A Van der Pol oscillator subjected to complicated excitations chaotic phenomenon
This paper deals with the Van der Pol oscillator subjected to complicated excitations. Section 1 presents the Van der Pol oscillator with a variable friction force . Section 2 is concerned with a Van der Pol oscillator under a simultaneous influence of forced and nonlinear parametric excitations....
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Інститут математики НАН України
2001
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| Цитувати: | A Van der Pol oscillator subjected to complicated excitations chaotic phenomenon / Nguen Van Dao, Nguyen Van Dinh, Tran Kim Chi // Нелінійні коливання. — 2001. — Т. 4, № 3. — С. 500-528. — Бібліогр.: 6 назв. — англ. |
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nasplib_isofts_kiev_ua-123456789-1747562025-02-09T22:49:42Z A Van der Pol oscillator subjected to complicated excitations chaotic phenomenon Осцилятор Ван дер Поля, що піддається хаотичному явищу складних збурень Осциллятор Ван дер Поля, подверженный хаотическому явлению сложных возмущений Nguen Van Dao Nguyen Van Dinh Tran Kim Chi This paper deals with the Van der Pol oscillator subjected to complicated excitations. Section 1 presents the Van der Pol oscillator with a variable friction force . Section 2 is concerned with a Van der Pol oscillator under a simultaneous influence of forced and nonlinear parametric excitations. Section 3 is devoted to the regular oscillation and chaotic phenomenon in a forced, strongly nonlinear Van der Pol’s oscillator. The stationary oscillations, their stability and transitional regimes and also chaotic phenomenon are of special interest. The asymptotic method of nonlinear mechanics and the method of harmonic balance, in combination with a computer, are used. This work was supported by the Council for Natural Science of Vietnam 2001 Article A Van der Pol oscillator subjected to complicated excitations chaotic phenomenon / Nguen Van Dao, Nguyen Van Dinh, Tran Kim Chi // Нелінійні коливання. — 2001. — Т. 4, № 3. — С. 500-528. — Бібліогр.: 6 назв. — англ. 1562-3076 AMS Subject Classification: 34E10 https://nasplib.isofts.kiev.ua/handle/123456789/174756 en Нелінійні коливання application/pdf Інститут математики НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
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English |
| description |
This paper deals with the Van der Pol oscillator subjected to complicated excitations. Section 1
presents the Van der Pol oscillator with a variable friction force . Section 2 is concerned with a
Van der Pol oscillator under a simultaneous influence of forced and nonlinear parametric excitations. Section 3 is devoted to the regular oscillation and chaotic phenomenon in a forced,
strongly nonlinear Van der Pol’s oscillator. The stationary oscillations, their stability and transitional regimes and also chaotic phenomenon are of special interest. The asymptotic method
of nonlinear mechanics and the method of harmonic balance, in combination with a computer,
are used. |
| format |
Article |
| author |
Nguen Van Dao Nguyen Van Dinh Tran Kim Chi |
| spellingShingle |
Nguen Van Dao Nguyen Van Dinh Tran Kim Chi A Van der Pol oscillator subjected to complicated excitations chaotic phenomenon Нелінійні коливання |
| author_facet |
Nguen Van Dao Nguyen Van Dinh Tran Kim Chi |
| author_sort |
Nguen Van Dao |
| title |
A Van der Pol oscillator subjected to complicated excitations chaotic phenomenon |
| title_short |
A Van der Pol oscillator subjected to complicated excitations chaotic phenomenon |
| title_full |
A Van der Pol oscillator subjected to complicated excitations chaotic phenomenon |
| title_fullStr |
A Van der Pol oscillator subjected to complicated excitations chaotic phenomenon |
| title_full_unstemmed |
A Van der Pol oscillator subjected to complicated excitations chaotic phenomenon |
| title_sort |
van der pol oscillator subjected to complicated excitations chaotic phenomenon |
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Інститут математики НАН України |
| publishDate |
2001 |
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https://nasplib.isofts.kiev.ua/handle/123456789/174756 |
| citation_txt |
A Van der Pol oscillator subjected to complicated excitations chaotic phenomenon / Nguen Van Dao, Nguyen Van Dinh, Tran Kim Chi // Нелінійні коливання. — 2001. — Т. 4, № 3. — С. 500-528. — Бібліогр.: 6 назв. — англ. |
| series |
Нелінійні коливання |
| work_keys_str_mv |
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2025-12-01T13:35:05Z |
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| fulltext |
Nonlinear Oscillations, Vol. 4, No. 4, 2001
A VAN DER POL OSCILLATOR SUBJECTED TO COMPLICATED
EXCITATIONS CHAOTIC PHENOMENON*
Nguyen Van Dao, Nguyen Van Dinh, Tran Kim Chi
Vietnam National University, Hanoi
144 Xuan Thuy, Cau Giay, Hanoi, Vietnam
e-mail: daonv@ vnuh.edu.vn
This paper deals with the Van der Pol oscillator subjected to complicated excitations. Section 1
presents the Van der Pol oscillator with a variable friction force . Section 2 is concerned with a
Van der Pol oscillator under a simultaneous influence of forced and nonlinear parametric ex-
citations. Section 3 is devoted to the regular oscillation and chaotic phenomenon in a forced,
strongly nonlinear Van der Pol’s oscillator. The stationary oscillations, their stability and tran-
sitional regimes and also chaotic phenomenon are of special interest. The asymptotic method
of nonlinear mechanics and the method of harmonic balance, in combination with a computer,
are used.
