Nonexistence theorem except the out-of-phase and in-phase solutions in the coupled van der Pol equation system
We consider a coupled van der Pol equation system. Our coupled system consists of two van der Pol equations that are connected with each other by linear terms. We assume that two distinctive solutions (out-of-phase and in-phase solutions) exist in the dynamical system of coupled equations and give a...
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| author | Nohara, B. T. Нохара, Б. Т. |
| author_facet | Nohara, B. T. Нохара, Б. Т. |
| author_sort | Nohara, B. T. |
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| datestamp_date | 2020-03-18T19:44:57Z |
| description | We consider a coupled van der Pol equation system. Our coupled system consists of two van der Pol equations that are connected with each other by linear terms. We assume that two distinctive solutions (out-of-phase and in-phase solutions) exist in the dynamical system of coupled equations and give answers to some problems. |
| first_indexed | 2026-03-24T02:35:57Z |
| format | Article |
| fulltext |
UDC 517.9
B. T. Nohara, A. Arimoto (Musashi Inst. Technol., Tokyo, Japan)
NON-EXISTENCE THEOREM
EXCEPT THE OUT-OF-PHASE AND IN-PHASE SOLUTIONS
IN THE COUPLED VAN DER POL EQUATION SYSTEM
ТЕОРЕМА ПРО НЕIСНУВАННЯ РОЗВ’ЯЗКIВ
ЗА ВИНЯТКОМ РЕЖИМУ СИНХРОННИХ КОЛИВАНЬ
ТА РЕЖИМУ КОЛИВАНЬ У ПРОТИФАЗI
ДЛЯ СИСТЕМИ З’ЄДНАНИХ РIВНЯНЬ ВАН ДЕР ПОЛЯ
We consider the coupled van der Pol equation system in this paper. Our coupled system consists of two van
der Pol equations which are connected by the linear terms with each other. In this paper, we consider that two
distinctive solutions (the out-of-phase and in-phase solutions) exist in the dynamical system of the coupled
equations and we give the answers to some of the problems.
Розглянуто систему з’єднаних рiвнянь Ван дер Поля. Ця система складається з двох рiвнянь Ван дер
Поля, що пов’язанi мiж собою лiнiйними членами. У статтi розглянуто випадок, коли динамiчна система
з’єднаних рiвнянь має два рiзних розв’язки (у режимi синхронних коливань та у режимi коливань у
протифазi), i дано вiдповiдi на деякi питання.
1. Introduction. In the course of studying the periodic solution of the differential
equation with nonlinear perturbed terms
x′′ + x = εf(x, x′),
where x = x(t) and ′ denotes the derivative with respect to time t (we use ′ for the
symbol of derivative hereinafter), the method of averaging (using Fourier series) was
established by Kryloff and Bogoliuboff of the Kiev school of mathematics after 1930 in
connection with the asymptotic methods [1 – 3]. The method of averaging was used for
the first time by van der Pol, and then Kryloff and Bogoliuboff gave the full justification
of the method. After that, Urabe considered more general forms than the above equation
using moving coordinate system [4, 5]. Hayashi studied nonlinear oscillations mainly
from the viewpoint of physics [6, 7] and Minorsky did it from the mechanical point of
view [8].
On the other hand, the group around Mitropolsky and Samoilenko [9] investi-
gated nonlinear systems of differential equations with lag and some classes of integro-
differential and difference equations, and they developed a method for the solution of
problems concerning the existence of periodic solutions and construction of algorithms
for calculating these solutions. However, there still exist difficult problems arising in
nonlinear coupled systems or in the case of a large number of degrees of freedom with
nonlinearity due to the inevitable computing complexity of the system.
We treat the van der Pol equation system with coupling presented below by the
positional difference. Let y = y(t) and z = z(t) be two real valued functions. We
consider the dynamical system
Σε,k
y
′′ − ε(1− y2)y′ + y = k(y − z),
z′′ − ε(1− z2)z′ + z = k(z − y), t0 ≤ t.
c© B. T. NOHARA, A. ARIMOTO, 2009
1106 ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 8
NON-EXISTENCE THEOREM EXCEPT THE OUT-OF-PHASE AND IN-PHASE SOLUTIONS ... 1107
Here, k, ε (> 0) are constants and t0 indicates an initial time. When k = 0, the
dynamical system Σε,0 turns into two independent van der Pol oscillators [10]. The
single van der Pol oscillator is a well-known classical problem. Many studies on the
van der Pol equation have been carried out, and the fact that the van der Pol equation
has a unique limit cycle is known and proved by the Poincaré – Bendixson theorem (see,
for example, [11]). However, the coupled van der Pol system, that is, the dynamical
system Σε,k, constructs a three-dimensional manifold. Therefore, we cannot apply the
Poincaré – Bendixson theorem to the dynamical system Σε,k, to analyze the system.
“Does there exist the limit cycle in Σε,k?” [12], “If there exists the limit cycle, how
many limit cycles are there?” [13] and “Are the limit cycles ‘stable’ or ‘completely
unstable’ or ‘semistable’?” are still open problems we have.
In this paper, we first show the generalized van der Pol equation and analyze it.
Then the analysis of the coupled van der Pol equation system is carried out based on the
formation of our method after defining the out-of-phase and in-phase solutions, which
are new concepts arising when the system is coupled. We consider that there exist two
distinctive solutions (out-of-phase and in-phase solutions) in the dynamical system Σε,k.
Finally, we give answers to some of the above open problems.
2. Preliminaries. In this section, we present the analysis of the system Σ0,k, that
is, a coupled harmonic oscillator with linear coupling in order to reveal the features of
the system Σε,k. Before discussing our problem, note that we show easily the existence
and uniqueness of the solution of our dynamical system Σε,k.
Proposition 2.1 (existence and uniqueness theorem: see [14], Section 4.6). In the
system of differential equations
x′ = f(x), x =
x1
...
xn
, f(x) =
f1(x1, . . . , xn)
...
fn(x1, . . . , xn)
,
let each of the functions f1(x1, . . . , xn), . . . , fn(x1, . . . , xn) have continuous partial
derivatives with respect to x1, . . . , xn. Then, the initial-value problem x′ = f(x),x(t0) =
= x0 has one, and only one, solution x = x(t), for every x0 in Rn.
Note that Σε,k has a fixed point: (0, 0, 0, 0) and the eigenvalues of the system are
r± =
ε±
√
4− ε2 i
2
, s± =
ε±
√
4− 8k − ε2 i
2
.
We now consider 0 < ε < 2 and 0 < k <
1
2
− ε2
8
so that the system is unstable.
Next we consider the system Σ0,k to clarify the nature of the system Σε,k. First
we give the following definitions. Let ξΣ(t) = col
(
y(t), y′(t), z(t), z′(t)
)
be a solution
of Σε,k.
Definition 2.1. If, in the dynamical system Σε,k, one has
y(t) + z(t) = 0,
where ξΣ(t) is not equivalent to 0, then the system is out-of-phase and the non-trivial
solutions of y(t) and z(t) are called the out-of-phase solutions.
Definition 2.2. If, in the dynamical system Σε,k, one has
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 8
1108 B. T. NOHARA, A. ARIMOTO
y(t)− z(t) = 0,
where ξΣ(t) is not equivalent to 0, then the system is in-phase and the non-trivial
solutions of y(t) and z(t) are called the in-phase solutions.
Proposition 2.2. Assume that k is irrational and such that 0 < k <
1
2
. The
dynamical system Σ0,k
Σ0,k
y
′′ + y = k(y − z),
z′′ + z = k(z − y), t0 ≤ t,
has only two families of periodic solutions. A family of periodic solutions is in-phase
and its period is τ = 2π. The other is the out-of-phase solution whose period is
τ =
2π√
1− 2k
. There exists no other family of periodic solutions.
Proof. Omitted.
