On invariant torus of weakly connected systems of differential equations
We consider a family of systems of differential equations depending on a sufficiently small parameter with zero value of which we obtained a couple of independent systems. We used the method of Green – Samoilenko function to construct an invariant manifold of the pertubed system and presented some...
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2005
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irk-123456789-1780182021-02-18T01:27:55Z On invariant torus of weakly connected systems of differential equations Elnazarov, A. We consider a family of systems of differential equations depending on a sufficiently small parameter with zero value of which we obtained a couple of independent systems. We used the method of Green – Samoilenko function to construct an invariant manifold of the pertubed system and presented some examples for application. Розглянуто сiм’ю систем диференцiальних рiвнянь, що залежать вiд достатньо малого параметра, яка є парою незалежних систем, якщо значення параметра дорiвнює нулю. Використано метод функцiї Грiна – Самойленка для побудови iнварiантного многовиду збуреної системи та наведено приклади. 2005 Article On invariant torus of weakly connected systems of differential equations / A. Elnazarov // Нелінійні коливання. — 2005. — Т. 8, № 4. — С. 468-489. — Бібліогр.: 12 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/178018 517.9 en Нелінійні коливання Інститут математики НАН України |
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We consider a family of systems of differential equations depending on a sufficiently small parameter
with zero value of which we obtained a couple of independent systems. We used the method of Green –
Samoilenko function to construct an invariant manifold of the pertubed system and presented some examples for application. |
| format |
Article |
| author |
Elnazarov, A. |
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Elnazarov, A. On invariant torus of weakly connected systems of differential equations Нелінійні коливання |
| author_facet |
Elnazarov, A. |
| author_sort |
Elnazarov, A. |
| title |
On invariant torus of weakly connected systems of differential equations |
| title_short |
On invariant torus of weakly connected systems of differential equations |
| title_full |
On invariant torus of weakly connected systems of differential equations |
| title_fullStr |
On invariant torus of weakly connected systems of differential equations |
| title_full_unstemmed |
On invariant torus of weakly connected systems of differential equations |
| title_sort |
on invariant torus of weakly connected systems of differential equations |
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Інститут математики НАН України |
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2005 |
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http://dspace.nbuv.gov.ua/handle/123456789/178018 |
| citation_txt |
On invariant torus of weakly connected systems of differential equations / A. Elnazarov // Нелінійні коливання. — 2005. — Т. 8, № 4. — С. 468-489. — Бібліогр.: 12 назв. — англ. |
| series |
Нелінійні коливання |
| work_keys_str_mv |
AT elnazarova oninvarianttorusofweaklyconnectedsystemsofdifferentialequations |
| first_indexed |
2025-07-15T16:22:52Z |
| last_indexed |
2025-07-15T16:22:52Z |
| _version_ |
1837730710303014912 |
| fulltext |
UDC 517 . 9
ON INVARIANT TORUS OF WEEKLY CONNECTED SYSTEMS
OF DIFFERENTIAL EQUATIONS
ПРО IНВАРIАНТНИЙ ТОР ДЛЯ СЛАБКОЗВ’ЯЗАНИХ СИСТЕМ
ДИФЕРЕНЦIАЛЬНИХ РIВНЯНЬ
A. Elnazarov
Khorog Univ., Tajikistan
We consider a family of systems of differential equations depending on a sufficiently small parameter
with zero value of which we obtained a couple of independent systems. We used the method of Green –
Samoilenko function to construct an invariant manifold of the pertubed system and presented some exam-
ples for application.
Розглянуто сiм’ю систем диференцiальних рiвнянь, що залежать вiд достатньо малого пара-
метра, яка є парою незалежних систем, якщо значення параметра дорiвнює нулю. Використано
метод функцiї Грiна – Самойленка для побудови iнварiантного многовиду збуреної системи та
наведено приклади.
Introduction. Let us consider a system of differential equations of the form
x′ = X(x, ε), (1)
where x = (x1, x2, . . . , xn) ∈ Rn, ε > 0 is a sufficiently small parameter.
Suppose that the unpertubed system
x′ = X(x, 0), (2)
has an invariant torus. The classical problem arising here is to find out what we can say about
the invariant manifold of the pertubed system?
The main tool of investigation of the above problem was the method of integral mapi-
folds of nonlinear mechanics by Krylov – Bogolyubov – Mitropolskiy, the method of Levinson –
Diliberto and others. Later, by using of the method of Green function, A. M. Samoilenko [1]
obtained new results on the theory of invariant torus. The method of Samoilenko was extended
to the other lasses of equations by Yu. V. Teplinskiy, D. I. Martynyuk, M. I. Ilolov [2 – 5] and
others. We use this approach in a practically interesting case when the dimension of the given
space may be reduced.
We shall consider a system of equations of the form
y′ = Y (y, z, ε),
(3)
z′ = Z(y, z, ε),
where y = (y1, y2, . . . , ym), z = (z1, z2, . . . , zk) are vectors in the Euclidean space Rn = Rm ⊕
⊕Rk, the parameter ε > 0 is a sufficiently small.
c© A. Elnazarov, 2005
468 ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 4
ON INVARIANT TORUS OF WEEKLY CONNECTED SYSTEMS OF DIFFERENTIAL EQUATIONS 469
Rewrite the system (3) to the following form:
y′ = Y0(y) + Y1(y, z, ε),
(4)
z′ = Z0(z) + Z1(z, y, ε),
where Y0(y) = Y (y, 0, 0), Z0(z) = Z(z, 0, 0), Y1(y, z, ε) = Y (y, z, ε) − Y0(y), Z1(z, y, ε) =
= Z(z, y, ε)− Z0(z).
Definition 1. A system of differential equations of the form (3) is called weakly connected if
for ε = 0 we obtain two independent systems, i.e.,
y′ = Y0(y), (5)
z′ = Z0(z). (6)
Let us give some examples of weakly connected systems of differential equations.
Consider weakly connected oscillators of the form
ẍi + Qi(xi) = εqi(t, x1, ẋ1, . . . , xN , ẋN ; ε), i = 1, 2, . . . , N, (7)
where ε is a sufficiently small parameter and qi are continuous periodic functions of t. This
system was investigated by L. D. Akulenko [6]. He constructed the stationary resonance rotating-
oscillating solutions of this system on infinite long period of time.
In [7] Yu. A. Mitropolsky and A. M. Samoilenko considered the system of the type
d2x
dt2
+ λ2x = εf
(
x,
dx
dt
)
,
where x = (x1, x2, . . . , xn), λ2 = diag (λ2
1, λ
2
2, . . . , λ
2
n), λj ≥ 0 (j = 1, n), f = (f1, f2, . . . , fn)
are polynomial function of its variables less than of order N ; ε is a sufficiently small parameter.