AMS Subject Classification: 34E10
1. Van der Pol Oscillator with Variable Friction Force
Let us consider a generalized Van der Pol equation:
ẍ+ ω2x = εf = ε
{
ω∆x− ωγx3 + h
[
1− k(x+ q cos νt)2
]
ẋ
}
, (1)
where dot denotes the derivative with respect to time t, h > 0, k > 0, ω, ν, γ, q are constants,
and ε is a small parameter, q is a parameter characterizing the intensity of the variable part of
the nonlinear friction. When q = ∆ = γ = 0 we have the well-known Van der Pol equation,
for which there exists only one stable stationary oscillation
x = 2 cos(ωt+ α).
To investigate the oscillations described by equation (1) we first transform it into the standard
form by means of the formulae
x = a cosψ,
ẋ = −aω sinψ, (2)
ψ = ωt+ θ,
* This work was supported by the Council for Natural Science of Vietnam.
500 c© Nguyen Van Dao, Nguyen Van Dinh, Tran Kim Chi, 2001
A VAN DER POL OSCILLATOR SUBJECTED TO COMPLICATED EXCITATIONS . . . 501
and then use the averaging method of nonlinear oscillations [1]. In the first approximation we
have the following equations for a and θ:
da
dt
= εaf1(a, θ),
(3)
adθ
dt
= εaf2(a, θ),
here
f1 = h
{
1
2
− k〈(a cosψ + q cos νt)2 sin2 ψ〉
}
,
f2 = −1
2
{
∆− 3
4
γa2 + 2hk〈(a cosψ + q cos νt)2 sinψ cosψ〉
}
,
and 〈f〉 denotes the averaged value of a function f .
Equations (3) have a zero solution a = a∗∗ = 0 and the other nontrivial stationary solutions
a = a∗ 6= 0, θ = θ∗ are determined by the system of equations
f1(a∗, θ∗) = 0, f2(a∗, θ∗) = 0.
The zero solution is stable if (ε > 0)
f1(0, θ∗∗) < 0,
θ∗∗ is arbitrary.
The conditions of stability for nontrivial solutions are
S = a∗
(
∂f1
∂a
)
∗
+
(
∂f2
∂θ
)
∗
< 0,
(4)
R =
(
∂f1
∂a
)
∗
(
∂f2
∂θ
)
∗
−
(
∂f1
∂θ
)
∗
(
∂f2
∂a
)
∗
> 0.
We shall now consider various resonance cases that correspond to concrete values of the fre-
quencies ω and ν.
The Subharmonic Resonance Case ν ≈ 3ω. In this case the averaged equations (3) take
the form
da
dt
=
εa
4
h
[
2− k
(
a2
2
+ q2
)
+ kaq cos 3θ
]
,
(5)
a
dθ
dt
= −εa
2
(
∆− 3
4
γa2 +
hk
2
qa sin 3θ
)
.
Stationary amplitudes are determined by
W = h2
[
2− k
(
a2
2
+ q2
)]2
+ 4
(
∆− 3
4
γa2
)2
− k2h2q2a2 = 0.
502 NGUYEN VAN DAO, NGUYEN VAN DINH, TRAN KIM CHI
The stability conditions (4) for stationary solutions (da/dt = dθ/dt = 0) of equations (5) are
S = h
[
1− k
2
(a2 + q2)
]
< 0 or a2 >
2
k
− q2
R =
3a
16
∂W
∂a2
> 0.
In the plane (a2,∆) the resonance curve W = 0 is an ellipse. Figure 1 shows the resonance
curve for the case q = 1.1, γ = 0.1, h = 0.1, k = 2 and Fig. 2 presents the case h = k =
1, q = 1.1, γ = 0.6.
Fig. 1. The resonance curve for
q = 1.1, γ = 0.1, h = 0.1, k = 2.