3. Analysis of the generalized van der Pol equation. In this section, we consider
the differential equation Wε,m,φ,
Wε,m,φ : w′′ − ε(w′ − φ) +mw = 0, (3.1)
where w = w(t), φ = φ(w,w′), 0 < ε < 2
√
m, and ′ denotes the derivative with respect
to t. We call this the generalized van der Pol equation since we obtain the ordinary van
der Pol equation if we set Wε,1,w2w′ , that is, m = 1, φ(t) = w2(t)w′(t). However, we
have no restriction regarding m ∈ R and φ = φ(w,w′) (but we simply write φ = φ(t)
instead of φ(w,w′)) in this section. We can write this in a matrix form as
x
′
w = Awxw − εξ, (3.2)
where
Aw =
(
0 1
−m ε
)
, xw =
(
w
w′
)
, ξ =
(
0
φ
)
.
We know that the solution xw(t) can be written as
xw(t) = eAw(t−t0)xw(t0)− ε
t∫
t0
eAw(t−s)ξ(s)ds. (3.3)
Here, Aw has two eigenvalues r and its complex conjugate r̄:
r =
ε+
√
4m− ε2 i
2
, r̄ =
ε−
√
4m− ε2 i
2
.
We now see that Aw has a spectral representation
Aw = rP1 + r̄P2,
E = P1 + P2, P1P2 = P2P1 = 0.
Hence, from the relation
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 8
NON-EXISTENCE THEOREM EXCEPT THE OUT-OF-PHASE AND IN-PHASE SOLUTIONS ... 1109
rP1 = Aw − r̄P2 = Aw − r̄(E − P1),
we obtain
P1 =
1
r − r̄
(Aw − r̄), P2 =
1
r − r̄
(r −Aw).
Using this, we simply write the exponential function of Aw as follows:
eAwt = ertP1 + er̄tP2 =
1
r − r̄
((
ert − er̄t
)
Aw +
(
rer̄t − r̄ert
))
.
We can easily obtain
ert − er̄t
r − r̄
= eεt/2
sin(ϑt)
ϑ
,
rer̄t − r̄ert
r − r̄
= eεt/2
(
cos(ϑt)− ε
2
sin(ϑt)
ϑ
)
,
where
ϑ =
√
4m− ε2
2
. (3.4)
Hence, we have
eAwt = eεt/2
(
sin(ϑt)
ϑ
Aw + cos(ϑt)− ε
2
sin(ϑt)
ϑ
)
and we see that, by virtue of equation (3.3), the solution of equation (3.2) satisfies
xw(t) = e
1
2 ε(t−t0)
sin
(
ϑ(t− t0)
)
ϑ
− ε/2 1
−m ε/2
+ cos
(
ϑ(t− t0)
)xw(t0)−
−ε
t∫
t0
eε(t−s)/2
sin
(
ϑ(t− s)
)
ϑ
−ε/2 1
−m ε/2
+ cos
(
ϑ(t− s)
)×
×
0
φ
(
s; t0, w(t0), w′(t0)
)
ds. (3.5)
In equation (3.5), φ
(
s; t0, w(t0), w′(t0)
)
means the function φ of s defined by the solution
with the initial condition of w(t0), w′(t0) at time t0.
Here, we define
Ut(ϑ) :=
cos(ϑt)
sin(ϑt)
ϑ
−ϑ sin(ϑt) cos(ϑt)
,
and give the following lemma:
Lemma 3.1. The following relation is true:
sin(ϑt)
ϑ
−ε/2 1
−m ε/2
+ cos(ϑt) =
−1/ε 0
−1/2 1/ε
−1
U−t(ϑ)
−1/ε 0
−1/2 1/ε
.
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 8
1110 B. T. NOHARA, A. ARIMOTO
Proof. We easily prove this by using the relation ofm = ϑ2+
ε2
4
from equation (3.4).
We can rewrite equation (3.5) using Lemma 3.1 as
xw(t) = eε(t−t0)/2
−1/ε 0
−1/2 1/ε
−1
U−(t−t0)(ϑ)
−1/ε 0
−1/2 1/ε
xw(t0)−
−ε
t∫
t0
eε(t−s)/2
−1/ε 0
−1/2 1/ε
−1
U−(t−s)(ϑ)×
×
−1/ε 0
−1/2 1/ε
0
φ
(
s; t0, w(t0), w′(t0)
)
ds.
Multiplying both sides of the above equation by e−ε(t−t0)/2Ut−t0(ϑ)
−1/ε 0
−1/2 1/ε
,
we obtain
e−ε(t−t0)/2Ut−t0(ϑ)
−1/ε 0
−1/2 1/ε
xw(t) =
=
−1/ε 0
−1/2 1/ε
xw(t0)− εU−t0(ϑ)
t∫
t0
e−ε(s−t0)/2Us(ϑ)×
×
−1/ε 0
−1/2 1/ε
0
φ
(
s; t0, w(t0), w′(t0)
)
ds =
=
−1/ε 0
−1/2 1/ε
xw(t0)− U−t0(ϑ)×
×
t∫
t0
e−ε(s−t0)/2
sin(ϑs)
ϑ
cos(ϑs)
φ
(
s; t0, w(t0), w′(t0)
)
ds.
Here, we set αw0 = w(t0) and βw0 = w′(t0) for simplicity and define the following
symbols:
Is(t, t0;αw0, βw0) :=
t∫
t0
e−ε(s−t0)/2 sin(ϑs)
ϑ
φ
(
s; t0, αw0, βw0
)
ds,
Ic(t, t0;αw0, βw0) :=
t∫
t0
e−ε(s−t0)/2 cos(ϑs)φ
(
s; t0, αw0, βw0
)
ds.
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 8
NON-EXISTENCE THEOREM EXCEPT THE OUT-OF-PHASE AND IN-PHASE SOLUTIONS ... 1111
Thus, we obtain the equation
e−ε(t−t0)/2Ut−t0(ϑ)
−1/ε 0
−1/2 1/ε
xw(t) =
=
−1/ε 0
−1/2 1/ε
αw0
βw0
− U−t0(ϑ)
Is(t, t0;αw0, βw0)
Ic(t, t0;αw0, βw0)
. (3.6)
We utilize the following relations in computing equation (3.6):
Ut(ϑ)Us(ϑ) = Ut+s(ϑ),
U−1
t (ϑ) = U−t(ϑ),
U0(ϑ) = E.
Remark 3.1. The multiplication of equation (3.6) by −ε yields
e−ε(t−t0)/2Ut−t0(ϑ)
1 0
ε/2 −1
xw(t) =
=
1 0
ε/2 −1
αw0
βw0
+ εU−t0(ϑ)
Is(t, t0;αw0, βw0)
Ic(t, t0;αw0, βw0)
. (3.7)
We simply substitute ε = 0 into equation (3.7) and obtain the solution of the harmonic
oscillator w′′ +mw = 0 as follows:
xw(t)
∣∣
ε=0
=
1 0
0 −1
Ut0−t(√m )
1 0
0 −1
α̂w0
β̂w0
=
= Ut−t0(
√
m)
α̂w0
β̂w0
. (3.8)
Here, α̂w0 and β̂w0 are arbitrary initial values, that are independent of ε and not necessari-
ly equal to αw0 and βw0, respectively. We establish that the period τ of a non-trivial
periodic solution of equation (3.8) is τ = 2π/
√
m from det
(
Uτ (
√
m)− 1
)
= 0.
Theorem 3.1. Suppose that limt→∞ e−εt/2xw(t) = 0. Then
lim
t→∞
Is(t, t0;αw0, βw0)
Ic(t, t0;αw0, βw0)
= Ut0(ϑ)
−1/ε 0
−1/2 1/ε
αw0
βw0
.
Proof. The above equation follows from directly equation (3.6).
Before stating the next theorem, we prepare the following proposition.
Proposition 3.1 (the property of autonomous systems) (for example, see [14]). The followi-
ng statements are equivalent:
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 8
1112 B. T. NOHARA, A. ARIMOTO
(1) there exists τ > 0 such that xw(t0 + τ) = xw(t0) for some t0,
(2) there exists τ > 0 such that xw(t+ τ) = xw(t) for any t.
Without warning, we often use this nature hereinafter.