They obtained results concerning the problems of existence, exponential stability and exponenti-
al dichotomy of the invariant torus in the resonance and nonresonance case.
Weakly connected networks of quasiperiodic oscillators of the form
Ẋi = Fi(Xi) + εGi(X1, X2, . . . , Xn, ε), Xi ∈ Rm, i = 1, 2, . . . , n, ε � 1,
were considered by E. M. Izhikevich [8], who proved that this system csn be transformed into a
phase (canonical) model,
θ̇i = Ωi + εhi(θ1, θ2, . . . , θn, ε), i = 1, . . . , n,
by a continuous, possibly noninvertible change of variables, where θi is a vector of phases
(angles), Ωi is a vector of frequencies of the ith oscillator Xi. It was shown also that whether
or not the oscillators interact depends not only on the existence of connections between them,
but also on their frequencies. One can find many examples of this system, especially in neural
networks.
ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 4
470 A. ELNAZAROV
Other type of weakly connected systems can be found in the works [9, 10]. In the present
work we assume that for ε = 0 each of the uncoupled systems of equations (5), (6) has an
asymptotically stable invariant torus of the following form:
M{y = f(ϕ), ϕ ∈ Tr}, N = {z = g(θ), θ ∈ Tl}, (8)
where f ∈ C1(Tr), g ∈ C1(Tl),
Tr = {ϕ(ϕ1, ϕ2, . . . , ϕr) : 0 ≤ ϕi ≤ 2π, 0 ≤ i ≤ r},
Tl = {θ(θ1, θ2, . . . , θl) : 0 ≤ θj ≤ 2π, 0 ≤ j ≤ l}.
We shall consider the problem, when pertubed system (3) has an invariant manifold. We used
the method described in [2]. To this end we have to transform the system about the invari-
ant manifolds M,N (8). We showed that under certain conditions the invariant torus of the
transformed system, which has dimension r + l, can be constructed from the invariant torus of
the unpertubed systems with less dimension r and l.
This paper is organized as follows. In Section 1 the local coordinate system about the
invariant torus of the uncoupled systems (5), (6) is introduced and the definition of an almost
independent function is given. The proof of the preliminary Lemma 2 is also included in this
section. The method for constructing the invariant torus of the system (3) and a proof of the
main result is given in Section 2. Section 3 contains two examples of weakly connected systems.
We apply the method described in Section 2 to this systems of differential equations.
1. A local coordinate system about the invariant torus. To introduce local coordinate we
suppose that the functions f, g are such that
f(ϕ) ∈ C1(Tr), rank
∂f(ϕ)
∂ϕ
= r ∀ ϕ ∈ Tr,
g(θ) ∈ C1(Tl), rank
∂g(θ)
∂θ
= l ∀ θ ∈ Tl
and also assume that the matrix
∂f(ϕ)
∂ϕ
and
∂g(θ)
∂θ
can be completed to a periodic basis in Rm,
Rk respectively, and the complement matrix B(ϕ) ∈ C1(Tr), W (θ) ∈ C1(Tl) is such that
det
∣∣∣∣∂f(ϕ)
∂ϕ
,B(ϕ)
∣∣∣∣ 6= 0,
(9)
det
∣∣∣∣∂f(θ)
∂θ
,W (θ)
∣∣∣∣ 6= 0.
Under the above assumptions we can define local coordinates ϕ, h (for (5)) and s, θ (for (6)) in
a neighborhood of the invariant manifolds M, N by
y = f(ϕ) + B(ϕ)h, (10)
z = g(θ + W (θ)s. (11)
ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 4
ON INVARIANT TORUS OF WEEKLY CONNECTED SYSTEMS OF DIFFERENTIAL EQUATIONS 471
Following lemma [2] (A. M. Samoilenko) we can apply the transformation (10), (11) to (3).
Lemma 1. For every sufficiently small δ > 0 there exists δ1(δ) → 0 as δ → 0 such that every
point y (or z) satisfying ρ(y, M) < δ (or ρ(z,N) < δ), has local coordinates ϕ, h (or s, θ ),
satisfying
‖h‖ ≤ δ1, ϕ ∈ Tr (or ‖s‖ ≤ δ1, θ ∈ Tl). (12)
By substituting (10), (11) into (3) we obtain[
∂f(ϕ)
∂ϕ
+
∂B(ϕ)
∂ϕ
h
]
dϕ
dt
+ B(ϕ)
dh
dt
= Y (f(ϕ) + B(ϕ)h, g(θ) + W (θ)s, ε),
[
∂g(θ)
∂θ
+
∂W (θ)
∂θ
s
]
dθ
dt
+ W (θ)
ds
dt
= Z(g(θ) + W (θ)s, f(ϕ) + B(ϕ)h, ε),
and condition (9) allows us to solve this system with respect to
dϕ
dt
,
dh
dt
,
dθ
dt
,
ds
dt
for all h, s, ϕ,
θ from the domain (12), so that the system of equations (3), after transformation, will have the
form
dϕ
dt
= a1(ϕ, θ, h, s, ε),
dh
dt
= P1(ϕ, θ, h, s, ε),
(13)
dθ
dt
= a2(ϕ, θ, h, s, ε),
ds
dt
= P2(ϕ, θ, h, s, ε),
where a1, a2, P1, P2 are periodic functions with respect to ϕi, i = 1, . . . , r, θj , j = 1, . . . , l,
defined and continuous with respect to ϕ, θ, h, s, ε in the domain
‖h‖ ≤ d, ϕ ∈ Tr, ‖s‖ ≤ d, θ ∈ Tl, ε ∈ (0, ε0], (14)
where d and ε0 are sufficiently small numbers.
It should be noted that for ε = 0 we have following systems:
dϕ
dt
= a∗1(ϕ, h),
(15)
dh
dt
= P ∗
1 (ϕ, h),
dθ
dt
= a∗2(θ, s),
(16)
ds
dt
= P ∗
2 (θ, s),
ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 4
472 A. ELNAZAROV
where a∗1(ϕ, h) = a1(ϕ, 0, h, 0, 0), P ∗
1 (ϕ, h) = P1(ϕ, 0, h, 0, 0), a∗2(θ, s) = a2(0, θ, 0, s, 0),
P ∗
2 (θ, s) = P2(0, θ, 0, s, 0).
By assuming that the systems of equations (5), (6) have the invariant manifolds (8) is equi-
valent to the systems (15) and (16) having the invariant torus
h = 0, ϕ ∈ Tr, s = 0, θ ∈ Tl.