Fig. 2. The resonance curve for
h = k = 1, q = 1.1, γ = 0.6.
Stationary Oscillations in the Case ν 6≈ ω, 3ω. For these cases the averaged equations
become quite simple,
da
dt
=
εa
2
h
[
1− k
(
a2
4
+
q2
2
)]
,
a
dθ
dt
= −εa
2
(
∆− 3
4
γa2
)
.
It is easy to show that the zero solution a = 0 is stable if q2 ≥ 2/k, and unstable if −
√
2/k <
q <
√
2/k.
The nontrivial stationary solution,
a2∗ = 4
(
1
k
− q2
2
)
,
is stable for the values q2 < 2/k. Thus, for q2 < 2/k the system oscillates with the stationary
amplitude a∗:
a2∗ =
4
k
− 2q2.
For q2 > 2/k the damping of oscillations takes place (Fig. 3).
A VAN DER POL OSCILLATOR SUBJECTED TO COMPLICATED EXCITATIONS . . . 503
Fig. 3. The dependence of the amplitude a on the parameter q.
Oscillations in the Case ν ≈ ω. Now, the averaged equations (3) are of the form
ȧ = −εa
2
f0, aθ̇ = −εa
2
g0, (6)
where
f0 = h
(
1
4
ka2 +
1
2
kq2 − 1
)
+
1
2
hkqa cos θ − 1
4
hkq2 cos 2θ,
g0 = ∆− 3
4
γa2 +
1
2
hkqa sin θ +
1
4
hkq2 sin 2θ.
The stationary values a0 of a and θ0 of θ are determined from the equations
f0 = 0, g0 = 0. (7)
We eliminate 2θ from (7) by using the combination
f = (2a+ q cos θ)f0 − q sin θg0 = A sin θ +B cos θ − E = 0,
(8)
g = −q sin θf0 + (2a− q cos θ)g0 = G sin θ +H cos θ −K = 0.
The resonance curves consist of two parts: the regular part C1 and the critical part C2. The
regular part C1 lies in the equivalence region, where equations (7) and (8) are equivalent, that
is, where the determinant of the transformation (8) is different from zero,∣∣∣∣∣2a+ q cos θ −q sin θ
−q sin θ 2a− q cos θ
∣∣∣∣∣ = 4a2 − q2 6= 0.
The critical part C2 lies in the nonequivalence region, where D = 0 and satisfies the compat-
ibility conditions D1 = EH − KB = 0, D2 = AK − BE = 0 and also the trigonometric
conditions A2 +B2 ≥ E2, G2 +H2 ≥ K2.
504 NGUYEN VAN DAO, NGUYEN VAN DINH, TRAN KIM CHI
Fig. 4. The resonance curve for q = 0.8. Fig. 5. The resonance curve for q = 0.94.
The regular part of resonance curves is obtained by eliminating the phase θ from equations
(8):
W (∆, a2) = 16a2h2q2
(
∆− 3
4
γa2
)2
(3T ∗ +X2)
+ 4a2q2
{(
∆− 3
4
γa2
)2
+ h2(T ∗ +X)(3T ∗ −X)
}2
− q4
{(
∆− 3
4
γa2
)2
− h2(5T ∗ +X)(3T ∗ −X)
}2
= 0, (9)
where
T ∗ =
k
16
(4a2 − q2), X =
9
16
(
kq2 − 16
9
)
.
The resonance curves are represented in Figs 4 – 14 for the case γ = 0.
For given values of h and k we increase q to observe a change in the forms of the resonance
curve. Taking, for example, h = k = 1, we have the level of the self-excited oscillation of the
original system: a20 = 4. In the plane R(∆, a2), a representative point of this oscillating regime
is the point I0
(
∆ = (3/4)γa2, a2 = a20 = 4/k = 4
)
.
If 0 < kq2 < 8/9, the critical points still do not appear. The resonance curve is a simple
oval encircling the point I0. In Fig. 4 the resonance curve corresponds to q = 0.8.
If kq2 = 8/9, the resonance curve is still a simple oval, but two critical points J1, J2 appear.
They are returning points as shown in Fig. 5 for q = 0.94 '
√
8/9.
If
8
9
< kq2 < 4/3, two critical points J1, J2 move down and become two nodes. The
resonance curve has two cycles which are tied at J1, J2, as shown in Fig. 6 for q = 1.1.
If kq2 = 4/3, critical point I2 appears at the origin. It is a degenerated node, at which two
branches of the resonance curve are tangential to each other and to the abscissa axis ∆. The
resonance curve in Fig. 7 corresponds to the value q = 1.155 '
√
4/3.
