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20. Investigation of Second-Order Oscillation Systems

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Consider a system of weakly connected oscillators with slowly varying parameters

of the form

d2xν

dt2 + ω2

ν(τ )xν = εfν

, τ

, (20.1)

where ν = 1,m, m ≥ 2, τ = εt is “slow” time, fν are polynomials in x =

(x1, . . . , xm) and

_dx1

, . . . ,

dxm

of degree not higher than N ≥ 0

with coefficients l ≥ 1 times continuously differentiable with respect to τ ∈

[0, L], ων(τ ) > 0 for any τ ∈ [0, L] and ν = 1,m, and ε is a small positive

Section 20 Investigation of Second-Order Oscillation Systems 225

parameter. Systems of the form (20.1) are encountered in numerous problems of

nonlinear mechanics [Mit3, Mit4].

In the present section, we apply the results obtained above to the investigation

of properties of solutions of Eqs. (20.1). For this purpose, we rewrite these

equations in the form of a system:

dxν

= yν,

dyν

= −ω2

ν(τ )xν + εfν(x, y, τ ), ν= 1,m.

(20.2)

Using the general scheme of the investigation of oscillations [BoM2, Mit2, MiS1–

MiS3], we pass to amplitude–phase variables rν and ϕν in Eqs. (20.2) according

to the formulas

xν = rν sin ϕν, yν = rνων(τ ) cos ϕν. ν= 1,m, (20.3)

As a result, we obtain the equations

drν

dτ

= −rν cos2 ϕν

dτ

ln ων(τ) + fν cos ϕν

ων(τ ) , ν= 1,m,

dϕν

dτ

= ων(τ )

+ sin ϕν cos ϕν

dτ

ln ων(τ ) − fν sin ϕν

rνων(τ ) , (20.4)

where

fν = fν(r sin ϕ, rω(τ ) cos ϕ, τ )

≡ fν(r1 sin ϕ1, . . . , rm sin ϕm, r1ω1(τ ) cos ϕ1, . . . , rmωm(τ ) cos ϕm, τ).

We construct the following system averaged over all angular variables ϕ =

(ϕ1, . . . , ϕm):

drν

dτ

= −rν

dτ

ln ων(τ) + gν(r, τ),

dϕν

dτ

= ων(τ )

+ _gν(r, τ)

rν

, (20.5)

where

gν =

(2π)−m

ων(τ )

_2π

. . .

_2π

fν(r sin ϕ, rω(τ ) cos ϕ, τ ) cos ϕνdϕ1 . . . dϕm,

_gν = −(2π)−m

ων(τ )

_2π

. . .

_2π

fν(r sin ϕ, rω(τ ) cos ϕ, τ ) sin ϕνdϕ1 . . . dϕm. (20.6)

226 Integral Manifolds Chapter 3

Analyzing relations (20.6), we establish that gν and _gν are not identically equal

to zero if and only if fν

, τ

have terms of the form

x2,

_dx

, τ

_dxν

+ fII

x2,

_dx

, τ

xν,

where

x2 = (x21

, . . . , x2

m),

_dx

__dx1

, . . . ,

_dxm

_2_

In this case, the function fI

ν introduces nonzero terms in gν, and the function

fII

ν introduces nonzero terms in _gν. Therefore,

gν(r, τ) = rνaν(r2, τ), _gν(r, τ) = rνbν(r2, τ),

where aν(z, τ) and bν(z, τ) are polynomials in z of degree not higher than

, and E{k} is the integer part of the number k. Thus, setting r2ν

= zν,

ν = 1,m, we can rewrite the averaged system (20.5) in the form

dzν

dτ

− d

dτ

ln ων(τ) + 2aν(z, τ)

zν,

dϕν

dτ

= ων(τ )

+ bν(z, τ), ν= 1,m.