Theorem 3.2. Let xw(t) be a solution of Wε,m,φ. Then the following statements
are equivalent:
(i) for some t0, one hasIs(t0 + τ, t0;αw0, βw0)
Ic(t0 + τ, t0;αw0, βw0)
=
= Ut0(ϑ)
(
1− e−ετ/2Uτ (ϑ)
)−1/ε 0
−1/2 1/ε
αw0
βw0
; (3.9)
(ii) for some t0, one hasIs(t+ τ, t0;αw0, βw0)
Ic(t+ τ, t0;αw0, βw0)
= Ut0(ϑ)
(
1− e−ετ/2Uτ (ϑ)
)−1/ε 0
−1/2 1/ε
αw0
βw0
+
+e−ετ/2Uτ (ϑ)
Is(t, t0;αw0, βw0)
Ic(t, t0;αw0, βw0)
for any t; (3.10)
(iii) xw(t) is periodic with period τ.
Proof. (ii)⇒(i). If we set t = t0 in equation (3.10), we obtain equation (3.9).
(i)⇒(iii). We assume that equation (3.9) is satisfied. Setting t = t0 + τ in equati-
on (3.6) and using equation (3.9), we obtain xw(t0 + τ) = xw(t0). Therefore, by
Proposition 3.1, we have xw(t+ τ) = xw(t).
(iii)⇒(ii). The substitution t+ τ instead of t in equation (3.6) leads to
e−ε(t+τ−t0)/2Ut+τ−t0(ϑ)
−1/ε 0
−1/2 1/ε
xw(t+ τ) =
=
−1/ε 0
−1/2 1/ε
αw0
βw0
− U−t0(ϑ)
Is(t+ τ, t0;αw0, βw0)
Ic(t+ τ, t0;αw0, βw0)
. (3.11)
Assuming xw(t+τ) = xw(t) and multiplying both sides of equation (3.11) by eετ/2U−τ (ϑ),
we have
e−ε(t−t0)/2Ut−t0(ϑ)
−1/ε 0
−1/2 1/ε
xw(t) =
= eετ/2U−τ (ϑ)
−1/ε 0
−1/2 1/ε
αw0
βw0
− eετ/2U−τ−t0(ϑ)
Is(t+ τ, t0;αw0, βw0)
Ic(t+ τ, t0;αw0, βw0)
.
(3.12)
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 8
NON-EXISTENCE THEOREM EXCEPT THE OUT-OF-PHASE AND IN-PHASE SOLUTIONS ... 1113
The equation of both right-hand sides of equations (3.6) and (3.12) yields equation (3.10).
Theorem is proved.
Here, we set m = 1 and φ = w2w′, that is, the ordinary van der Pol equation is
considered. Then we give the next proposition, which states how the orbit of ε → 0
shapes. This problem was studied in [1, 4], where the periodic orbit was obtained. For
example, in [15], this orbit was reduced under the assumption that the requirements of
the Poincaré expansion theorem are satisfied. However, we give a new proof without
this assumption.
Proposition 3.2. Let the periodic solution of the van der Pol equation be w(t, ε).
The orbit of Wε,1,w2w′ (the van der Pol equation) as ε→ 0 is presented by
w2(t, 0) + w′2(t, 0) = 4,
where w(t, 0) = lim
ε→0
w(t, ε). The period of Wε,1,w2w′ , which is denoted by τ(ε), is
represented as
τ(ε) = 2π + o(ε). (3.13)
Proof. We rewrite equation (3.9) for the peridic condition of the van der Pol equation
as follows:
Ut0(θ)
(
1− e−ετ(ε)/2Uτ(ε)(θ)
)−1/ε 0
−1/2 1/ε
w(t0, ε)
w′(t0, ε)
=
=
Is(t0 + τ(ε), t0;w(t0, ε), w′(t0, ε))
Ic(t0 + τ(ε), t0;w(t0, ε), w′(t0, ε))
, (3.14)
where θ =
√
4− ε2
2
. In the van der Pol equation, the solution w and its period τ depend
on the paremater ε so that we denote them by w(t, ε) and τ(ε), respectively. From
equation (3.14), we obtain{
cos θt0
ε
− e−(ε/2)τ(ε) cos θ
(
t0 + τ(ε)
)
ε
−
−1
2
(
e−(ε/2)τ(ε) sin θ
(
t0 + τ(ε)
)
θ
− sin θt0
θ
)}
w(t0, ε)+
+
{
e−(ε/2)τ(ε) sin θ
(
t0 + τ(ε)
)
θε
− sin θt0
θε
}
w′(t0, ε) =
= −
t0+τ(ε)∫
t0
e−ε(s−t0)/2 sin θs
θ
(
w2(s, ε)w′(s, ε)
)
ds,
{
−θ sin θt0
ε
+ e−(ε/2)τ(ε) θ sin θ
(
t0 + τ(ε)
)
ε
+
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 8
1114 B. T. NOHARA, A. ARIMOTO
+
1
2
(
−e(ε/2)τ(ε) cos θ
(
t0 + τ(ε)
)
+ cos θt0
)}
w(t0, ε)+
+
{
e−(ε/2)τ(ε) cos θ
(
t0 + τ(ε)
)
ε
− cos θt0
ε
}
w′(t0, ε) =
= −
t0+τ(ε)∫
t0
e−ε(s−t0)/2 cos θs
(
w2(s, ε)w′(s, ε)
)
ds.
We assume that ε→ 0 in the above equation. Let τ1 =
dτ(ε)
dε
∣∣∣∣
ε=0
. Note that w(t, 0) =
= cos(t0− t)w(t0, 0)− sin(t0− t)w′(t0, 0) and θ =
√
4− ε2
2
→ 1, τ(ε)→ 2π, ε→ 0.
Then we have
(π cos t0 + τ1 sin t0)w(t0, 0) + (−π sin t0 + τ1 cos t0)w′(t0, 0) =
= −π
4
(
sin t0w′(t0, 0)− cos t0w(t0, 0)
)(
w2(t0, 0) + w
′2(t0, 0)
)
,
(−π sin t0 + τ1 cos t0)w(t0, 0) + (−π cos t0 − τ1 sin t0)w′(t0, 0) =
= −π
4
(
sin t0w(t0, 0) + cos t0w′(t0, 0)
)(
w2(t0, 0) + w
′2(t0, 0)
)
.
From the above relations, we have
w2(t0, 0) + w′2(t0, 0) = 4, τ1 = 0.
Since t0 is an arbitrary initial time, we finally obtain the orbit as ε → 0 considered in
the proposition. We also have equation (3.13) from τ1 = 0.
Proposition is proved.
Remark 3.2. Proposition 3.2 is consistent with the earlier results (see, for example,
[4, p. 104] and [15, p. 133]).
4. Analysis of the coupled van der Pol equation system. 4.1. Formation of the
fundamental equations for the analysis. We now set
y(t0) = α0, y′(t0) = β0, z(t0) = λ0, z′(t0) = µ0,
and define some new symbols as follows:
φ±(t) := y2(t)y′(t)± z2(t)z′(t),
θ+ :=
√
4− ε2
2
,
θ− :=
√
4− ε2 − 8k
2
,
I±s (t, t0;α0, β0, λ0, µ0) :=
:=
t∫
t0
eε(t−s)/2
sin
(
θ±(t− s)
)
θ±
φ±(s; t0, α0, β0, λ0, µ0)ds,
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NON-EXISTENCE THEOREM EXCEPT THE OUT-OF-PHASE AND IN-PHASE SOLUTIONS ... 1115
I±c (t, t0;α0, β0, λ0, µ0) :=
:=
t∫
t0
eε(t−s)/2 cos
(
θ±(t− s)
)
φ±(s; t0, α0, β0, λ0, µ0)ds,
where a double sign ± in equations corresponds in order.
Let x+(t) = y(t) + z(t) for y, z ∈ Σε,k. Then we have the differential equation
Wε,1,φ+ corresponding to equation (3.1) of the previous section, that is,
Wε,1,φ+ : x′′+ − ε(x′+ − φ+) + x+ = 0.
We again define other symbols as follows:
Is±(t, t0;α0, β0, λ0, µ0) :=
:=
t∫
t0
e−ε(s−t0)/2 sin
(
θ±s
)
θ±
φ±(s; t0, α0, β0, λ0, µ0)ds,
Ic±(t, t0;α0, β0, λ0, µ0) :=
:=
t∫
t0
e−ε(s−t0)/2 cos
(
θ±s
)
φ±(s; t0, α0, β0, λ0, µ0)ds.
Before obtaining the fundamental equations for the analysis, we prepare the next
lemma.