Definition 2. A function a(ϕ, θ, σ) ∈ Cp(Tl × Tr) is called almost independent of θ of the
order p if for any ε > 0 sufficiently small there exists σ > 0 such that∣∣a(ϕ, θ, σ)− a(ϕ, 0, 0)
∣∣
p
< ε.
Remark. For the class of this functions we may put, for example,
H(ϕ, θ) = F (ϕ) + σG(ϕ, θ),
where F (ϕ), G(ϕ, θ) are periodic functions with respect to each component of ϕ and θ; σ is a
sufficiently small parameter.
Definition 3. According to [2] (A. M. Samoilenko), the Green function G0(τ, ϕ) of the system
dϕ
dt
= a(ϕ),
dh
dt
= P (ϕ)h
is said to be rough if there exists δ = const > 0 and an integer number p ≥ 0 such that the system
of equations
dϕ
dt
= a(ϕ) + a1(ϕ),
dh
dt
= P (ϕ)h, (17)
for a1 ∈ Cp0(Tm) (p0 = max (1, p)) and
|a1|p0 ≤ δ, (18)
has a Green function G0(τ, ϕ) satisfying the inequality∣∣G0(τ, ϕ)f(ϕr(ϕ))
∣∣
p
≤ Ke−γ|τ ||f |p (19)
where f is a function in Cp0(Tm), ϕτ (ϕ) is a solution of the first equation (17); K, γ are positive
constants independent of ϕ, δ, f.
The following lemma takes a central place in the future investigations.
Lemma 2. Assume that the functions a1(ϕ, θ, σ), P1(ϕ, θ, σ) and a2(ϕ, θ, σ), P2(ϕ, θ, σ) are
almost independent of θ and ϕ, respectively. Let each of the systems of equations
dϕ
dt
= a1(ϕ, 0, 0),
dθ
dt
= a2(0, θ, 0), (20)
dh
dt
= P1(ϕ, θ, σ)h
ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 4
ON INVARIANT TORUS OF WEEKLY CONNECTED SYSTEMS OF DIFFERENTIAL EQUATIONS 473
and
dϕ
dt
= a1(ϕ, 0, 0),
dθ
dt
= a2(0, θ, 0), (21)
ds
dt
= P2(ϕ, θ, σ)s
have a rough Green function. Then there exist µ = µ(δ) (µ → 0, δ → 0) and σ0 > 0 such that
for every σ ∈ (0, σ0], functions a•1, P •
1 , a•2, P •
2 ∈ Cp(Tl × Tr) satisfying the inequalities
|a•1|p + |a•2|p + |P •
1 |p < µ,
(22)
|a•1|p + |a•2|p + |P •
2 |p < µ,
and functions f1, f2 ∈ Cp(Tr × Tl), each of the systems of equations
dϕ
dt
= a1(ϕ, θ, σ) + a•1(ϕ, θ),
dθ
dt
= a2(ϕ, θ, σ) + a•2(ϕ, θ), (23)
dh
dt
=
[
P1(ϕ, θ, σ) + P •
1 (ϕ, θ)
]
h + f1(ϕ, θ)
and
dϕ
dt
= a1(ϕ, θ, σ) + a•1(ϕ, θ),
dθ
dt
= a2(ϕ, θ, σ) + a•2(ϕ, θ), (24)
ds
dt
=
[
P2(ϕ, θ, σ) + P •
2 (ϕ, θ)
]
s + f2(ϕ, θ)
has an invariant torus of the type
h = u(ϕ, θ, σ), s = v(ϕ, θ, σ), ϕ ∈ Tr, θ ∈ Tl, σ ∈ (0, σ0],
satisfying the inequality
|u(ϕ, θ, σ)|p ≤ K1|f1|p,
(25)
|v(ϕ, θ, σ)|p ≤ K2|f2|p,
where K1, K2 are positive constants independent of µ, f1, f2.
ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 4
474 A. ELNAZAROV
Proof. Let us rewrite the system of equations
dϕ
dt
= a1(ϕ, θ, σ),
dθ
dt
= a2(ϕ, θ, σ),
dh
dt
= P1(ϕ, θ, σ)h
in the following form:
dϕ
dt
= a1(ϕ, 0, 0) + a11(ϕ, θ, σ),
dθ
dt
= a2(0, θ, 0) + a21(ϕ, θ, σ),
dh
dt
=
[
P1(ϕ, 0, 0) + P11(ϕ, θ, σ)
]
h,
where a11(ϕ, θ, σ) = a1(ϕ, θ, σ)−a1(ϕ, 0, 0), a21(ϕ, θ, σ) = a2(ϕ, θ, σ)−a2(0, θ, 0), P11(ϕ, θ, σ) =
= P1(ϕ, θ, σ)−P1(ϕ, 0, 0). Since the functions a1, P1 and a2 are almost independent of θ and ϕ,
respectively, for every ε > 0 we can choose σ0 > 0 such that for any σ ∈ (0, σ0] the following
inequalities hold:
|a11(ϕ, θ, σ)|p < ε, |a21(ϕ, θ, σ)|p < ε, |P11(ϕ, θ, σ)|p < ε, (26)
ϕ ∈ Tr, θ ∈ Tl.
By using the fact that the system (23) has a rough Green function we can choose δ > 0 in (18)
and ε > 0 in (26) such that the system of equations
dϕ
dt
= a1(ϕ, θ, σ) + a•1(ϕ, θ),
dθ
dt
= a2(ϕ, θ, σ) + a•2(ϕ, θ),
dh
dt
=
[
P1(ϕ, θ, σ) + P •
1 (ϕ, θ)
]
h + f1(ϕ, θ)
with the functions a•1(ϕ, θ), a•2(ϕ, θ), P •
1 (ϕ, θ) satisfying the inequality
|a•1|p + |a11|p + |a•2|p + |a21|p + |P •
1 |p + |P11|p < δ, ϕ ∈ Tr, θ ∈ Tl, (27)
has a Green function G0(τ, ϕ, θ) which satisfies∣∣G0(τ, ϕ, θ)f1(ϕτ (ϕ), θτ (θ))
∣∣
p
≤ K1e
−γ1|τ ||f1|p. (28)
ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 4
ON INVARIANT TORUS OF WEEKLY CONNECTED SYSTEMS OF DIFFERENTIAL EQUATIONS 475
Let us denote by
Gf(ϕ, θ) =
∞∫
−∞
G0(τ, ϕ, θ)f(ϕτ (ϕ), θτ (θ))dτ
the operator defined on f ∈ Cp(Tr × Tl) and let consider the following equation:
u = GP •
1 u + Gf1.
