.39).

As in the case of Theorem 26.1, Theorem 26.3 can be formulated in the form

of a statement related to the system of equations (26.24) and its invariant torus

M(ε), namely

y = f(ϕ) + u(ϕ, ε), ϕ∈ Tm, ε∈ [0, ε0].

Hence, one can easily establish the principle of reducibility in problems of

stability of solutions of system (26.24) originating on M(ε). According to this

principle, the stability or the asymptotic stability of these solutions is determined

by their stability or asymptotic stability on M(ε).

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