A VAN DER POL OSCILLATOR SUBJECTED TO COMPLICATED EXCITATIONS . . . 505
Fig. 6. The resonance curve for q = 1.1. Fig. 7. The resonance curve for q = 1.155.
Fig. 8. The resonance curve for q = 1.25. Fig. 9. The resonance curve for q = 1.33.
If 4/3 < kq2 < 16/9, the critical point I2 moves up while the two other critical points
continue to move down. All these three points are nodes and the resonance curve has the form
shown in Fig. 8 for q = 1.25.
If kq2 = 16/9, three critical nodes I2, J1, J2 coincide at a special critical point I of the
nonequivalence line. The resonance curve has the form shown in Fig. 9 for q = 1.33 ' 4/3.
If 16/9 < kq2 < 8/3, three nodes appear again, but I2 lies higher and J1, J2 lie lower. The
resonance curve has the form shown in Fig. 10 for q = 1.45.
If kq2 = 8/3, two nodes J1 and J2 lie on the abscissa axis and become two returning points
as shown in Fig. 11 for q = 1.63 ≈
√
8/3.
If 8/3 < kq2 < 4, the resonance curve has only one node I2 and takes the form in Fig. 12
for q = 1.75.
If kq2 = 4, we have the resonance curve in the form of the number eight which is tangential
to the abscissa axis as shown in Fig. 13 for q = 2.
If kq2 > 4, the resonance curve still has the form of the number eight but lies above the
abscissa axis as shown in Fig. 14 for q = 2.8.
506 NGUYEN VAN DAO, NGUYEN VAN DINH, TRAN KIM CHI
Fig. 10. The resonance curve for q = 1.45. Fig. 11. The resonance curve for q = 1.63.
Fig. 12. The resonance curve for q = 1.75.
Fig. 13. The resonance curve for q = 2. Fig. 14. The resonance curve for q = 2.8.
A VAN DER POL OSCILLATOR SUBJECTED TO COMPLICATED EXCITATIONS . . . 507
Stability of Stationary Oscillations. The stability conditions obtained from the original equa-
tions have a well-known form. However, for the system under consideration, the transforma-
tion of stability conditions is more complicated.
The first stability condition is
S1 = a
∂f0
∂a
+
∂g0
∂θ
= a
{
1
2
(hka+ hkq cos θ)
}
+
1
2
{
hkqa cos θ + hkq2 cos 2θ
}
> 0.
Based on the original equations (6), we can elimitate cos 2θ,
S1 = h{(ka2 + kq2 − 2) + 2kqa cos θ} > 0. (10)
For the regular oscillation in the equivalence region, from the associated equations we get
cos θ = D2/D. Substituting this value into the last equation we have
S1 =
h
D
{(ka2 + kq2 − 2)D + 2kqaD2} =
hq2
D
S̄1 > 0, (11)
where
S̄1 = (ka2 + kq2 − 2)
{
∆2 − h2
(
5
4
ka2 +
1
4
kq2 − 1
)(
34ka2 − 3
4
kq2 + 1
)}
+ 4ka2
{
∆2 + h2
(
3
4
ka2 − 3
4
kq2 + 1
)(
1
4
ka2 +
1
2
kq2 − 1
)}
.
Hence, in the planeR(∆, a2) and in the equivalence region, based on the sign of the functionsD
and S̄1, we can identify the branches which satisfy and do not satisfy the first stability condition
for the regular part C1 of the resonance curves.
For the critical points in the equivalence region, we use form (10) if the phase is known.
However, since the critical phase is the limit of the phase on the regular branch leading to the
critical point, we can use form (11). More simply, we can deduce the first stability condition at
the critical point from the first stability condition on the regular branch which is considered as
containing the critical point.
For the regular oscillation in the nonequivalence region, we can use initial forms (10), (11).
For the critical oscillation in the nonequivalence region, i.e., at the special critical point, we can
use either initial form (10) or form (11).
For the critical point I
(
∆ = 0, ka2 = 4/9
)
corresponding to kq2 = 16/9 in the equivalence
region, we have three phases θ = 0, θ = ±2π/3. Substituting these values into (10) we obtain
for θ = 0, S1 = h
{
4/9 + 16/9− 2 + 2a
√
kq
√
k cos θ
}
= 2h > 0,
for θ = ±2π/3, S1 = −2h/3 < 0.
Thus, only the oscillation with the phase θ = 0 satisfies the first stability condition and here the
critical point I is considered as belonging to the horizontal resonance branch.