A solution of this equation exists if
|GP •
1 |p = d < 1, (29)
but we can choose a sufficiently small δ > 0 such that (29) holds and we obtain a unique
solution in the form
u =
∞∑
k=0
(GP •
1 )k Gf1, (30)
which satisfies the inequality
|u(ϕ, θ, σ)|p ≤ K1|f1|p
for any σ ∈ (0, σ0]. Obviously, u = u(ϕ, θ, σ) defined by (30) for any σ ∈ (0, σ0] is an invariant
torus of the system
dϕ
dt
= a1(ϕ, θ, σ) + a•1(ϕ, θ),
dθ
dt
= a2(ϕ, θ, σ) + a•2(ϕ, θ),
dh
dt
=
[
P1(ϕ, θ, σ) + P •
1 (ϕ, θ)
]
h + f1(ϕ, θ).
Lemma 2 is proved for h = u(ϕ, θ, σ), σ ∈ (0, σ0], ϕ ∈ Tr, θ ∈ Tl. For the case when s =
v(ϕ, θ, σ), ϕ ∈ Tr, θ ∈ Tl, we can prove it in the same way.
2. Constructing an invariant torus. Let us, by using the method of Samoilenko, construct an
invariant torus of system (13). To this end we select the «linear» part of P1(ϕ, θ, h, s, ε), P2(ϕ, θ,
h, s, ε) and obtain
dϕ
dt
= a1(ϕ, θ, h, s, ε),
dh
dt
= P ∗
1 (ϕ, θ, h, s, ε)h + f1(ϕ, θ, ε),
dθ
dt
= a2(ϕ, θ, h, s, ε),
ds
dt
= P ∗
2 (ϕ, θ, h, s, ε)s + f2(ϕ, θ, ε),
ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 4
476 A. ELNAZAROV
where
P ∗
1 (ϕ, θ, h, s, ε) =
1∫
0
∂[P1(ϕ, θ, th, s, ε)− P1(ϕ, θ, 0, 0, ε)]
∂(th)
dt,
P ∗
2 (ϕ, θ, h, s, ε) =
1∫
0
∂[P2(ϕ, θ, h, ts, ε)− P2(ϕ, θ, 0, 0, ε)]
∂(ts)
dt,
f1(ϕ, θ, ε) = P1(ϕ, θ, 0, 0, ε), f2(ϕ, θ, ε) = P2(ϕ, θ, 0, 0, ε). When ε = 0, system (13) has the
invariant torus h = 0, s = 0, therefore f1(ϕ, 0, 0) = 0, f2(ϕ, 0, 0) = 0.
We can also rewrite the above system in the form
dϕ
dt
= a10(ϕ) + a11(ϕ, θ, h, s, ε),
dθ
dt
= a20(θ) + a22(ϕ, θ, h, s, ε),
dh
dt
=
[
P ∗
10(ϕ) + P ∗
11(ϕ, θ, h, s, ε)
]
h + f1(ϕ, θ, ε),
ds
dt
=
[
P ∗
20(θ) + P ∗
21(ϕ, θ, h, s, ε)
]
s + f2(ϕ, θ, ε),
where
a10(ϕ) = a1(ϕ, 0, 0, 0, 0), a11(ϕ, θ, h, s, ε) = a1(ϕ, θ, h, s, ε)− a10(ϕ),
a20(θ) = a2(0, θ, 0, 0, 0), a22(ϕ, θ, h, s, ε) = a2(ϕ, θ, h, s, ε)− a20(θ),
P ∗
10(ϕ) = P ∗
1 (ϕ, 0, 0, 0, 0), P ∗
11(ϕ, θ, h, s, ε) = P ∗
1 (ϕ, θ, h, s, ε)− P ∗
10(ϕ),
P ∗
20(θ) = P ∗
2 (0, θ, 0, 0, 0), P ∗
21(ϕ, θ, h, s, ε) = P ∗
2 (0, θ, h, s, ε)− P ∗
20(θ).
Consider the systems of equations of the type
dϕ
dt
= a10(ϕ),
dθ
dt
= a20(θ),
dh
dt
=
[
P ∗
10(ϕ) + P101(ϕ, θ, ε)
]
h,
and
dϕ
dt
= a10(ϕ),
dθ
dt
= a20(θ),
ds
dt
=
[
P ∗
20(θ) + P201(ϕ, θ, ε)
]
s.
(31)
We can prove the following theorem for the system (13).
Theorem. Suppose that for every P101(ϕ, θ, ε), P201(ϕ, θ, ε), almost independent of θ and ϕ
respectively, each of the system of equations (31) has a rough Green function of order p, satisfying
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ON INVARIANT TORUS OF WEEKLY CONNECTED SYSTEMS OF DIFFERENTIAL EQUATIONS 477
inequality (19). Assume that the functions a1, a2, P ∗
1 , P ∗
2 , in the domain
‖h‖ ≤ d, ϕ ∈ Tr, ‖s‖ ≤ d, θ ∈ Tl, ε ∈ (0, ε0],
are continuous together with all derivative with respect to ϕ, θ, h, s up to the order p. If p ≥ 1
then there exists a sufficiently small ε0 > 0 such that for every ε ∈ (0, ε0] the system of equations
(13) has an invariant torus,
H = U(ϕ, θ, ε) =
(
u(ϕ, θ, ε)
v(ϕ, θ, ε)
)
, ϕ ∈ Tr, θ ∈ Tl,
where u(ϕ, θ, ε), v(ϕ, θ, ε) ∈ Cp−1(Tr × Tl) and satisfy the inequality
|u(ϕ, θ, ε)|p−1 ≤ K1|f1|p−1,
|v(ϕ, θ, ε)|p−1 ≤ K2|f2|p−1,
where K1, K2 are positive constants independent of ε.
Proof. We may find the invariant torus by an iteration process. To this end for the zero
approximation we put
u0(ϕ, θ) ≡ 0, v0(ϕ, θ) ≡ 0, ϕ ∈ Tr, θ ∈ Tl,
and the first approximation u1(ϕ, θ, ε) can be obtained from
dϕ
dt
= a10(ϕ) + a11(ϕ, θ, 0, 0, ε),
dθ
dt
= a20(θ), (32)
dh
dt
=
[
P ∗
10(ϕ) + P ∗
11(ϕ, θ, 0, 0, ε)
]
h + f1(ϕ, θ, ε)
and v1(ϕ, θ, ε) from
dϕ
dt
= a10(ϕ),
dθ
dt
= a20(θ) + a21(ϕ, θ, 0, 0, ε), (33)
ds
dt
=
[
P ∗
20(θ) + P ∗
21(ϕ, θ, 0, 0, ε)
]
s + f1(ϕ, θ, ε).