(20.7)

For Eqs. (20.1), we introduce the initial conditions

xν|t=0 = x0ν

dxν

|t=0 = ˙x0ν

, (x0ν

)2 + (˙ x0ν

)2 > 0, ν= 1,m. (20.8)

Then the corresponding initial conditions for the amplitude–phase variables take

the form

zν|τ=0 = z0

ν , ϕν

|τ=0 = ϕ0ν

, ν= 1,m, (20.9)

where z0

ν = (x0ν

)2 +

_ 1

ων(0)

˙ x0ν

> 0 and ϕ0ν

is one of solutions of the system

of equations

x0ν

ν sin ϕ0ν

, ˙ x0ν

νων(0) cos ϕ0ν

. (20.10)

Note that the averaged Cauchy problem (20.7), (20.9) decomposes into the following

two problems:

dzν

dτ

− d

dτ

ln ων(τ) + 2aν(z, τ)

zν, zν|τ=0 = z0

ν, ν= 1,m, (20.11)

dϕν

dτ

= ων(τ )

+ bν(z, τ), ϕν|τ=0 = ϕ0ν

, ν= 1,m. (20.12)

Section 20 Investigation of Second-Order Oscillation Systems 227

z(τ, z0) = (z1(τ, z0), . . . , zm(τ, z0)), z0 = (z0

1, . . . , z0m

is a solution of the Cauchy problem (20.11), then a solution of the Cauchy problem

(20.12) is given by the formula

ϕν(τ, z0, ϕ0ν

, ε) = ϕ0ν

_τ

[ων(τ) + εb(z(τ, z0), τ)]dτ, ν = 1,m.

Theorem 20.1. Suppose that the following conditions are satisfied:

(i) ω(τ ) ∈ Cp−1

[0,L], p ≥ m;

(ii) det (WT

p (τ )Wp(τ )) _= 0 ∀ τ ∈ [0, L];

(iii) there exists a solution z = z(τ, z0) of problem (20.11) defined on [0, L].

Then one can find constants c1 and ε0 > 0 such that a solution x = x(t, ε)

of the Cauchy problem (20.1), (20.8) is defined for all t ∈ [0, Lε−1] and ε ∈

(0, ε0] and

|xν(t, ε) −

zν(εt, z0) sin ϕν(εt, z0, ϕ0ν

, ε)|

___

dxν(t, ε)

−

zν(εt, z0)ων(εt) cos ϕν(εt, z0, ϕ0ν

, ε)

___

≤ c1ε

p, ν= 1,m. (20.13)

Proof. It is obvious that each component zν(τ, z0) of the solution z =

z(τ, z0) of the Cauchy problem (20.11) does not vanish on the segment [0, L].

Denote

min

1≤ν≤m

min

τ∈[0,L]

zν(τ, z0) = 2ρ > 0,

max

1≤ν≤m

max

τ∈[0,L]

zν(τ, z0) = Δ.

This implies that the curve z = z(τ, z0) lies in the cube

Π = {z : z ∈ Rm, ρ ≤ zν ≤ Δ+ρ, ν = 1,m}

228 Integral Manifolds Chapter 3

together with its ρ-neighborhood for any τ ∈ [0, L]. Moreover, the right-hand

side of system (20.4) satisfies all conditions of Theorem 2.1. Therefore, the solution

(r(τ,

√

z0, ϕ0, ε); ϕ(τ,

√

z0, ϕ0, ε)) of the Cauchy problem (20.4), (20.9)

is defined for all τ ∈ [0, L] and ε ∈ (0, ε0] (ε0 > 0 is sufficiently small) and

satisfies the estimates

|rν(τ,

√

z0, ϕ0, ε) −

zν(τ, z0)| + |ϕν(τ,

√

z0, ϕ0, ε) − ϕν(τ, z0, ϕ0ν

, ε)|

≤ cε

p, ν= 1,m, (20.14)

where c is a certain constant independent of ε. Relations (20.3) and (20.14) yield

inequality (20.13) with

c1 = c(1 + Δ)(1 + max

1≤ν≤m

max

τ∈[0,L]

ων(τ )).

Theorem 20.1 is proved.