Lemma 4.1. The following relation is true:Is±(t, t0;α0, β0, λ0, µ0)
Ic±(t, t0;α0, β0, λ0, µ0)
= Ut0(θ±)
Is±(t− t0, 0;α0, β0, λ0, µ0)
Ic±(t− t0, 0;α0, β0, λ0, µ0)
.
Proof. For Is±, we have
Is±(t, t0;α0, β0, λ0, µ0) =
=
t∫
t0
e−ε(s−t0)/2 sin(θ±s)
θ±
φ±(s; t0, α0, β0, λ0, µ0)ds =
=
t−t0∫
0
e−εs
′/2 sin
(
θ±(s′ + t0)
)
θ±
φ±(s′ + t0; t0, α0, β0, λ0, µ0)ds′ =
(by virtue of the property of autonoumous systems)
=
t−t0∫
0
e−εs
′/2 sin
(
θ±(s′ + t0)
)
θ±
φ±(s′; 0, α0, β0, λ0, µ0)ds′ =
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1116 B. T. NOHARA, A. ARIMOTO
= cos(θ±t0)
t−t0∫
0
e−εs/2
sin(θ±s)
θ±
φ±(s; 0, α0, β0, λ0, µ0)ds+
+
sin(θ±t0)
θ±
t−t0∫
0
e−εs/2 cos(θ±s)φ±(s; 0, α0, β0, λ0, µ0)ds =
= cos(θ±t0) Is±(t− t0, 0;α0, β0, λ0, µ0)+
+
sin(θ±t0)
θ±
Ic±(t− t0, 0;α0, β0, λ0, µ0). (4.1)
For Ic±, we have
Ic±(t, t0;α0, β0, λ0, µ0) = −θ± sin(θ±t0) Is±(t− t0, 0;α0, β0, λ0, µ0)+
+ cos(θ±t0) Ic±(t− t0, 0;α0, β0, λ0, µ0). (4.2)
From equations (4.1) and (4.2) we obtainIs±(t, t0;α0, β0, λ0, µ0)
Ic±(t, t0;α0, β0, λ0, µ0)
=
=
cos(θ±t0)
sin(θ±t0)
θ±
−θ± sin(θ±t0) cos(θ±t0)
Is±(t− t0, 0;α0, β0, λ0, µ0)
Ic±(t− t0, 0;α0, β0, λ0, µ0)
.
Using the definition of the rotational matrix U, we prove the lemma.
As the fundamental equation for x+(t), that is, y(t) + z(t), we have the follo-
wing linear system of integral equations using integral symbols defined above, which
corresponds to equation (3.6):
e−ε(t−t0)/2Ut−t0(θ+)
−1/ε 0
−1/2 1/ε
x+(t)
x′+(t)
=
=
−1/ε 0
−1/2 1/ε
x+(t0)
x′+(t0)
− U−t0(θ+)
Is+(t, t0;α0, β0, λ0, µ0)
Ic+(t, t0;α0, β0, λ0, µ0)
. (4.3)
By applying Lemma 4.1 to the above equation, we obtain
e−ε(t−t0)/2Ut−t0(θ+)
−1/ε 0
−1/2 1/ε
x+(t)
x′+(t)
=
=
−1/ε 0
−1/2 1/ε
x+(t0)
x′+(t0)
−
Is+(t− t0, 0;α0, β0, λ0, µ0)
Ic+(t− t0, 0;α0, β0, λ0, µ0)
. (4.4)
If we set x−(t) = y(t)− z(t) for y, z ∈ Σε,k, then we obtain Wε,1−2k,φ− , that is,
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NON-EXISTENCE THEOREM EXCEPT THE OUT-OF-PHASE AND IN-PHASE SOLUTIONS ... 1117
Wε,1−2k,φ− : x′′− − ε(x′− − φ−) + (1− 2k)x− = 0.
In the same way, we obtain the linear system of integral equations for x−(t) = y(t) −
− z(t) :
e−ε(t−t0)/2Ut−t0(θ−)
−1/ε 0
−1/2 1/ε
x−(t)
x′−(t)
=
=
−1/ε 0
−1/2 1/ε
x−(t0)
x′−(t0)
−
Is−(t− t0, 0;α0, β0, λ0, µ0)
Ic−(t− t0, 0;α0, β0, λ0, µ0)
.
4.2. Necessary and sufficient condition for the periodicity of the coupled van der
Pol equation system. We give the necessary and sufficient condition for the periodicity
of the solutions of the coupled van der Pol equation system in this subsection. First, the
following theorem holds in the same way as Theorem 3.1.
Theorem 4.1. Suppose that limt→∞ e−εt/2col
(
x±(t), x′±(t)
)
= 0. Then
lim
t→∞
Is±(t, t0;α0, β0, λ0, µ0)
Ic±(t, t0;α0, β0, λ0, µ0)
= Ut0(θ±)
−1/ε 0
−1/2 1/ε
x±(t0)
x′±(t0)
.
In this theorem, a double sign ± corresponds in order.
Proof. Omitted.
Below, we state some properties for the case where the system has the periodicity.
Remember that ξΣ(t) = col
(
y(t), y′(t), z(t), z′(t)
)
.
Theorem 4.2. Suppose that ξΣ(t + τ) = ξΣ(t), then the following relations are
equivalent for a fixed t0 :
(i) x+(t0) = 0, x′+(t0) = 0;
(ii) Is+(t0 + nτ, t0;α0, β0, λ0, µ0) = 0, Ic+(t0 + nτ, t0;α0, β0, λ0, µ0) = 0, n =
= 1, 2, . . . .
Proof. (i)⇒ (ii). Substituting t = t0 + nτ into equation (4.3), we obtain
e−εnτ/2Unτ (θ+)
−1/ε 0
−1/2 1/ε
x+(t0 + nτ)
x′+(t0 + nτ)
=
=
−1/ε 0
−1/2 1/ε
x+(t0)
x′+(t0)
− U−t0(θ+)
Is+(t0 + nτ, t0;α0, β0, λ0, µ0)
Ic+(t0 + nτ, t0;α0, β0, λ0, µ0)
.
From the assumption of the theorem, we have
y(t0 + nτ)
y′(t0 + nτ)
z(t0 + nτ)
z′(t0 + nτ)
=
y(t0)
y′(t0)
z(t0)
z′(t0)
=
α0
β0
λ0
µ0
.
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1118 B. T. NOHARA, A. ARIMOTO
The substitution of this result yields
e−εnτ/2Unτ (θ+)
−1/ε 0
−1/2 1/ε
x+(t0)
x′+(t0)
=
=
−1/ε 0
−1/2 1/ε
x+(t0)
x′+(t0)
−
− U−t0(θ+)
Is+(t0 + nτ, t0;α0, β0, λ0, µ0)
Ic+(t0 + nτ, t0;α0, β0, λ0, µ0)
. (4.5)
Using the relation from (i), that is, x+(t0) = 0 and x′+(t0) = 0, we arrive at (ii).
(ii)⇒ (i). The substitution of (ii) into equation (4.5) leads to
x+(t0) = 0,
x′+(t0) = 0,
which means that (i) holds.
In the same manner, we obtain the next theorem.
Theorem 4.3. Suppose that ξΣ(t + τ) = ξΣ(t). Then the following relations are
equivalent for a fixed t0 :
(i) x−(t0) = 0, x′−(t0) = 0;
(ii) Is−(t0 + nτ, t0;α0, β0, λ0, µ0) = 0, Ic−(t0 + nτ, t0;α0, β0, λ0, µ0) = 0, n =
= 1, 2, . . . .
Proof. We can prove this theorem by the same manner as Theorem 4.2.
Lemma 4.2. The following relations are equivalent:
(i) ξΣ(t+ τ) = ξΣ(t),
(ii) x±(t+ τ) = x±(t).