We have the following second stability condition:
S2 =
∂f0
∂a
∂g0
∂θ
− ∂f0
∂θ
∂g0
∂a
=
(
hka
2
+
hk
2
q cos θ
)(
hk
2
qa cos θ +
hk
2
q2 cos 2θ
)
−
(
−hka
2
qa sin θ +
hk
2
q2 sin 2θ
)
hk
2
q sin θ > 0.
508 NGUYEN VAN DAO, NGUYEN VAN DINH, TRAN KIM CHI
Fig. 15. Stability (st) branches of resonance
curves in the case q = 0.94.
Using original equations (6) we can eliminate sin 2θ, cos 2θ to get
S2 =
hk
4
{
4ah
(
k
4
a2 +
k
2
q2 − 1
)
+ 3hkq2a+ 4∆q sin θ
+
[
3hkqa2 + 4hq
(k
4
a2 +
k
2
q2 − 1
)]
cos θ
}
> 0. (12)
By substituting sin θ = D1/D, cos θ = D2/D, we obtain the second stability condition for
the regular oscillation in the equivalence region. For other oscillations, we can use either form
(12) or its corresponding limit form. For example, for special critical points corresponding to
θ = 0, the second stability condition (12) gives
S2 =
4
3
h2
√
k > 0.
However, it is more convenient to use the abbreviated form [2] of the second stability condition
for the regular oscillation in the equivalence region:
S2 =
a
TD
∂W
∂a2
> 0,
or
S2 =
ka
16D
∂W0
∂a2
> 0.
In Figs 15, 16, 17, the branches marked with ”st” correspond to stable oscillations.
A VAN DER POL OSCILLATOR SUBJECTED TO COMPLICATED EXCITATIONS . . . 509
Fig. 16. Stability (st) branches of resonance
curves in the case q = 1.1.
Fig. 17. Stability (st) branches of resonance
curves in the case q = 1.63.
2. Van der Pol Oscillator under External and Quadratic Parametric Excitations
Let us consider a quasilinear system described by the differential equation
ẍ+ ω2x = ε{∆x− γx3 + h(1− kx2)ẋ+ 2px2 cosωt+ e cos(ωt+ σ)}. (13)
The solution of equation (13) will be found in form (2). Applying the asymptotic method [1]
510 NGUYEN VAN DAO, NGUYEN VAN DINH, TRAN KIM CHI
we obtain the following averaged differential equations:
ȧ =
−ε
2ω
f =
−ε
2ω
{
hω
(
k
4
a2 − 1
)
a+
1
2
pa2 sin θ + e sin(θ − σ)
}
,
aθ̇ =
−ε
2ω
g =
−ε
2ω
{
∆a− 3
4
γa3 +
3
2
pa2 cos θ + e cos(θ − σ)
}
.
The constant amplitude and dephase of stationary oscillations satisfy the equations:
f = A sin θ +B cos θ − E = 0,
g = G sin θ +H cos θ −K = 0,
where
A =
1
2
pa2 + e cosσ, B = −e sinσ, E = −hω
(
k
4
a2 − 1
)
,
G = e sinσ, H =
3
2
pa2 + e cosσ, K = −∆a+
3
4
γa3.
Eliminating the phase θ gives the relationship for the frequency amplitude,
W (∆, a2) = D2
1 +D2
2 −D2 = a2
{(
3
2
pa2 + e cosσ
)
hω
(
k
4
a2 − 1
)
+
(
∆− 3
4
γa2
)
e sinσ
}2
+ a2
{(
1
2
pa2 + e cosσ
)(
∆− 3
4
γa2
)
− hω
(
k
4
a2 − 1
)
e sinσ
}2
−
{(
1
2
pa2 + e cosσ
)(
3
2
pa2 + e cosσ
)
+ e2 sin2 σ
}2
= 0, (14)
where
D =
(
1
2
pa2 + e cosσ
)(
3
2
pa2 + e cosσ
)
+ e2 sin2 σ,
D1 = −a
{(
3
2
pa2 + e cosσ
)
hω
(
k
4
a2 − 1
)
+
(
∆− 3
4
γa2
)
e sinσ
}
,
D2 = −a
{(
1
2
pa2 + e cosσ
)(
∆− 3
4
γa2
)
− hω
(
k
4
a2 − 1
)
e sinσ
}
.
The resonance curve consists of two parts: the regular part C1 and the critical one C2.
The regular part C1 satisfies (14) and lies in the regular region R1,
D(ω, a) 6= 0.