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478 A. ELNAZAROV
We may choose ε0 sufficiently small such that for ε ∈ (0, ε0] the following inequalities hold:
|a11(ϕ, θ, u(ϕ, θ, ε), v(ϕ, θ, ε), ε)|p + |a21(ϕ, θ, u(ϕ, θ, ε), v(ϕ, θ, ε), ε)|p+
+ |P ∗
11(ϕ, θ, u(ϕ, θ, ε), v(ϕ, θ, ε), ε)|p < δ, (34)
|a11(ϕ, θ, u(ϕ, θ, ε), v(ϕ, θ, ε), ε)|p + |a21(ϕ, θ, u(ϕ, θ, ε), v(ϕ, θ, ε), ε)|p+
+ |P ∗
21(ϕ, θ, u(ϕ, θ, ε), v(ϕ, θ, ε), ε)|p < δ (35)
for every u(ϕ, θ, ε), v(ϕ, θ, ε) ∈ Cp(Tr × Tl) satisfying the inequality
|u(ϕ, θ, ε)|p ≤ K1|f1|p,
|v(ϕ, θ, ε)|p ≤ K2|f2|p.
By using Lemma 2 we obtain that (32), (33) have an invariant torus satisfying
|u1(ϕ, θ, ε)|p ≤ K1|f1|p,
|v1(ϕ, θ, ε)|p ≤ K2|f2|p.
Therefore, we can continue the iteration process and at the (i + 1)th approximation
Ui+1(ϕ, θ, ε) =
(
ui+1(ϕ, θ, ε)
vi+1(ϕ, θ, ε)
)
,
the ui+1(ϕ, θ, ε) are obtained from the following system:
dϕ
dt
= a10(ϕ) + a11(ϕ, θ, ui(ϕ, θ, ε), vi(ϕ, θ, ε), ε),
dθ
dt
= a20(θ) + a21(ϕ, θ, ui−1(ϕ, θ, ε), vi−1(ϕ, θ, ε), ε), (36)
dh
dt
=
[
P ∗
10(ϕ) + P ∗
11(ϕ, θ, ui(ϕ, θ, ε), vi(ϕ, θ, ε), ε)
]
h + f1(ϕ, θ, ε),
and vi+1(ϕ, θ, ε) from
dθ
dt
= a20(θ) + a21(θ, ϕ, ui(ϕ, θ, ε), vi(ϕ, θ, ε), ε),
dϕ
dt
= a10(ϕ) + a11(ϕ, θ, ui−1(ϕ, θ, ε), vi−1(ϕ, θ, ε), ε), (37)
ds
dt
=
[
P ∗
20(θ) + P ∗
21(θ, ϕ, ui(ϕ, θ, ε), vi(ϕ, θ, ε), ε)
]
s + f2(θ, ϕ, ε).
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ON INVARIANT TORUS OF WEEKLY CONNECTED SYSTEMS OF DIFFERENTIAL EQUATIONS 479
Since the functions a11, a21, P ∗
11, P ∗
21 for all ε ∈ (0, ε0] satisfy inequalities (34), (35), the
invariant torus of (36) and (37) exists and satisfies
|ui+1(ϕ, θ, ε)|p ≤ K1|f1|p,
(38)
|vi+1(ϕ, θ, ε)|p ≤ K2|f2|p.
By using the induction we can prove that each iteration for all ε ∈ (0, ε0] is defined and
satisfies an inequality of type (38).
Now, we need to prove the uniform convergence of the sequence of the invariant tori. To
this end we consider the following functions:
ωi(ϕ, θ, ε) = ui+1(ϕ, θ, ε)− ui(ϕ, θ, ε), (39)
νi(ϕ, θ, ε) = vi+1(ϕ, θ, ε)− vi(ϕ, θ, ε). (40)
After the calculation of the differences between
∂ui+1(ϕ, θ, ε)
∂ϕ
[a10(ϕ) + a11(ϕ, θ, ui(ϕ, θ, ε), vi(ϕ, θ, ε), ε)]+
+
∂ui+1(ϕ, θ, ε)
∂θ
[a20(θ) + a21(ϕ, θ, ui−1(ϕ, θ, ε), vi−1(ϕ, θ, ε), ε)] =
=
[
P ∗
10(fi) + P ∗
11(ϕ, θ, ui(ϕ, θ, ε), vi(ϕ, θ, ε), ε)
]
ui+1(ϕ, θ, ε) + f1(ϕ, θ, ε)
and
∂ui(ϕ, θ, ε)
∂ϕ
[a10(ϕ) + a11(ϕ, θ, ui−1(ϕ, θ, ε), vi−1(ϕ, θ, ε), ε)]+
+
∂ui(ϕ, θ, ε)
∂θ
[a20(θ) + a21(ϕ, θ, ui−2(ϕ, θ, ε), vi−2(ϕ, θ, ε), ε)] =
=
[
P ∗
10(ϕ) + P ∗
11(ϕ, θ, ui−1(ϕ, θ, ε), vi−1(ϕ, θ, ε), ε)
]
ui(ϕ, θ, ε) + f1(ϕ, θ, ε)
we obtain following expression:
∂ωi+1(ϕ, θ, ε)
∂ϕ
[a10(ϕ) + a11(ϕ, θ, ui(ϕ, θ, ε), vi(ϕ, θ, ε), ε)]+
+
∂ωi+1(ϕ, θ, ε)
∂θ
[a20(θ) + a21(ϕ, θ, ui−1(ϕ, θ, ε), vi−1(ϕ, θ, ε), ε)] =
=
[
P ∗
10(ϕ) + P ∗
11(ϕ, θ, ui(ϕ, θ, ε), vi(ϕ, θ, ε), ε)
]
ωi+1(ϕ, θ, ε) + fi(ϕ, θ, ε),
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480 A. ELNAZAROV
where
fi(ϕ, θ, ε) = [P ∗
11(ϕ, θ, ui(ϕ, θ, ε), vi(ϕ, θ, ε), ε)−
− P ∗
11(ϕ, θ, ui−1(ϕ, θ, ε), vi−1(ϕ, θ, ε), ε)]ui(ϕ, θ, ε)+
+
∂ui(ϕ, θ, ε)
∂ϕ
[a11(ϕ, θ, ui(ϕ, θ, ε), vi(ϕ, θ, ε), ε)−
− a11(ϕ, θ, ui−1(ϕ, θ, ε), vi−1(ϕ, θ, ε), ε)]+
+
∂ui(ϕ, θ, ε)
∂θ
[a21(ϕ, θ, ui−1(ϕ, θ, ε), vi−1(ϕ, θ, ε), ε)−