Theorem 4.4 (necessary and sufficient condition for the periodicity). The solution of
the dynamical system Σε,k with the initial condition
y(t0) = α0, y′(t0) = β0, z(t0) = λ0, z′(t0) = µ0
has a period τ if and only if
F±(ε) = 0, (4.6)
where
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NON-EXISTENCE THEOREM EXCEPT THE OUT-OF-PHASE AND IN-PHASE SOLUTIONS ... 1119
F±(ε) =
(
1− e−ετ/2Uτ (θ±)
) 1 0
ε/2 −1
x±(t0)
x′±(t0)
+
+ε
Is±(τ, 0;α0, β0, λ0, µ0)
Ic±(τ, 0;α0, β0, λ0, µ0)
. (4.7)
Proof. Necessity. Σε,k has a period, that is, ξΣ(t + τ) = ξΣ(t) for some τ > 0,
because x±(t + τ) = x±(t) and x′±(t + τ) = x′±(t) from Lemma 4.2. Therefore, we
have the following equation by the same procedure which yields equation (3.9):
(
1− e−ετ/2Uτ (θ±)
) 1 0
ε/2 −1
x±(t0)
x′±(t0)
+
+ εU−t0(θ±)
Is±(t0 + τ, t0;α0, β0, λ0, µ0)
Ic±(t0 + τ, t0;α0, β0, λ0, µ0)
= 0. (4.8)
The second term is computed by Lemma 4.1 as
U−t0(θ±)
Is±(t0 + τ, t0;α0, β0, λ0, µ0)
Ic±(t0 + τ, t0;α0, β0, λ0, µ0)
=
Is±(τ, 0;α0, β0, λ0, µ0)
Ic±(τ, 0;α0, β0, λ0, µ0)
, (4.9)
and the substitution of this result into eqaution (4.8) leads to equations (4.6) and (4.7).
Sufficiency. Here, we prove that F± = 0 ⇒ x±(t0 + τ) = x±(t0), which is
equivalent to x±(t+ τ) = x±(t). Using equation (4.9) in equation (4.7), we have
e−ετ/2Ut0+τ (θ±)
1 0
ε/2 −1
x±(t0)
x′±(t0)
=
= Ut0(θ±)
1 0
ε/2 −1
x±(t0)
x′±(t0)
+
+ε
Is±(t0 + τ, t0;α0, β0, λ0, µ0)
Ic±(t0 + τ, t0;α0, β0, λ0, µ0)
. (4.10)
On the other hand, the substitution of t = t0 + τ into equation (4.3) yields
e−ετ/2Uτ (θ+)
1 0
ε/2 −1
x+(t0 + τ)
x′+(t0 + τ)
=
1 0
ε/2 −1
x+(t0)
x′+(t0)
+
+εU−t0(θ+)
Is+(t0 + τ, t0;α0, β0, λ0, µ0)
Ic+(t0 + τ, t0;α0, β0, λ0, µ0)
,
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1120 B. T. NOHARA, A. ARIMOTO
that is,
e−ετ/2Ut0+τ (θ+)
1 0
ε/2 −1
x+(t0 + τ)
x′+(t0 + τ)
=
= Ut0(θ+)
1 0
ε/2 −1
x+(t0)
x′+(t0)
+
+ε
Is+(t0 + τ, t0;α0, β0, λ0, µ0)
Ic+(t0 + τ, t0;α0, β0, λ0, µ0)
. (4.11)
Similarly, we have
e−ετ/2Ut0+τ (θ−)
1 0
ε/2 −1
x−(t0 + τ)
x′−(t0 + τ)
=
= Ut0(θ−)
1 0
ε/2 −1
x−(t0)
x′−(t0)
+
+ε
Is−(t0 + τ, t0;α0, β0, λ0, µ0)
Ic−(t0 + τ, t0;α0, β0, λ0, µ0)
. (4.12)
The subtraction of equation (4.10) from equations (4.11) and (4.12) leads to
e−ετ/2Ut0+τ (θ±)
1 0
ε/2 −1
x±(t0 + τ)
x′±(t0 + τ)
−
x±(t0)
x′±(t0)
= 0.
Therefore, we obtain
x±(t0 + τ) = x±(t0),
x′±(t0 + τ) = x′±(t0).
Consequently, F±(ε) = 0⇒ x±(t0 + τ) = x±(t0) is proved.
Theorem is proved.
5. Non-existence theorem of periodic solutions except the out-of-phase and in-
phase solutions in Σε,k. Let y = y(t, ε) and z = z(t, ε) be two real-valued functions
depending on the parameter ε and 0 < ε < 2, 0 < k <
1
2
− ε2
8
. Our objective equation
system Σε,k is as follows:
Σε,k
y
′′ − ε(1− y2)y′ + y = k(y − z),
z′′ − ε(1− z2)z′ + z = k(z − y), t0 ≤ t,
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NON-EXISTENCE THEOREM EXCEPT THE OUT-OF-PHASE AND IN-PHASE SOLUTIONS ... 1121
with the initial condition
y(t0, ε) = α0(ε), y′(t0, ε) = β0(ε),
z(t0, ε) = λ0(ε), z′(t0, ε) = µ0(ε),
where the initial condition also depends on the parameter ε because we write α0(ε),
β0(ε), λ0(ε) and µ0(ε) deliberately.
Here, we give the assumption on periodic solutions of the dynamical system Σε,k.
Assumption 5.1 (periodic solutions of Σε,k). Periodic solutions of Σε,k satisfy
y
(
t+ τ(ε), ε
)
= y(t, ε), z
(
t+ τ(ε), ε
)
= z(t, ε), |τ(ε)| < T, (5.1)
where τ indicates a period of Σε,k and T is independent of the parameter ε. Moreover,
periodic solutions and their derivatives satisfy
|y(t, ε)| < M, |y′(t, ε)| < M, |z(t, ε)| < M, |z′(t, ε)| < M, (5.2)
where M is independent of the parameter ε and t.
Hereinafter, we consider only periodic solutions restricted by Assumption 5.1. Before
stating the main theorem, we prepare the following lemma.
Lemma 5.1. Let y(t, ε), z(t, ε) be a periodic solution of Σε,k satisfying Assump-
tion 5.1. Assume that there exists lim
ε→0
x±(t0, ε) and lim
ε→0
x±(t0, ε) = x±(t0, 0). Then
there exists a solution y(t) and z(t) of the degenerated system Σ0,k such that
lim
ε→0
x±(t, ε) = x±(t, 0) = y(t) ± z(t) and lim
ε→0
x′±(t, ε) = x′±(t, 0) = y′(t) ± z′(t).
Let τ±(ε) and τ±(0) be periods of x±(t, ε) created by Σε,k and x±(t, 0) by Σ0,k,
respectively. Then lim
ε→0
τ±(ε) = τ±(0).
Proof. We only show that lim
ε→0
x+(t, ε) = x+(t, 0) = y(t) + z(t) and lim
ε→0
τ+(ε) =
= τ+(0). From equation (4.4), x+(t, ε) and x′+(t, ε) are represented asx+(t, ε)
x′+(t, ε)
= eε(t−t0)/2
1 0
ε/2 −1
−1
×
×Ut0−t(θ+)
1 0
ε/2 −1
x+(t0, ε)
x′+(t0, ε)
+
+εeε(t−t0)/2
1 0
ε/2 −1
−1
Ut0−t(θ+)
Is+(t− t0, 0;α0, β0, λ0, µ0)
Ic+(t− t0, 0;α0, β0, λ0, µ0)
=
= eε(t−t0)/2
cos θ+(t0 − t) +
ε
2
sin θ+(t0 − t)
θ+
− sin θ+(t0 − t)
θ+
θ+ sin θ+(t0−t)+
ε2
4
sin θ+(t0−t)
θ+
cos θ+(t0−t)−
ε
2
sin θ+(t0−t)
θ+
×
×
x+(t0, ε)
x′+(t0, ε)
+ εeε(t−t0)/2
1 0
ε/2 −1
−1
×
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1122 B. T. NOHARA, A. ARIMOTO
×Ut0−t(θ+)
Is+(t− t0, 0;α0, β0, λ0, µ0)
Ic+(t− t0, 0;α0, β0, λ0, µ0)
.
We take ε→ 0 in both sides of the above equation. By virtue of Assumption 5.1, i.e., by
virtue of the relations |y(t, ε)| < M, |y′(t, ε)| < M, |z(t, ε)| < M, and |z′(t, ε)| < M,
the second term vanishes. Therefore, we have
lim
ε→0
x+(t, ε) = x+(t, 0) = cos(t0 − t)x+(t0, 0)− sin(t0 − t)x′+(t0, 0),
lim
ε→0
x′+(t, ε) = x′+(t, 0) = sin(t0 − t)x+(t0, 0) + cos(t0 − t)x′+(t0, 0).