A VAN DER POL OSCILLATOR SUBJECTED TO COMPLICATED EXCITATIONS . . . 511
The critical part C2 lies in the critical region R2,
D(ω, a) = 0 or
3
2
p2a4 + 2pa2e cosσ + e2 = 0,
and satisfies the following:
the compatibility conditions,
D1 = 0 or
(
3
2
pa2 + e cosσ
)
hω
(
k
4
a2 − 1
)
+
(
∆− 3
4
γa2
)
e sinσ = 0,
D2 = 0 or
(
1
2
pa2 + e cosσ
)(
∆− 3
4
γa2
)
− hω
(
k
4
a2 − 1
)
e sinσ = 0;
the trigonometrical conditions,
A2 +B2 ≥ E2 or
(
1
2
pa2 + e cosσ
)2
+ e2 sin2 σ ≥ h2ω2
(
k
4
a2 − 1
)2
a2,
G2 +H2 ≥ K2 or e2 sin2 σ +
(
3
2
pa2 + e cosσ
)2
≥
(
∆− 3
4
γa2
)2
a2.
The critical region R2 exists (in the upper half-plane R(∆, a2 > 0) if
5π
6
≤ σ ≤ 7π
6
,
and consists of two straight lines D′ and D′′, respectively, of the ordinates:
D
′
, D
′′
: a2 = a21,2 =
4e
3p
{
− cosσ ∓
√
cos2 σ − 3
4
}
,
D′ and D′′ coincide if σ = 5π/6 or σ = 7π/6.
There exists another interesting property: the resonance curve always contains two points
— denoted by J−0 , J
+
0 — of abscissa not depending on h and situated on the line a2 = a20 = 4/k
(the amplitude of the purely self-excited oscillation).
Indeed, substituting a2 = a20 into (14) we obtain a quadratic equation for unknown ∆,
W (∆, a20) = a20
{
e2 sin2 σ +
(
1
2
pa20 + e cosσ
)2
}(
∆− 3
4
γa20
)2
−
{(
1
2
pa20 + e cosσ
)(
3
2
pa20 + e cosσ
)
+ e2 sin2 σ
}2
= 0, (15)
from which it follows that
∆ =
3
4
γa20 ± ∆̄ =
3
4
γa20 ±
∣∣∣∣(1
2
pa20 + e cosσ
)(
3
2
pa20 + e cosσ
)
+ e2 sin2 σ
∣∣∣∣
a0
√
e2 sin2 σ +
(
1
2
pa20 + e sinσ
)2
,
512 NGUYEN VAN DAO, NGUYEN VAN DINH, TRAN KIM CHI
3γa20/4± ∆̄ are thus the abscissa of J−0 and J+
0 respectively. By J0 we denote the segment
J−0 J
+
0 ≡ J0 :
3
4
γa2 − ∆̄ ≤ ∆ ≤ 3
4
γa2 + ∆̄, a2 = a20.
If a20 6= a21, a
2
2, the two ends J−0 , J
+
0 are located in the regular region R1, thus they belong to
the regular part C1 of the resonance curve. If a20 = a21 or a20 = a22, the two points J−0 and J+
0
coincide, and the segment J0 is reduced to a critical point (of C2).
In the particular case σ = π, a20 = a22 = 2e/p = 4/k if we substitute k with its value 2π/e,
the frequency amplitude relationship (14) can be written as
W (∆, a2) =
1
e2
(
1
2
pa2 − e
)2
×
{
a2
(
3
2
pa2 − e
)2
h2ω2 + a2e2
(
∆− 3
4
γa2
)2
− e2
(
3
2
pa2 − e
)2
}
= 0. (16)
Since
(
1/2pa2 − e
)2
is a factor of W , the line a2 = a20 forms a branch of the curve W =
0. As will be seen below, on this line there exists a critical segment J1J2(|∆ − (3/4)γa2| ≤√
2ep; a2 = a20) which can be regarded as the aforesaid segment J0 (substituting σ = π into
(15), in the limit a2 → a20 = 2e/p, we obtain ∆̄ =
√
2ep). Other branches of the resonance
curve are given by
a2
(
3
2
pa2 − e
)2
h2ω2 + a2e2
(
∆− 3
4
γa2
)2
− e2
(
3
2
pa2 − e
)2
= 0.
Since h is small enough, these branches intersect the segment J1J2 at two points J ′, J ′′ of the
abscissa determined by the equation
(
∆− 3
4
γa2
)2
+ 4h2
(
∆− 3
4
γa2
)
+ 4h2 − 2ep = 0 or
∆ =
3
4
γa2 − 2h2 ∓
√
4h4 − 4h2 + 2ep.
On the basis of formula (14) the resonance curves are plotted in Figs 18 – 27 for γ = 0.15
and in Fig. 28 for γ = −0.15. With the positive (negative ) value of γ the resonance curve leans
toward the right (left).