For Eqs. (20.1), we now introduce boundary conditions of the form

xν|t=tν = x0ν

dxν

|t=tν = ˙x0ν

, ν= 1,m, (20.15)

where 0 ≤ t1 < t2 < ... < tm ≤ Lε−1 and (x0ν

)2 + (˙ x0ν

)2 > 0. Problem

(20.1), (20.15) may arise in the case where, in the course of the investigation of

properties of a system of oscillators, one can determine the values of x and

at given time for only one oscillator (e.g., the number of measuring devices is

insufficient).

In the variables r and ϕ, conditions (20.15) can be rewritten as follows:

rν|τ=τν =

ν, ϕν|τ=τν = ϕ0ν

, ν= 1,m, (20.16)

where τν = εtν, and z0

ν and ϕ0ν

have the same meaning as in (20.9).

The multipoint problem (20.1), (20.15) generates the averaged problem

dzν

dτ

− d

dτ

ln ων(τ) + 2aν(z, τ)

zν, zν|τ=τν = z0

ν, ν= 1,m, (20.17)

dϕν

dτ

= ων(τ )

+ bν(z, τ), ϕν|τ=τν = ϕ0ν

, ν= 1,m, (20.18)

which is obviously much simpler than problem (20.4), (20.16). Indeed, if a solution

z = z(τ, y0), z(0, y0) = y0 of problem (20.17) is obtained, then problem

Section 20 Investigation of Second-Order Oscillation Systems 229

(20.18) decomposes into m mutually independent Cauchy problems whose solutions

ϕν = ϕν(τ, y0, ψ0

ν, ε), ϕν(0, y0, ψ0

ν, ε) = ψ0

ν are determined as follows:

ϕν(τ, y0, ψ0

ν, ε) = ψ0

ν +

_τ

[ων(τ) + εbν(z(τ, y0), τ)]dτ,

where

ψ0

ν = ϕ0ν

− 1

_τν

[ων(τ) + εbν(z(τ, y0), τ)]dτ.

Theorem 20.2. Suppose that conditions (i) and (ii) of Theorem 20.1 are satisfied

and there exists a solution z = z(τ, y0), y0 = (y0

1, . . . , y0m

), of problem

(20.17) defined on [0, L] and such that the matrix

A =

_∂zν(τν, y0)

∂y0μ

ν,μ=1

is nondegenerate. Then, for every ε ∈ (0, ε1], where ε1 > 0 is sufficiently small,

there exists a unique solution x = x(t, ε) of problem (20.1), (20.15) defined for

all t ∈ [0, Lε−1] and such that

|xν(t, ε) −

zν(εt, y0) sin ϕν(εt, y0, ψ0

ν, ε)|

___

dxν(t, ε)

−

zν(εt, y0)ων(εt) cos ϕν(εt, y0, ψ0

ν, ε)

___

≤ c1ε

p, ν= 1,m, (20.19)

where c1 is a certain constant independent of ε.

Proof. By analogy with the proof of Theorem 20.1, we establish that the

curve z = z(τ, y0) lies in the cube Π together with its ρ-neighborhood. Further,

we use Theorem 8.2. The matrix S defined by (8.20) for problem (20.17), (20.18)

has the form

S =

A 0

B Em

where Em is the m-dimensional identity matrix, 0 is the zero matrix, and

B =

__τν

∂bν(z(τ, y0), τ)

∂z

∂z(τ, y0)

∂y0μ

dτ

ν,μ=1

230 Integral Manifolds Chapter 3

Since det S = detA _= 0, i.e., _S−1_ ≤ c2 = const, we conclude that,

according to Theorem 8.2, for every ε ∈ (0, ε1], where ε1 is sufficiently small,

there exists a unique solution (r(τ, ε); ϕ(τ, ε)) of problem (20.4), (20.16) that

satisfies the inequality

ν=1

[|rν(τ, ε) −

zν(τ, y0)| + |ϕν(τ, ε) − ϕν(τ, y0, ψ0

ν, ε)|] ≤ c2ε

for all τ ∈ [0, L] and ε ∈ (0, ε1]. Taking into account formulas (20.3) and the

last inequality, we get estimate (20.19). Theorem 20.2 is proved.

