24. Variational Equation and Theorem on Attraction to Quasiperiodic Solutions

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Consider the following variational equation for a solution (22.3) of system

(22.2):

dy

dt

= ∂X(f(λt + ψ0))

∂x

y. (24.1)

Assume that f ∈ C1(Tm) and the matrix

∂f(ϕ)

∂ϕ

can be complemented to a

2π-periodic basis in Rn. It follows from Eq. (24.1) that the system of functions

∂f(λt + ψ0)

∂ϕν

, ν = 1,m, forms a system of linearly independent solutions of

Eq. (24.1). We perform a change of variables in (24.1), namely, instead of y, we

introduce new variables (c, h) = (c1, . . . , cm, h1, . . . , hn−m) according to the

formulas

y = ∂f(λt + ψ0)

∂ϕ

c + B(λt + ψ0)h, (24.2)

where B(ϕ) is the matrix from formula (22.7). As a result, we obtain the system

of equations

dc

dt

= Q(λt + ψ0)h,

dh

dt

= P(λt + ψ0)h, (24.3)

where P(ϕ) is matrix (22.17). The relationship between systems (24.3) and

(22.16) is obvious, namely, the equations for h in system (24.3) are obtained

from (22.16) for ϕ = λt+ψ0, and, vice versa, the closure of the second equation

in system (24.3) with respect to t leads to system (22.16).

In what follows, the system of equations (22.16) is called the variational system

of equations for the invariant torus (22.4) of the dynamical system (22.2) that

is filled with the quasiperiodic trajectory of this system. The statement below

describes the dependence of the variational equation (22.16) on the matrix B(ϕ).

262 Investigation of a Dynamical System Chapter 4

Theorem 24.1. Under the assumptions made above, two arbitrary equations

(22.16) are Cp−1-equivalent.

Proof. Consider two changes of variables of the form (24.2) defined by matrices

B ∈ Cp(Tm) and B1 ∈ Cp(Tm), respectively. Denote by P(ϕ) and P1(ϕ)

the matrices of Eq. (24.3) obtained as a result of these changes of variables. Since

both systems of equations are obtained with the use of changes of variables of the

form (24.2) on the basis of the same equation (24.1), their fundamental matrices

of solutions are related by the identity

_∂f(ψt)

∂ϕ

,B(ψt)

_&

E Q

0 Ωt

0(P)

'

=

_∂f(ψt)

∂ϕ

,B1(ψt)

_&

E Q1

0 Ωt

0(P1)

'

C1, (24.4)

where C1 is a nondegenerate constant matrix.

Let &

L1(ψt)

L2(ψt)

'

be the matrix inverse to

_∂f(ψt)

∂ϕ

,B(ψt)

_

, i.e.,

L1(ψt)∂f(ψt)

∂ϕ

= E, L1(ψt)B(ψt) = O,

L2(ψt)∂f(ψt)

∂ϕ

= O, L2(ψt)B(ψt) = E,

where E and O are, respectively, the identity matrix and the zero matrix of

the corresponding dimensions. Using (24.4), one can easily obtain the following

identity:

&

E Q

0 Ωt

0(P)

'

=

&

E L1(ψt)B1(ψt)

0 L2(ψt)B1(ψt)

'&

E Q1

0 Ωt

0(P1)

'

C1. (24.5)

Analyzing relation (24.5), we conclude that the matrix L2(ψt)B1(ψt) is nondegenerate

and

Ωt

0(P) = L2(ψt)B1(ψt)Ωt

0(P1)C2, (24.6)

Section 24 Variational Equation and Theorem on Attraction 263

where C2 is a nondegenerate constant matrix. Identity (24.6) proves that the

change of variables

h = L2(ϕt)B1(ϕt)h1 (24.7)

transforms system (22.16) with the matrix P(ϕ) to system (22.16) with the matrix

P1(ϕ). Since L2B1 ∈ Cp−1(Tm), the change of variables (24.7) realizes a

Cp−1-homeomorphism from (ϕ, h) onto (ϕ, h1).