− a11(ϕ, θ, ui−2(ϕ, θ, ε), vi−2(ϕ, θ, ε), ε)],
which means that the function ωi+1(ϕ, θ, ε) is an invariant torus of the following system:
dϕ
dt
= a10(ϕ) + a11(ϕ, θ, ui(ϕ, θ, ε), vi(ϕ, θ, ε), ε),
dθ
dt
= a20(θ) + a21(ϕ, θ, ui−1(ϕ, θ, ε), vi−1(ϕ, θ, ε), ε),
dh
dt
=
[
P ∗
10(ϕ) + P ∗
11(ϕ, θ, ui(ϕ, θ, ε), vi(ϕ, θ, ε), ε)
]
h + fi(ϕ, θ, ε),
and, as mentioned above, the functions a11, a21, P ∗
11 satisfy all conditions of Lemma 2. So we
have an invariant torus of the above system which satisfies an inequality of type (38) with p− 1,
|ωi(ϕ, θ, ε)|p−1 ≤ K1|fi|p−1, (41)
and for p = 1 we have
|ωi+1(ϕ, θ, ε)|0 ≤ K1
[
L1{|ωi(ϕ, θ, ε)|+ |νi(ϕ, θ, ε)|}|ui|0+
+ L2{|ωi(ϕ, θ, ε)|+ |νi(ϕ, θ, ε)|}|ui|1+
+ L3{|ωi−1(ϕ, θ, ε)|+ |νi−1(ϕ, θ, ε)|}|ui|1
]
. (42)
We may choose L = max(L1, L2, L3) and obtain from (42)
|ωi+1(ϕ, θ, ε)|0 ≤ K1L
{
{|ωi(ϕ, θ, ε)|+ |νi(ϕ, θ, ε)|}|ui|0+
+ {|ωi(ϕ, θ, ε)|+ |νi(ϕ, θ, ε)|}|ui|1+
+ {|ωi−1(ϕ, θ, ε)|+ |νi−1(ϕ, θ, ε)|}|ui|1
}
. (43)
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ON INVARIANT TORUS OF WEEKLY CONNECTED SYSTEMS OF DIFFERENTIAL EQUATIONS 481
Analogously, we can show that the function νi(ϕ, θ, ε) is an invariant torus of the system
dϕ
dt
= a10(ϕ) + a11(ϕ, θ, ui−1(ϕ, θ, ε), vi−1(ϕ, θ, ε), ε),
dθ
dt
= a20(θ) + a21(ϕ, θ, ui(ϕ, θ, ε), vi(ϕ, θ, ε), ε), (44)
ds
dt
=
[
P ∗
20(θ) + P ∗
21(ϕ, θ, ui(ϕ, θ, ε), vi(ϕ, θ, ε), ε)
]
s + gi(ϕ, θ, ε),
where
gi(ϕ, θ, ε) =
[
P ∗
21(ϕ, θ, ui(ϕ, θ, ε), vi(ϕ, θ, ε), ε)−
− P ∗
21(ϕ, θ, ui−1(ϕ, θ, ε), vi−1(ϕ, θ, ε), ε)
]
vi(ϕ, θ, ε)+
+
∂vi(ϕ, θ, ε)
∂θ
[a21(ϕ, θ, ui(ϕ, θ, ε), vi(ϕ, θ, ε), ε)−
− a21(ϕ, θ, ui−1(ϕ, θ, ε), vi−1(ϕ, θ, ε), ε)]+
+
∂vi(ϕ, θ, ε)
∂ϕ
[a11(ϕ, θ, ui−1(ϕ, θ, ε), vi−1(ϕ, θ, ε), ε)−
− a11(ϕ, θ, ui−2(ϕ, θ, ε), vi−2(ϕ, θ, ε), ε)],
and, by using the inequality of type (38), we have
|νi+1(ϕ, θ, ε)|0 ≤ K1L
{
{|ωi(ϕ, θ, ε)|0 + |νi(ϕ, θ, ε)|0}|vi|0+
+ {|ωi(ϕ, θ, ε)|0 + |νi(ϕ, θ, ε)|0}|vi|1+
+ {|ωi−1(ϕ, θ, ε)|0 + |νi−1(ϕ, θ, ε)|0}|vi|1
}
. (45)
Adding expression (43) and (45) we obtain
|ωi+1(ϕ, θ, ε)|0 + |νi+1(ϕ, θ, ε)|0 ≤ M
{
{|ωi(ϕ, θ, ε)|0 + |νi(ϕ, θ, ε)|0}{|vi|0 + |ui|0}+
+ {|ωi(ϕ, θ, ε)|0 + |νi(ϕ, θ, ε)|0}{|vi|1 + |ui|1}+
+ {|ωi−1(ϕ, θ, ε)|0 + |νi−1(ϕ, θ, ε)|0}{|vi|1 + |ui|1}
}
, (46)
where M is a positive constant independent of ε. It is known that
|ui|1 ≤ K1|f1|1, |v1|1 ≤ K1|f2|1 (47)
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482 A. ELNAZAROV
and, by denoting
αi = |ωi(ϕ, θ, ε)|0 + |νi(ϕ, θ, ε)|0,
it is easy to show that (46) is equivalent to
αi+1 ≤ K1M{|f1|1 + |f2|1}{α1 + αi−1}. (48)
The property of the functions f1, f2 allows us to choose a sufficiently small ε0 such that
K1M{|f1|1 + |f2|1} ≤
1
2
,
but then
lim
i→∞
αi = 0,
which means that for every ε ∈ (0, ε0] there exist functions u(ϕ, θ, ε), v(ϕ, θ, ε) in C(Tr × Tl)
for which
lim
i→∞
ui(ϕ, θ, ε) = u(ϕ, θ, ε),
(49)
lim
i→∞
vi(ϕ, θ, ε) = v(ϕ, θ, ε)
uniformly with respect to ϕ, θ.
Since the sequences ui(ϕ, θ, ε), vi(ϕ, θ, ε), i = 1, 2, . . . , are bounded in the space Cp(Tr×Tl),
they are compact in the Cp−1(Tr × Tl) which means that
lim
i→∞
|ui(ϕ, θ, ε)− u(ϕ, θ, ε)|p−1 = 0,
(50)
lim
i→∞
|vi(ϕ, θ, ε)− v(ϕ, θ, ε)|p−1 = 0,
and taking limits in (38) we obtain
|u(ϕ, θ, ε)|p−1 ≤ K1|f1|p−1,
|v(ϕ, θ, ε)|p−1 ≤ K1|f2|p−1.