A VAN DER POL OSCILLATOR SUBJECTED TO COMPLICATED EXCITATIONS . . . 513
Fig. 18. The resonance curves for γ = 0.15,
σ = π, k = 3.2, and different
values of h(0, 0.03, 0.05).
Fig. 19. The resonance curves for γ = 0.15,
σ = π, k = 8, and different
values of h(0, 0.05, 0.1).
The case σ = π. If k = 3.2, we have a20 = 1.25 > 1 = a22 and the segment J0 lies above
the second critical line D′′ . In Fig. 18 the curve 1 including its nodal point I together with
the segment J1J2 form the resonance curve corresponding to h = 0, the resonance curves
2, 3 correspond to h = 0, 0.03, 0.05, respectively. We see that under the action of the self-
excitation, the critical segment disappears and the resonance curve is divided into two branches,
separated and located, respectively, above and under the critical line D′′. Passing through two
points J−0 , J
+
0 and encircling the segment J0, the upper branch, as h increases, is contracted
and ”tends” to J0. For small h, the lower branch has a loop, tied at I . As h increases, this
loop, always tied at I , becomes narrower. For h = h̄ ≈ 0.039, the loop disappears, the nodal
point I is degenerated into a returning point. Further increasing h, the lower branch leaves
I (which now becomes an isolated point, not belonging to the resonance curve), moves down
and approaches the abscissa axis ∆. Thus, when h is large enough, the amplitude level of
the combined oscillation is the same as that of the purely self - excited one (the effect of the
external and parametric excitations only results in the phase determination and the expansion
of the resonance zone).
For k = 8, we have a20 = 1/2, the segment J0 is situated between the two critical lines
D′, D′′. When h = 0, the form of the resonance curve is like that of its corresponding curve in
the above presented case. When h > 0, the resonance curve is also divided into two branches,
separated and respectively located above and under the critical line D′′. The upper branch
moves up as h increases. Various forms of the lower branch as shown in Fig. 19 (1, 2, 3) corre-
spond to h = 0, 0.05, 0.1, respectively. We see that for small h, there is also a loop; for large h,
the lower branch divides itself into two sub-branches, the upper branch is a single closed curve
encircling and tending to the segment J0.
For k = 12, we have a20 = 1/2 = a21; the segment J0 is reduced to nodal point I . In Fig. 20
the curves 1, 2, 3 are lower branches of the resonance curve corresponding to h = 0, 0.005, 0.1.
We see that, for a large enough h, the upper sub-branch takes the form of the number eight,
tied at I .
514 NGUYEN VAN DAO, NGUYEN VAN DINH, TRAN KIM CHI
Fig. 20. The resonance curves for γ = 0.15,
σ = π, k = 12, and different
values of h(0, 0.005, 0.1).
Fig. 21. The resonance curves for γ = 0.15,
σ = π, k = 20, and different
values of h(0, 0.02, 0.1, 0.138).
Fig. 22. The resonance curve for γ = 0.15,
σ = π, k = 4, and different values
of h(0, 0.02, 0.03, 0.05).
For k = 20, we have a20 = 0, 2 < a21 = 1/3, the segment J0 is located under the first
critical line D
′
. In Fig. 21 (1, 2, 3, 4) are lower branches of the resonance curve corresponding
to h = 0, 0.02, 0.1, 0.138, respectively.
Finally, let us examine a particular case k = 4. We have a20 = a22 = 1, the line a2 = a22
coincides with the critical line D
′′
and the segment J0 coincides with the critical segment J1J2.
In Fig. 22, the resonance curves 1, 2, 3, 4 correspond to h = 0, 0.02, 0.03, 0.05, respectively.
The critical segment J1J2 is a part of all the resonance curves. For small h, the segment J1J2
intersects other branches of the resonance curve at two points J
′
, J
′′
of abscissae determined
by (2.14). For large h, there remains only J1J2 (the amplitude is absolutely constant for the
whole resonance zone).
For the case π 6= σ ∈ [5π/6, 7π/6], let us choose σ = 17π/8, p = 0.05, e = 0, 025. We
have a21 ≈ 0.34, a22 ≈ 0.97 and two critical points I1 and I2 on D′ and D′′ appear.
In Fig. 23 , for k = 3.2, the resonance curves 1, 2, 3, 4 correspond to h = 0, 0.006, 0.02, 0.05,
respectively; J0 is above D
′′
, I1 and I2 move to the right.
A VAN DER POL OSCILLATOR SUBJECTED TO COMPLICATED EXCITATIONS . . . 515
Fig. 23. The resonance curves for γ = 0.15,
σ = 17π/8, k = 3.2, and different
values of h(0, 0.006, 0.02, 0.05).