Theorem 24.1 is proved.

Let us clarify the behavior of solutions originating in a neighborhood of the

torus M: x = f(ϕ), ϕ ∈ Tm. For this purpose, we define the distance from the

point y0 to M by the formula

ρ(y0,M) = inf

x∈M

_y0 − x_.

Theorem 24.2. Suppose that the smoothness conditions for the function X =

X(x) presented in Section 22 are satisfied and the system of equations (22.2) has

a quasiperiodic solution x = f(λt) ∈ Cs(λ) for r ≥ s ≥ 2. Also assume that

the matrix

∂f(ϕ)

∂ϕ

can be complemented to a 2π-periodic basis in Rn, and the

variational equation for the invariant torus M satisfies the condition of exponential

stability (22.19).

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

satisfying the inequality ρ(y0,M) ≤ δ, there exist ϕ0 ∈ Tm and ψ0 ∈ Tm such

that

_x(t, y0) − f(ϕt)_ ≤ Ke

−γ1t_y0 − f(ϕ0)_ (24.8)

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

_ψ0 − ϕ0_ → 0 as δ → 0.

Proof. We choose δ > 0 so small that the inequality ρ(y0,M) ≤ δ yields

the inequality _h0_ ≤ μ, where μ is the constant from Theorem 23.1 and

(ϕ0, h0) are the local coordinates of the point y0, i.e.,

y0 = f(ϕ0) + B(ϕ0)h0. (24.9)

The possibility of such a choice of δ is guaranteed by inequality (22.10). Let

ψ0 = ϕ0 + v(ϕ0, h0), (24.10)

264 Investigation of a Dynamical System Chapter 4

where the function v = v(ϕ, h) is defined by (23.36), and let ψt = λt + ψ0,

where ψ0 is defined by (24.10).

Let us estimate the difference x(t, y0) − f(ϕt). It follows from (24.9) that

x(t, y0) = f(ϕt) + B(ϕt)ht, t∈ R+, (24.11)

where ϕt, ht is a solution of system (22.12) that takes the value ϕ0, h0 for t = 0.

It follows from (24.10) that

ϕ0 = ψ0 + u(ϕ0, h0),

where u(ϕ, h) is the function of the change of variables (23.1), which reduces

system (22.20) to the form (23.2).

According to Theorem 23.1, the solutions ϕt, h(t) and ψt, ht are related as

follows:

ϕt = ψt + u(ψt, ht), h(t) = ht, t∈ R+. (24.12)

Then relations (24.11) and (24.12) yield the following estimate for all t ∈ [0,∞):

_x(t, y0) − f(ψt)_ ≤ _f(ψt + u(ψt, ht) − f(ψt)_ + _B(ϕt)__ht_

≤ K_u(ψt, ht)_ + c1_ht_ ≤ c2_ht_

where c2 is a certain positive constant.

Since ht = XΦ

t h0, where XΦ

t is a solution of Eq. (23.4) for F = Φ(ψ, g),

g = h, we conclude that ht satisfies the following estimate of the form (23.8):

_ht_ ≤ Le

−γ1t_h0_.

This yields

_x(t, y0) − f(ψt)_ ≤ Lc2e

−γ1t_h0_, t ∈ R+. (24.13)

It follows from (24.9) that

_h0_ = _Γ−1

0 (ϕ0)BT (ϕ0)[y0 − f(ϕ0)]_ ≤ c3_y0 − f(ϕ0)_, (24.14)

where c3 is a certain positive constant. Inequalities (24.13) and (24.14) yield

estimate (24.8). Since γ1 = γ1(μ) → 0 and _ψ0 − ϕ0_ → 0 as μ → 0, we

have γ1 → γ and _ψ0 − ϕ0_ → 0 as δ → 0, which completes the proof of

Theorem 24.2.

Section 24 Variational Equation and Theorem on Attraction 265

Below, we present two corollaries of Theorem 24.2.