Corollary 6. If the polynomials fν

, τ

do not contain terms of the

form f

x2,

_dx

, τ

_dxν

, i.e., aν(z, τ) ≡ 0 ∀ν = 1,m, then problem

(20.17) decomposes into m Cauchy problems whose solutions are given by the

formulas

zν(τ, y0) = y0

νων(0)

ων(τ ) , y0

ν = z0

νων(τν)

ων(0) , ν= 1,m.

In this case, we have

detA = det diag

_ ω1(0)

ω1(τ1), . . . ,

ωm(0)

ωm(τm)

ν=1

ων(0)

ων(τν)

_= 0.

We have considered above one of the simplest versions of boundary conditions.

Note that Theorem 8.2 can also be used in the case where conditions (20.15)

are replaced by the more general conditions

x|t=t1 ,

|t=t1, . . . , x|t=tk ,

|t=tk, ε

= 0,

where 0 ≤ t1 < t2 < ... < tk ≤ Lε−1, k ≥ 2, and Φ is a 2m-dimensional

vector function.

Now assume that the following conditions are satisfied:

(1◦) the coefficients of the polynomials fν

, τ

are defined and bounded

together with their derivatives with respect to τ up to an order l ≥ 2 for

all τ ∈ R;

Section 20 Investigation of Second-Order Oscillation Systems 231

(2◦) the frequencies ων(τ ), ν = 1,m, and their derivatives up to an order

p ≥ m are uniformly bounded for all τ ∈ R and

det (WT

p (τ )Wp(τ )) ≥ c3 = const > 0, ων(τ ) ≥ c4 = const > 0;

(3◦) the averaged equations (20.5) for the slow variables r have a bounded solution

r = ξ(τ) = (ξ1(τ ), . . . , ξm(τ )) defined on the entire axis and such

that ξν(τ ) ≥ 2ρ = const > 0, ν = 1,m;

(4◦) the normal fundamental matrix Q(τ, τ ) of the variational system corresponding

to the solution r = ξ(τ ) satisfies the estimate

_Q(τ, τ )_ ≤ Ke

−γ(τ−τ) ∀τ ≥ τ ∈ R,

where K ≥ 1 and γ > 0 are certain constants;

(5◦) the following inequality is true:

K sup

ϕ,τ

___

∂

∂r

_a(ξ(τ ), ϕ, τ)

___

< 1,

where

_a(r, ϕ, τ) = (_a1(r, ϕ, τ ), . . . , _am(r, ϕ, τ )),

_aν(r, ϕ, τ) = −rν

cos2 ϕν − 1

_ d

dτ

ln ων(τ )

ων(τ )fν(r sin ϕ, rω(τ ) cos ϕ, τ ) − gν(r, τ ).

Under these conditions, in Sections 12–17 we have proved the existence and

studied properties of the asymptotically stable integral manifold r = R(ψ, τ, ε) =

(R1(ψ, τ, ε), . . . , Rm(ψ, τ, ε)) of system (20.4), on which the equations for the

fast variables ϕν, ν = 1,m, have the form

dϕν

dτ

ων(τ ) + sin ϕν cos ϕν

dτ

ln ων(τ )

− sin ϕν

Rν(ϕ, τ, ε)ων(τ )fν(R(ϕ, τ, ε) sin ϕ,R(ϕ, τ, ε)ω(τ ) cos ϕ, τ ). (20.20)

232 Integral Manifolds Chapter 3

It follows from the definition of integral manifold [MiLy] that if ϕ = ϕττ

0(ψ, ε),

ϕτ0

τ0(ψ, ε) = ψ ∈ Rm, is a solution of Eqs. (20.20), then

r = R(ϕττ

0(ψ, ε), τ, ε), ϕ= ϕττ

0(ψ, ε) (20.21)

is a solution of system (20.4) for all τ ∈ R. In this case, in view of formula

(20.3), we have the bounded solution

x = R(ϕττ

0(ψ, ε), τ, ε) sin ϕττ

0(ψ, ε),

y = R(ϕττ

0(ψ, ε), τ, ε)ω(τ ) cos ϕττ

0(ψ, ε)

of system (20.2), which is defined for τ ∈ R, ψ ∈ Rm, and ε ∈ (0, ε2], where

ε2 > 0 is sufficiently small. Thus, in the (2m+2)-dimensional space of variables

x, y, τ, and ε, the relation

(x; y) = Γ(ψ, τ, ε) ≡ (R(ψ, τ, ε) sin ψ;R(ψ, τ, ε)ω(τ ) cos ψ) (20.22)

is the equation of a surface that possesses the following property: if (x0; ˙ x0) ∈