(5.3)
In the same manner, we obtain
lim
ε→0
x−(t, ε) = x−(t, 0) = cos
√
1− 2k(t0 − t)x−(t0, 0) −
− sin
√
1− 2k(t0 − t)√
1− 2k
x′−(t0, 0),
lim
ε→0
x′−(t, ε) = x′−(t, 0)
√
1− 2k sin
√
1− 2k(t0 − t)x−(t0, 0) +
+ cos
√
1− 2k(t0 − t)x′−(t0, 0).
(5.4)
From equations (5.3) and (5.4), we construct y and z as follows:
y(t) =
x+(t, 0) + x−(t, 0)
2
, z(t) =
x+(t, 0)− x−(t, 0)
2
. (5.5)
We easily find that y and z satisfy Σ0,k. From equations (5.3), (5.4) and (5.5), we obtain
lim
ε→0
x±(t, ε) = x±(t, 0) = y(t)± z(t),
lim
ε→0
x′±(t, ε) = x′±(t, 0) = y′(t)± z′(t).
(5.6)
Furthermore, using the assumption on periodic solutions, we have
x±(t+ τ±(ε), ε) = x±(t, ε), (5.7)
x±(t+ τ±(0), 0) = x±(t, 0). (5.8)
From equations (5.6), (5.7) and (5.8), we get
lim
ε→0
τ±(ε) = τ±(0).
Also we obtain τ+(0) = 2π and τ−(0) =
2π√
1− 2k
.
We give the next main theorem for Σε,k.
Theorem 5.1 (non-existence of periodic solutions except the out-of-phase and in-
phase solutions). Let y(t, ε) and z(t, ε) be a periodic solution of Σε,k, which is analytic
with respect to ε on the segment [0, ε0), where 0 < ε0 < 2, 0 < k <
1
2
− ε2
0
8
, and k is
irrational. Then this solution is either out-of-phase or in-phase.
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NON-EXISTENCE THEOREM EXCEPT THE OUT-OF-PHASE AND IN-PHASE SOLUTIONS ... 1123
Preparations for the proof. We assume that the periodicity is built up and a period
(but unknown) is τ(ε) depending on ε. Then we have the following relation from
Theorem 4.4:
F±(ε) = 0,
where
F±(ε) =
(
1− e−ετ(ε)/2Uτ(ε)(θ±)
) 1 0
ε/2 −1
x±(t0, ε)
x′±(t0, ε)
+
+ε
Is±(τ(ε), 0;α0(ε), β0(ε), λ0(ε), µ0(ε))
Ic±(τ(ε), 0;α0(ε), β0(ε), λ0(ε), µ0(ε))
.
First, we take ε→ 0 in F+(ε) = 0. Then we have1− cos(τ(0)) sin(τ(0))
− sin(τ(0)) 1− cos(τ(0))
x+(t0, 0)
x′+(t0, 0)
= 0. (5.9)
Here, τ(0) = lim
ε→0
τ(ε).
On the other hand, taking ε→ 0 in F−(ε) = 0, we have 1− cos(
√
1− 2kτ(0))
sin(
√
1− 2kτ(0))√
1− 2k
−
√
1− 2k sin(
√
1− 2kτ(0)) 1− cos(
√
1− 2kτ(0))
x−(t0, 0)
x′−(t0, 0)
= 0. (5.10)
Equations (5.9) and (5.10) must hold simultaneously because we have the following
results for each t0 :
(i) Equation (5.9) implies
x+(t0, 0)
x′+(t0, 0)
= 0 or τ(0) = 2π. In the latter case, we
set τ−(0) = 2π for the sake of convenience.
(ii) Similarly, equation (5.10) implies
x−(t0, 0)
x′−(t0, 0)
= 0 or τ(0) =
2π√
1− 2k
. In
the latter case, we set τ+(0) =
2π√
1− 2k
for the sake of convenience.
(iii) If k is irrational and satisfies 0 < k <
1
2
− ε2
8
, then jτ+(0) 6= lτ−(0),
j, l = 1, 2, 3, . . . , j 6= l. Therefore, we obtain following two conditions: a condition
is
x+(t0, 0)
x′+(t0, 0)
= 0 and τ+(0) =
2π√
1− 2k
and another condition is
x−(t0, 0)
x′−(t0, 0)
= 0
and τ−(0) = 2π, since (i) and (ii) must hold simultaneously. We take some t0 in the
above consideration, but we find that t0 can be taken arbitrary in this stage. Consequently,
the former condition means out-of-phase and the latter in-phase.
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 8
1124 B. T. NOHARA, A. ARIMOTO
Note that the condition of
x+(t0, 0)
x′+(t0, 0)
= 0 and
x−(t0, 0)
x′−(t0, 0)
= 0 is α0(0) =
= β0(0) = λ0(0) = µ0(0), that is, the origin.
Summarizing above, when ε = 0, there exists no periodic solutions except the out-
of-phase and in-phase solutions, in which periods are τ+(0) =
2π√
1− 2k
and τ−(0) =
= 2π, respectively. This fact is consistent with Proposition 2.2. Before proving the main
theorem, we prepare two propositions and give the following definitions in order to
prove the propositions using the inductive method.
Definition 5.1. The statement P+(ν), ν = 1, 2, 3, . . . , is defined as follows:
If
x+(t0, 0)
x′+(t0, 0)
= 0, then there exist derivatives
∂νx+(t, ε)
∂εν
and
∂νx′+(t, ε)
∂εν
, and
∂νx+(t, ε)
∂εν
= 0 and
∂νx′+(t, ε)
∂εν
= 0 at ε = 0.
Definition 5.2. The statement P−(ν), ν = 1, 2, 3, . . . , is defined as follows:
If
x−(t0, 0)
x′−(t0, 0)
= 0, then there exist derivatives
∂νx−(t, ε)
∂εν
and
∂νx′−(t, ε)
∂εν
, and
∂νx−(t, ε)
∂εν
= 0 and
∂νx′−(t, ε)
∂εν
= 0 at ε = 0.
Proposition 5.1. P+(ν) is true for ν = 1, 2, 3, . . . .
Proposition 5.2. P−(ν) is true for ν = 1, 2, 3, . . . .
Proof. We prove only Proposition 5.1 using the inductive method because Proposi-
tion 5.2 can be proved by the same manner.
(i) x+(t, 0) defined in equation (5.3) satisfies the differential equations x′′+(t, 0) +
+ x+(t, 0) = 0 with the initial conditions x+(t0, 0) and x′+(t0, 0). By uniqueness of
the solution, we must have
x+(t, 0)
x′+(t, 0)
≡ 0 for
x+(t0, 0)
x′+(t0, 0)
= 0. Hence,
lim
ε→0
x+(t, ε)
x′+(t, ε)
= 0 from Lemma 5.1. Then we have
y2(s, ε)y′(s, ε) + z2(s, ε)z′(s, ε) =
=
(
x+(s, ε)− z(s, ε)
)2
x′+(s, ε)−
−z′(s, ε)
(
y(s, ε)− z(s, ε)
)
x+(s, ε)→ 0, ε→ 0. (5.11)
Since we have F+(ε) = 0 by the periodicity condition, i.e.,
(
1− e−ετ(ε)/2Uτ(ε)(θ+)
) 1 0
ε/2 −1
x+(t0, ε)
x′+(t0, ε)
+
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 8
NON-EXISTENCE THEOREM EXCEPT THE OUT-OF-PHASE AND IN-PHASE SOLUTIONS ... 1125
+ ε
τ(ε)∫
0
e−εs/2
sin(θ+s)
θ+
(
y2(t0 + s, ε)y′(t0 + s, ε) +
+ z2(t0 + s, ε)z′(t0 + s, ε)
)
ds
τ(ε)∫
0
e−εs/2 cos(θ+s)
(
y2(t0 + s, ε)y′(t0 + s, ε) +
+ z2(t0 + s, ε)z′(t0 + s, ε)
)
ds
= 0. (5.12)
Dividing equation (5.12) by ε, we obtain
(
1− e−ετ(ε)/2Uτ(ε)(θ+)
) 1 0
ε/2 −1
x+(t0, ε)
ε
x′+(t0, ε)
ε
+
+
τ(ε)∫
0
e−εs/2
sin(θ+s)
θ+
(
y2(t0 + s, ε)y′(t0 + s, ε) +
+ z2(t0 + s, ε)z′(t0 + s, ε)
)
ds
τ(ε)∫
0
e−εs/2 cos(θ+s)
(
y2(t0 + s, ε)y′(t0 + s, ε) +
+ z2(t0 + s, ε)z′(t0 + s, ε)
)
ds
= 0. (5.13)
We take ε → 0 in equation (5.13). Then the second term vanishes from equati-
on (5.11) and there exist the derivatives
∂x+(t0, 0)
∂ε
= lim
ε→0
x+(t0, ε)− x+(t0, 0)
ε
and
∂x′+(t0, 0)
∂ε
= lim
ε→0
x′+(t0, ε)− x′+(t0, 0)
ε
. Here we can take arbitrary t0, therefore, we
have the derivatives
∂x+(t, 0)
∂ε
and
∂x′+(t, 0)
∂ε
. Furthermore, we obtain
∂x+(t, 0)
∂ε
= 0
and
∂x′+(t, 0)
∂ε
= 0.