Corollary 1. Under the conditions of Theorem 24.2, the quasiperiodic solutions

x = f(λt + ψ), ψ ∈ Tm, are Lyapunov stable.

Indeed, let y0 be an arbitrary point of the ball _y − x0_ < δ, where δ

is a sufficiently small positive number and x0 = f(ψ0). The local coordinates

(ϕ0, h0) of the point y0 are determined from the equation

y0 − f(ψ0) = f(ϕ) − f(ψ0) + B(ψ0)h +

_

B(ϕ) − B(ψ0)

_

h.

We rewrite this equation in the form

y0 − f(ψ0) = ∂f(ψ0)

∂ϕ

(ϕ − ψ0) + B(ψ0)h + D(ϕ, h), (24.15)

where D(ϕ, h) denotes a value of higher order of smallness as compared with

_ϕ − ψ0_ + _h_. It follows from the inequality _y0 − x0_ ≤ δ, where δ is

small, that the solution ϕ = ϕ0, h = h0 of Eq. (24.15) satisfies the conditions

_ϕ0 − ψ0_ ≤ δ1(δ), _y0 − f(ϕ0)_ ≤ δ1(δ),

where δ1(δ) → 0 as δ → 0.

According to Theorem 24.2, the following inequality holds:

_x(t, y0) − f(λt + ψ

0)_ ≤ Ke

−γ1t_y0 − f(ϕ0)_, t∈ R+,

where _ψ

0 − ϕ0_ → 0 as δ → 0.

This yields the following estimate for t ∈ R+:

_x(t, y0) − x(t, x0)_ ≤ _x(t, y0) − f(λt + ψ

0)_ + _f(λt + ψ

0)

− f(λt + ϕ0)_ + _f(λt + ϕ0) − f(λt + ψ0)_

≤ K_y0 − f(ϕ0)_ + K1(_ψ

0 − ϕ0_ + _ϕ0 − ψ0_)

≤ (K + K1)δ1(δ) + K1_ψ

0 − ϕ0_, (24.16)

where K1 is a constant that depends only on the first-order derivatives of the

function f(ϕ). Taking into account that _ψ

0 − ϕ0_ → 0 as δ → 0 and using

inequality (24.16), for any ε > 0 one can choose δ = δ(ε) > 0 so small that the

right-hand side of (24.16) is less than ε. Thus, the stability of the quasiperiodic

solution x(t, x0) = f(λt + ψ0) is proved.

266 Investigation of a Dynamical System Chapter 4

Corollary 2. Under the conditions of Theorem 24.2, for an arbitrary function

F = F(x) satisfying the HЁolder condition and an arbitrary solution x = x(t, y0)

for which ρ(y0,M) ≤ δ, the following limit relation holds uniformly in t ∈ R+:

lim

T→∞

1

T

_t+T

t

F(x(t, y0))dt = (2π)−m

_2π

0

. . .

_2π

0

F(f(ϕ))dϕ1 . . . dϕm. (24.17)

Note that the property of the dynamical system (22.2) expressed by equality

(24.17) characterizes the ergodicity of semitrajectories in a neighborhood of the

manifold M.

To prove relation (24.17), it is necessary to use estimate (24.8), which implies

that

_F(x(t, y0)) − F(f(ψt))_ ≤ c4e

−γ1βt, t∈ R+, (24.18)

where β is the HЁolder index of the function F(x). Inequality (24.18) yields

1

T

_t+T

t

F(x(t, x0))dt =

1

T

_t+T

t

F(f(ψt))dt + α(t, T), (24.19)

where

_α(t, T)_ ≤ c4

T

_t+T

t

e

−γ1βtdt = c4

T

e−γ1βt(1 − e−γ1βT )

γ1β

. (24.20)

Inequality (24.20) implies that the following relation holds uniformly in t ∈ R+:

lim

T→∞

α(t, T) = 0.

Equality (24.17) can easily be obtained from relations (24.19) and (24.20), and

the mean-value theorem for a quasiperiodic function.

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