Γ(ψ, τ0, ε), then the solution (xtt

0(x0, x˙ 0, ε); yt

t0(x0, x˙ 0, ε)) of the Cauchy problem

dxν

= yν,

dyν

= −ω2

ν(τ )xν + εfν(x, y, τ ), ν= 1,m,

xν|t=t0 = x0, yν|t=t0 = ˙x0, t0 = τ0ε

−1, (20.23)

is defined for all t ∈ R and ε ∈ (0, ε2], bounded, and lying on the surface Γ.

The asymptotic stability of the integral manifold r = R(ψ, τ, ε) of system

(20.4) means (see Theorem 15.1) that if r|τ=τ0 = r0 lies in a certain small

neighborhood of the point R(ψ, τ0, ε), then, as τ → ∞, the slow variables

rτ

τ0(r0, ψ0, ε) of every solution

(rτ

τ0(r0, ψ0, ε); ϕττ

0(r0, ψ0, ε)),

rτ0

τ0 (r0, ψ0, ε) = r0; ϕτ0

τ0(r0, ψ0, ε) = ψ0, ψ0 ∈ Rm,

of system (20.4) tend exponentially to the curve r = R(ϕττ

0(r0, ψ0, ε), τ, ε),

which lies on the surface r = R(ψ, τ, ε). Taking into account formula (20.3),

we establish that, as t → ∞, every solution (xtt

0(x0, x˙ 0, ε); yt

t0(x0, x˙ 0, ε)) of

the Cauchy problem (20.23) tends exponentially to the curve

(x; y) = (R(ϕττ

0(r0, ψ0, ε), τ, ε) sin ϕττ

0(r0, ψ0, ε),

R(ϕττ

0(r0, ψ0, ε), τ, ε)ω(τ ) cos ϕττ

0(r0, ψ0, ε), τ = εt,

Section 21 Weakening of Conditions in the Theorem on Integral Manifold 233

which lies on the surface Γ under the condition that the point (x0, x˙ 0) lies in a

small neighborhood of the point Γ(ψ0, τ0, ε). Here,

r0 = (r0

1, . . . , r0m

), ψ0 = (ψ0

1, . . . , ψ0m

ν =

(x0ν

)2 + (˙ x0ν

−1

ν (τ0))2,

and ψ0

ν is one of solutions of the system of equations

x0ν

= r0

ν sin ψ0

ν, x˙ 0ν

= r0

νων(τ0) cos ψ0

ν.

This reasoning enables us to apply Theorems 15.1 and 16.3 to system (20.2) and

obtain the following corollary of these statements:

Theorem 20.3. If conditions (1◦)–(5◦) are satisfied, then one can find a sufficiently

small ε2 > 0 and a sufficiently large constant c5 such that, for all

(τ, ε) ∈ R × (0, ε2], there exists the asymptotically stable integral manifold

(x; y) = Γ(ψ, τ, ε) of Eqs. (20.2) for which the function Γ is 2π-periodic in ψ,

k = min{p; l} − 1 times continuously differentiable with respect to (ψ, τ, ε) ∈

Rm × R × (0, ε2], and such that

__ _

Dsψ

∂q

∂τq

∂q

∂εq (Γ(ψ, τ, ε) − _Γ(ψ, τ))

___

≤ c5ε

−q−2q,

where 0 ≤ s + q + q ≤ k and _Γ(ψ, τ) = (ξ(τ ) sin ψ; ξ(τ )ω(τ ) cos ψ).

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