The final stage of the proof is to show that the function
U(ϕ, θ, ε) =
(
u(ϕ, θ, ε)
v(ϕ, θ, ε)
)
is an invariant torus of the system (13).
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ON INVARIANT TORUS OF WEEKLY CONNECTED SYSTEMS OF DIFFERENTIAL EQUATIONS 483
Let us consider the system of equations
dϕ
dt
= a(ϕ, h),
dh
dt
= P (ϕ, h)h + f(ϕ) (51)
and define a sequence of invariant tori
h = ui+1(ϕ), ϕ ∈ Tr, i = 0, 1, . . . , (52)
each of which is an invariant torus of the system
dϕ
dt
= a(ϕ, ui(ϕ)),
dh
dt
= P (ϕ, ui(ϕ))h + f(ϕ). (53)
The following lemma is proved in [2].
Lemma 3. Let functions a, P, f are defined and continuous with respect to ϕ, h in the domain
‖h‖ ≤ d, ϕ ∈ Tr (54)
and be periodic with respect to ϕν , ν = 0, 1, . . . , r, with the period 2π. Suppose that for every
i = 1, 2, . . . the system (46) has an invariant torus of the type (45) belonging to (47). If
lim
i→∞
ui(ϕ) = u(ϕ) (55)
uniformly convergent with respect to ϕ ∈ Tr, then the function u(ϕ) defines an invariant torus of
the system (44).
Now, by applying Lemma 3 we finish the prove of the theorem.
3. Application. 1. The interaction of chemical reactors, described by a plane autonomous
system of the type
xi = Fi(x1, x2),
yi = Gi(y1, y2),
i = 1, 2,
each admitting an exponentially attractive limit cycle, was considered by J. C. Neu [10], and
a further investigation on bifurcation of periodic orbits was made by U. Kirchgraber [11] and
A. Freidli, U. Kirchgraber, J. Waldvogel [12]. To get nearly identical reactors they put
F1(x1, x2) = F (x1, x2) + Λf(x1, x2), F2(x1, x2) = G(x1, x2) + Λg(x1, x2),
G1(y1, y2) = F (y1, y2), G2(y1, y2) = G(y1, y2),
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484 A. ELNAZAROV
where Λ is a small parameter. If this reactors are separated from each other by a membrane
which allows for diffusion from one reactor to the other, they obtained coupled chemical reactors
of the form
ẋ1 = F1(x1, x2) + K(y1 − x1),
ẋ2 = F2(x1, x2) + K(y2 − x2),
(56)
ẏ1 = G1(y1, y2) + K(x1 − y1),
ẏ2 = G2(y1, y2) + K(x2 − y2),
where K is a small coupling parameter.
Let
x1 = X1(ϕ1), x2 = X2(ϕ1),
y1 = Y1(ϕ2), y2 = Y2(ϕ2)
be limit cycles of the uncoupled systems with periods T1, T2, respectively. We can transform
them into the unit circle
u = u(φi) =
(
u1(ϕi)
u2(ϕi)
)
=
(
cos λiϕi
− sinλiϕi
)
, i = 1, 2,
0 ≤ ϕi ≤ Ti, λi =
2π
Ti
, i = 1, 2,
and by using
x =
(
x1
x2
)
= u(ϕ1)(1 + h),
y =
(
y1
y2
)
= u(ϕ2)(1 + s),
the system (56) can be reduced into the form
dh
dt
= P1(ϕ1, h)h−K(1 + h) + K(1 + s) cos(λ1ϕ1 − λ2ϕ2),
dϕ1
dt
= λ1 −K
1 + s
λ1(1 + h)
sin(λ1ϕ1 − λ2ϕ2),
ds
dt
= P2(ϕ2, s)s + K(1 + s)−K(1 + h) cos(λ2ϕ2 − λ1ϕ1),
dϕ2
dt
= λ2 + K
1 + h
λ2(1 + s)
sin(λ2ϕ2 − λ1ϕ1),
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ON INVARIANT TORUS OF WEEKLY CONNECTED SYSTEMS OF DIFFERENTIAL EQUATIONS 485
where
P (ϕ1, h)h =
[
F̃ (ϕ1, h) + λf̃(ϕ1, h)
]
cos λ1ϕ1 −
[
G̃(ϕ1, h) + λg̃(ϕ1, h)
]
sinλ1ϕ1,
P (ϕ2, s)s = F̃ (ϕ2, s) cos λ2ϕ2 − G̃(ϕ2, s) sinλ2ϕ2,
F̃ (ϕ1, h) = F (x1, x2), F̃ (ϕ2, s) = F (y1, y2), f̃(ϕ1, h) = f(x1, x2),
G̃(ϕ1, h) = G(x1, x2), G̃(ϕ2, s) = G(y1, y2), g̃(ϕ1, h) = g(x1, x2).
Let us introduce the notations
L(h, s) =
1 + h
1 + s
, K = εk, Λ = ελ,
where ε is a sufficiently small parameter; k, λ are fixed numbers. Then
dh
dt
= P1(ϕ1, h)h + εk[L(s, 0) cos(λ1ϕ1 − λ2ϕ2)− L(h, 0)],
dϕ1
dt
= λ1 − ε
k
λ1
L(s, h) sin(λ1ϕ1 − λ2ϕ2),
ds
dt
= P2(ϕ2, s)s− εk[L(h, 0) cos(λ2ϕ2 − λ1ϕ1)− L(s, 0)],
dϕ2
dt
= λ2 + ε
k
λ2
L(h, s) sin(λ2ϕ2 − λ1ϕ1).
Let ϕ1t(ϕ1) = λ1t + ϕ1, ϕ2t(ϕ) = λ2t + ϕ2, P01(ϕ1) = P1(ϕ1, 0) and suppose that G01(τ, ϕ1)
is a Green function of the equation
dh
dt
= P01(ϕ1t(ϕ1))h.
Then we can write
G01(τ, ϕ1) =
e−
R 0
τ P01(ϕ1t(ϕ1))dt, τ ≤ 0,
0, τ > 0,
and the first approximation is obtained from
h(1) = u(ϕ1, ϕ2, ε) = εk
+∞∫
−∞
G01(τ, ϕ1)[cos(λ1ϕ1 − λ2ϕ2)− 1]dτ.
Since the limit cycle is exponentially attractive, the function G01(τ, ϕ1) will satisfy the inequality
|G01(τ, ϕ1)| ≤ e−γ|τ |, γ > 0,
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486 A. ELNAZAROV
and, obviously, this function is a rough Green function.
Analogously we can find s(1) and other iterations of the torus of the system (57).