Note that, in the computation of the limit, we can exchange the limit and the integral.
We show below this fact. The integral of equation (5.13) is written as follows using T
defined in equation (5.1):
τ(ε)∫
0
e−εs/2
sin(θ+s)
θ+
(
y2(t0 + s, ε)y′(t0 + s, ε) + z2(t0 + s, ε)z′(t0 + s, ε)
)
ds =
=
T∫
0
1τ(ε)(s)e−εs/2
sin(θ+s)
θ+
(
y2(t0 + s, ε)y′(t0 + s, ε)+
+z2(t0 + s, ε)z′(t0 + s, ε)
)
ds,
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 8
1126 B. T. NOHARA, A. ARIMOTO
where
1τ(ε)(s) =
1, for s ≤ τ(ε),
0, for s > τ(ε).
Now we find ∣∣∣∣∣1τ(ε)(s)e−εs/2
sin(θ+s)
θ+
(
y2(t0 + s, ε)y′(t0 + s, ε)+
+z2(t0 + s, ε)z′(t0 + s, ε)
)∣∣∣∣∣ ≤
≤ 1
θ+
∣∣∣y2(t0 + s, ε)y′(t0 + s, ε) + z2(t0 + s, ε)z′(t0 + s, ε)
∣∣∣ ≤M
for 0 < s < T.
Here, for the sake of convenience, we use the same symbol M as in relations (5.2), but
they are different from each other. Then we can apply the bounded convergence theorem
and we obtain
lim
ε→0
τ(ε)∫
0
e−εs/2
sin(θ+s)
θ+
(
y2(t0 + s, ε)y′(t0 + s, ε)+
+z2(t0 + s, ε)z′(t0 + s, ε)
)
ds =
=
T∫
0
lim
ε→0
1τ(ε)(s)e−εs/2
sin(θ+s)
θ+
(
y2(t0 + s, ε)y′(t0 + s, ε)+
+z2(t0 + s, ε)z′(t0 + s, ε)
)
ds =
=
T∫
0
1τ(0)(s) sin s lim
ε→0
(
y2(t0 + s, ε)y′(t0 + s, ε)+
+z2(t0 + s, ε)z′(t0 + s, ε)
)
ds = 0.
In the above equation, we use the relation τ(ε) → τ(0) as ε → 0. In fact, we have
lim
ε→0
x±
(
t+ τ(ε), ε
)
= x±
(
t+ τ(0), 0
)
from the assumption lim
ε→0
x±(t, ε) = x±(t, 0) =
= y(t)± z(t) and the periodicity conditions lim
ε→0
x±
(
t+ τ(ε), ε
)
= x±(t, 0) and x±
(
t+
+ τ(0), 0
)
= x±(t, 0).
(ii) We assume that P+(ν), ν ≤ n, is true, i.e., there exist
∂νx+(t0, 0)
∂εν
and
∂νx′+(t0, 0)
∂εν
and
∂νx+(t0, 0)
∂εν
= 0,
∂νx′+(t0, 0)
∂εν
= 0, ν = 0, 1, 2, . . . , n. Then we
show that P+(n+ 1) is true. Dividing equation (5.12) by εn+1, we obtain
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 8
NON-EXISTENCE THEOREM EXCEPT THE OUT-OF-PHASE AND IN-PHASE SOLUTIONS ... 1127
(
1− e−ετ(ε)/2Uτ(ε)(θ+)
) 1 0
ε/2 −1
x+(t0, ε)−
n∑
ν=1
∂νx+(t0, 0)
∂εν
εν
εn+1
x′+(t0, ε)−
n∑
ν=1
∂νx′+(t0, 0)
∂εν
εν
εn+1
+
+
τ(ε)∫
0
e−εs/2
sin(θ+s)
θ+
{(
x+(t0 + s, ε)− z(t0 + s, ε)
)2x′+(t0 + s, ε)
εn
−
−z′(t0 + s, ε)
(
y(t0 + s, ε)− z(t0 + s, ε)
)x+(t0 + s, ε)
εn
}
ds
τ(ε)∫
0
e−εs/2 cos(θ+s)
{(
x+(t0 + s, ε)− z(t0 + s, ε)
)2x′+(t0 + s, ε)
εn
−
−z′(t0 + s, ε)
(
y(t0 + s, ε)− z(t0 + s, ε)
)x+(t0 + s, ε)
εn
}
ds
= 0.
Here, if we take ε → 0, then the second term vanishes since lim
ε→0
x+(t0 + s, ε)
εn
=
= 0, lim
ε→0
x′+(t0 + s, ε)
εn
= 0. Therefore, we find that there exist the derivatives
∂n+1x+(t0, 0)
∂εn+1
and
∂n+1x′+(t0, 0)
∂εn+1
. Since t0 is arbitrary, we have the existence of
∂n+1x+(t, 0)
∂εn+1
and
∂n+1x′+(t, 0)
∂εn+1
. Furthermore, we obtain
∂n+1x+(t, 0)
∂εn+1
= 0,
∂n+1x′+(t, 0)
∂εn+1
= 0.
(iii) From (i) and (ii), we obtain that P+(ν) is true for any ν ∈ N .
The fact that the – part, i.e., P−(ν), is true for any ν ∈ N can also be proved in the
same way using the relation
y2(s, ε)y′(s, ε)− z2(s, ε)z′(s, ε) =
=
(
x−(s, ε) + z(s, ε)
)2
x′−(s, ε)+
+z′(s, ε)
(
y(s, ε) + z(s, ε)
)
x−(s, ε).
We obtain the following lemma from Propositions 5.1 and 5.2.
Lemma 5.2. We assume that y(t, ε) and z(t, ε) are analytic with respect to the
parameter ε. If
x+(t0, 0)
x′+(t0, 0)
= 0, then
x+(t0, ε)
x′+(t0, ε)
= 0.Moreover, If
x−(t0, 0)
x′−(t0, 0)
=
= 0, then
x−(t0, ε)
x′−(t0, ε)
= 0.
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 8
1128 B. T. NOHARA, A. ARIMOTO
Proof of Theorem 5.1. From Lemma 5.2, if
x+(t0, ε)
x′+(t0, ε)
6= 0 and
x−(t0, ε)
x′−(t0, ε)
6=
6= 0, then
x+(t0, 0)
x′+(t0, 0)
6= 0 and
x−(t0, 0)
x′−(t0, 0)
6= 0. However, this is inconsistent
with Proposition 2.2 under the assumption that k is irrational, which states that the
dynamical system Σ0,k does not have solutions except the out-of-phase and in-phase
ones. Therefore, we have
x+(t, ε)
x′+(t, ε)
= 0 or
x−(t, ε)
x′−(t, ε)
= 0. Consequently, the
dynamical system Σε,k does not have any other periodic solutions except the out-of-
phase and in-phase solutions.
We give the following consideration for Theorem 5.1.
Remark 5.1. We consider the averaged system of Σε,k. First, using the symbols
x+ and x−, we transform Σε,k into
x
′′
+ + x+ =
ε
4
(
x
′
+(4− x2
+ − x2
−)− 2x
′
−x+x−
)
,
x
′′
− + (1− 2k)x− =
ε
4
(
x
′
−(4− x2
+ − x2
−)− 2x
′
+x+x−
)
.
Passing to polar coordinates x+ = a+ sin θ+, x
′
+ = a+ cos θ+, x− = a− sin θ−, x
′
− =
= a−
√
1− 2k cos θ− and averaging the right-hand side of the obtained system with
respect to the phase variable θ+, θ−, we obtain the following averaged system:
a
′
1 =
ε
32
(16a1 − 2a1a
2
2 − a3
1),
a
′
2 =
ε
32
(16a2 − 2a2a
2
1 − a3
2),
θ
′
1 = 1,
θ
′
2 =
√
1− 2k,
(5.14)
where a1, a2, θ1, θ2 denote the averaged counterparts of a+, a−, θ+, θ−. From the
two dimensional system given by the first two equations of equation (5.14), we find
four fixed points: (0, 0), (4, 0), (0, 4),
(
4√
3
,
4√
3
)
. The first three are focuses, one
unstable and two stable, while the last one is a saddle. Paying attention to the phase
variables, we conclude that the averaged system has four invariant tori: one unstable
zero-dimensional (the zero solution, i.e., the origin), two stable one-dimensional (the
limit of the out-of-phase and in-phase solutions), one semistable two-dimensional.
According to a theorem from [16], for small enough ε, in proximity of the above-
listed invariant tori of the averaged system, the corresponding analytically smooth invari-
ant tori of the system Σε,k lie, which have the same dimensions and stability. For small
enough ε, periodic trajectories on the semistable two-dimensional torus cannot be put in
the form of analytic functions in ε satisfying the condition of periodicity.
We also present the next theorem, which shows that the orbits of Σε,k as ε → 0
become the specific orbits in Σ0,k.
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 8
NON-EXISTENCE THEOREM EXCEPT THE OUT-OF-PHASE AND IN-PHASE SOLUTIONS ... 1129
Theorem 5.2. Let k be irrational and let 0 < k <
1
2
− ε2
8
. The orbit of Σε,k as
ε→ 0 is presented by
y2(t, 0) + y′2(t, 0) = 4,
z2(t, 0) + z′2(t, 0) = 4
at in-phase, i.e., y(t, 0)− z(t, 0) = 0, and
y2(t, 0) +
y′2(t, 0)
1− 2k
= 4,
z2(t, 0) +
z′2(t, 0)
1− 2k
= 4
at out-of-phase, i.e., y(t, 0) + z(t, 0) = 0.
The period of in-phase and out-of-phase, which are denoted by τ+(ε) and τ−(ε), are
represented as
τ+(ε) = 2π + o(ε), τ−(ε) =
2π
1− 2k
+ o(ε).
Proof. We prove this theorem as the same procedure of Proposition 3.2 without any
special assumptions.
Acknowledgment. The authors are gratefull to academician A. M. Samoilenko,
NAS of Ukraine, for the opportunity of presentation of this paper at BOGOLYUBOV
READINGS-2008, Kirillovka, Ukraine, 2008.
1. Kryloff N., Bogoliuboff N. The application of methods of nonlinear mechanics to the theory of stationary
oscillations (in Russian). – Kiev: Publ. Ukr. Acad. Sci., 1934. – 8.
2. Kryloff N., Bogoliuboff N. N. Introduction to nonlinear mechanics // Ann. Math. Stud. – Princeton, N.J.:
Princeton Univ. Press, 1947. – № 11.
3. Bogoliuboff N. N., Mitropolski Yu. A. Asymptotic methods in the theory of nonlinear oscillations. –
Moscow, 1958 (Engl. ver. New York: Gordon and Breach, 1962).
4. Urabe M. Nonlinear problems — autonomous oscillations (in Japanese). – Tokyo: Kyoritsushuppan,
1957, 1968 (revised ver.).
5. Urabe M. Nonlinear autonomous oscillations // Anal. Theory. – New York: Acad. Press, 1967.
6. Hayashi C. Nonlinear oscillations in physical systems. – New York: McGraw-Hill, 1964.
7. Furuya S., Nagumo J. Nonlinear oscillations // Ser. Modern Appl. Math. – Tokyo: Iwanamishoten, 1957
(in Japanese).
8. Minorsky N. Nonlinear oscillations. – Princeton: Van Nostrand, 1962.
9. Mitropolsky Yu. A., Samoilenko A. M., Martynyuk D. I. Systems of evolution equations with periodic
and quasiperiodic coefficients. – Kluwer Acad. Publ., 1993.
10. van der Pol B. On relaxation oscillations // Phil. Mag. – 1926. – 2. – P. 978 – 992.
11. Guckenheimer H., Holmes P. Nonlinear oscillations // Dynam. Systems, Appl. Math. Sci. – Berlin:
Springer, 1983. – 42.
12. Nohara B. T., Arimoto A. Non-existence theorem except in-phase and out-of-phase solutions in the
coupled van der Pol equation system // Int. Sci. Conf.: Different. Equat., Theory Functions and Their
Appl. (Melitopol, Ukraine, June 16 – 21, 2008). – P. 86 – 87.
13. Nohara B. T., Arimoto A. Limit cycles of the coupled van der Pol equation system // Proc. Annual Conf.
Jap. Math. Soc. (Tokyo, Japan, September 24 – 28, 2008).
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Received 24.12.08
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 8
|
| id | umjimathkievua-article-3084 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:35:57Z |
| publishDate | 2009 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/d5/c6c085592e878eb00302fe17cf04a9d5.pdf |
| spelling | umjimathkievua-article-30842020-03-18T19:44:57Z Nonexistence theorem except the out-of-phase and in-phase solutions in the coupled van der Pol equation system Теорема про неіснування розв'язків за винятком режиму синхронних коливань та режиму коливань у протифазі для системи з'єднаних рівнянь Ван Дер Поля Nohara, B. T. Нохара, Б. Т. We consider a coupled van der Pol equation system. Our coupled system consists of two van der Pol equations that are connected with each other by linear terms. We assume that two distinctive solutions (out-of-phase and in-phase solutions) exist in the dynamical system of coupled equations and give answers to some problems. Розглянуто систему з'єднаних рівнянь Ван дер Поля. Ця система складається з двох рівнянь Ван дер Поля, що пов'язані між собою лінійними членами. У статті розглянуто випадок, коли динамічна система з'єднаних рівнянь має два різних розв'язки (у режимі синхронних коливань та у режимі коливань у протифазі), і дано відповіді на деякі питання. Institute of Mathematics, NAS of Ukraine 2009-08-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3084 Ukrains’kyi Matematychnyi Zhurnal; Vol. 61 No. 8 (2009); 1106-1129 Український математичний журнал; Том 61 № 8 (2009); 1106-1129 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3084/2917 https://umj.imath.kiev.ua/index.php/umj/article/view/3084/2918 Copyright (c) 2009 Nohara B. T. |
| spellingShingle | Nohara, B. T. Нохара, Б. Т. Nonexistence theorem except the out-of-phase and in-phase solutions in the coupled van der Pol equation system |
| title | Nonexistence theorem except the out-of-phase and in-phase solutions in the coupled van der Pol equation system |
| title_alt | Теорема про неіснування розв'язків за винятком режиму синхронних коливань та режиму коливань у протифазі для системи з'єднаних рівнянь Ван Дер Поля |
| title_full | Nonexistence theorem except the out-of-phase and in-phase solutions in the coupled van der Pol equation system |
| title_fullStr | Nonexistence theorem except the out-of-phase and in-phase solutions in the coupled van der Pol equation system |
| title_full_unstemmed | Nonexistence theorem except the out-of-phase and in-phase solutions in the coupled van der Pol equation system |
| title_short | Nonexistence theorem except the out-of-phase and in-phase solutions in the coupled van der Pol equation system |
| title_sort | nonexistence theorem except the out-of-phase and in-phase solutions in the coupled van der pol equation system |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3084 |
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