2. Consider two harmonic weakly connected oscillators
dx1
dt
= x2 + x1(1− x2
1 − x2
2) + εf1(x1, x2, y1, y2),
dx2
dt
= −x1 + x2(1− x2
1 − x2
2) + εf2(x1, x2, y1, y2),
(57)
dy1
dt
= y2 + y1(1− y2
1 − y2
2) + εg1(x1, x2, y1, y2),
dy2
dt
= −y1 + y2(1− y2
1 − y2
2) + εg2(x1, x2, y1, y2),
where f1, f2, g1, g2 are continuous with respect to all variables.
Obviously, for ε = 0 we have two independent systems, each of which has a limit cycle since
the invariant manifold has the type
u = u(φ) =
(
u1(ϕ)
u2(ϕ)
)
=
(
cos ϕ
− sinϕ
)
, 0 ≤ ϕ ≤ 2π.
For the transformation we may use
x =
(
x1
x2
)
= u(ϕ)(1 + h),
y =
(
y1
y2
)
= u(θ)(1 + s),
and by substituting it to (57) we obtain
dϕ
dt
= 1− ε
1 + h
[f̃1 sin ϕ + f̃2 cos ϕ],
dθ
dt
= 1− ε
1 + s
[g̃1 sin θ + g̃2 cos θ],
(58)
dh
dt
= −(1 + h)(2 + h)h− ε[f̃2 sinϕ− f̃1 cos ϕ],
ds
dt
= −(1 + s)(2 + s)s− ε[g̃2 sin θ − g̃1 cos θ],
where f̃i(ϕ, θ, h, s) = fi(x1, x2, y1, y2), g̃i(ϕ, θ, h, s) = gi(x1, x2, y1, y2), i = 1, 2.
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ON INVARIANT TORUS OF WEEKLY CONNECTED SYSTEMS OF DIFFERENTIAL EQUATIONS 487
By substituting the zero approximation h = 0, s = 0 to (58) we can find the first approxi-
mation from
dθ
dt
= 1,
dϕ
dt
= 1− ε[f̃0
1 sinϕ + f̃0
2 cos ϕ], (59)
dh
dt
= −2h− ε[f̃0
2 sinϕ− f̃0
1 cos ϕ],
and
dϕ
dt
= 1,
dθ
dt
= 1− ε[g̃0
1 sin θ + g̃0
2 cos θ], (60)
ds
dt
= −2s− ε[g̃0
2 sin θ − g̃0
1 cos θ],
where f̃0
i (ϕ, θ) = f̃i(ϕ, θ, 0, 0), g̃ 0
i (ϕ, θ) = g̃i(ϕ, θ, 0, 0), i = 1, 2.
The Green function has the form
G0(τ) =
{
e2τ , τ ≤ 0,
0 τ > 0.
We can explicitly find an invariant torus of this equations,
h(1)(ϕ, θ) = ε
+∞∫
−∞
G0(τ)
[
f̃0
2 (ϕτ (ϕ), θτ (θ)) sinϕτ (ϕ)− f̃0
1 (ϕτ (ϕ), θτ (θ)) cos ϕτ (ϕ)
]
dτ,
s(1)(ϕ, θ) = ε
+∞∫
−∞
G0(τ)
[
g̃0
1(ϕτ (ϕ), θτ (θ)) sin θτ (θ)− g̃0
1(ϕτ (ϕ), θτ (θ)) cos θτ (θ)
]
dτ,
where ϕτ (ϕ), θτ (θ) are solutions of the first equations of (59) for h(1)(ϕ, θ), and solutions of the
first equations of (60) for s(1)(ϕ, θ) and satisfying the initial conditions ϕ0(ϕ) = ϕ, θ0(θ) = θ.
The (i + 1)th approximation ui+1(ϕ, θ, ε) = (hi+1(ϕ, θ, ε), si+1(ϕ, θ, ε)) can be found from
ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 4
488 A. ELNAZAROV
following two systems of equations:
dϕ
dt
= 1− ε
1 + hi(ϕ, θ, ε)
[
f̃1(ϕ, θ, hi(ϕ, θ, ε), si(ϕ, θ, ε)) sinϕ+
+ f̃2(ϕ, θ, hi(ϕ, θ, ε), si(ϕ, θ, ε)) cos ϕ
]
,
dθ
dt
= 1− ε
1 + si−1(ϕ, θ, ε)
[
g̃1(ϕ, θ, hi−1(ϕ, θ, ε), si−1(ϕ, θ, ε)) sin θ+
+ g̃2(ϕ, θ, hi−1(ϕ, θ, ε), si−1(ϕ, θ, ε)) cos θ
]
,
dh
dt
= −(1 + hi(ϕ, θ, ε))(2 + hi(ϕ, θ, ε))h−
− ε
[
f̃2(ϕ, θ, hi(ϕ, θ, ε), si(ϕ, θ, ε)) sinϕ− f̃1(ϕ, θ, hi(ϕ, θ, ε), si(ϕ, θ, ε)) cos ϕ
]
,
and
dϕ
dt
= 1− ε
1 + hi−1(ϕ, θ, ε)
[
f̃1(ϕ, θ, hi−1(ϕ, θ, ε), si−1(ϕ, θ, ε)) sinϕ+
+ f̃2(ϕ, θ, hi−1(ϕ, θ, ε), si−1(ϕ, θ, ε)) cos ϕ
]
,
dθ
dt
= 1− ε
1 + si(ϕ, θ, ε)
[
g̃1(ϕ, θ, hi(ϕ, θ, ε), si(ϕ, θ, ε)) sin θ+
+ g̃2(ϕ, θ, hi(ϕ, θ, ε), si(ϕ, θ, ε)) cos θ
]
,
ds
dt
= −(1 + si(ϕ, θ, ε))(2 + si(ϕ, θ, ε))s−
− ε
[
g̃2(ϕ, θ, hi(ϕ, θ, ε), si(ϕ, θ, ε)) sin θ − g̃1(ϕ, θ, hi(ϕ, θ, ε), si(ϕ, θ, ε)) cos θ
]
.
It should be noted that the Green function in this case is rough with for order p, where 0 ≤
≤ p ≤ ∞. Now, by using the main result from Section 2 we may choose ε0 > 0 such that for
every ε ∈ (0, ε0] there exists an invariant torus u(ϕ, θ, ε) of (57), which we can find by the above
process as a limit function, i.e.,
u(ϕ, θ, ε) = lim
k→∞
u(k)(ϕ, θ, ε) = lim
k→∞
(
h(k)(ϕ, θ, ε), s(k)(ϕ, θ, ε)
)
.
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ON INVARIANT TORUS OF WEEKLY CONNECTED SYSTEMS OF DIFFERENTIAL EQUATIONS 489
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Received 19.09.2005
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