Fig. 24. The resonance curves for γ = 0.15,
σ = 17π/8, k = 5.3, and different
values of h(0, 0.006, 0.02, 0.05).
In Fig. 24, for k = 5.3, the resonance curves 1, 2, 3, 4 correspond to h = 0, 0.006, 0.03, 0.06,
respectively; J0 is situated between D
′
and D
′′
, I1 moves to the right while I2 moves to the left.
In Fig. 25 for k = 20, the resonance curves 1, 2, 3, 4 correspond to h = 0, 0.01, 0.08, 0.14,
respectively; J0 is under D
′
, I1 and I2 move to the left.
Fig. 25. The resonance curves for γ = 0.15,
σ = 17π/8, k = 20, and different
values of h(0, 0.01, 0.08, 0.14).
Fig. 26. The resonance curves for γ = 0.15,
σ = 5π/6, k = 5.3, and different
values of h(0, 0.01, 0.05, 0.08).
In the case σ = 5π/6, we have a ”double” critical line D
′ ≡ D
′′
and two points I1 and I2
coincide at I .
In Fig. 26, for k = 5.3, the resonance curves 1, 2, 3, 4 correspond to h = 0, 0.01, 0.05, 0.08,
respectively; J0 is above the double critical line, the critical point I moves to the right.
516 NGUYEN VAN DAO, NGUYEN VAN DINH, TRAN KIM CHI
Fig. 27. The resonance curves for γ = 0.15,
σ = 2π/3, k = 5.3, and different
values of h(0, 0.01, 0.05, 0.07).
Fig. 28. The resonance curves for γ = − 0.15,
σ = 2π/3, k = 5.3, and different
values of h(0, 0.01, 0.05, 0.07).
For the case σ /∈ [5π/6, 7π/6] the critical region, and consequently, the critical part C2
does not exist. For σ = 2π/3, p = 0.05, e = 0.025, k = 5.3 in Fig. 27 (γ = 0.15) and Fig. 28
(γ = − 0.15) the resonance curves 1, 2, 3, 4 correspond to h = 0, 0.01, 0.05, 0.07. When h is
small, the resonance curve consists of two branches separated by the back-bone ∆ = 3γa2/4.
These two branches approach each other as h increases. At a certain value h̄, they will be
joined at a singular point (belonging to C1, not to C2). As h exceeds h̄, the above mentioned
ordinary singular joint disappears at once, the resonance curve once again divides itself into two
branches: the upper, encircling J0, becomes increasing by narrower; the lower, moving down,
runs near the ∆-axis.
Stability conditions. The two stability conditions are of well-known forms. The first stability
condition can be transformed into
S1 = a
∂f
∂a
+
∂g
∂θ
= hωa(ka2 − 2) > 0, i.e., a2 >
2
k
=
a20
2
.
The second stability condition can be written as
S2 =
∂f
∂a
∂g
∂θ
− ∂f
∂θ
∂g
∂a
=
{
hω
(
3
4
ka2 − 1
)
+ pa sin θ
}{
−pa2 sin θ + hωa
(
k
4
a2 − 1
)}
+
{
pa2 cos θ +
(
∆− 3
4
γa2
)
a
}{
∆− 3
4
γa2 + 3pa cos θ
}
> 0.
This form can be used directly for studying the stability of critical stationary oscillations. For
example, for the segment J1J2, among the critical stationary oscillations represented by J1J2
and having sin θ < 0, only those corresponding to subsegments J1J
′
and J
′′
J2 are stable. For
large h, J
′
and J
′′
disappear, the whole segment J1J2 with sin θ < 0 is stable.
A VAN DER POL OSCILLATOR SUBJECTED TO COMPLICATED EXCITATIONS . . . 517
For regular stationary oscillations, it is better to use the abbreviated form of the second
stability condition [2],
S2 =
a
D
∂W
∂a2
> 0.
Stable and unstable portions of the resonance curve can thus be obtained by examining the sign
distribution of two functions D(∆, a2) and W (∆, a2) on the plane (∆, a2) [1].
We have identified various forms of the resonance curve of a system of the Van der Pol
type, simultaneously subjected to an external and quadratic parametric excitations. Critical
representative points corresponding to critical stationary oscillations have been shown to be
useful to this end.
3. Regular Oscillation and Chaotic Phenomenon in a Forced Strongly Nonlinear Van
der Pol’s Oscillator
Let us consider a system described by the following differential equation:
ẍ− δ(1− βx2)ẋ+ x+ γx3 = Qν2 cos νt, (17)
where parameters δ > 0, β > 0 and γ,Q are constants and are not necessarily small quantities.
The self-excited oscillation in the system under consideration, when the external force is
absent Q = 0, can be found by putting in (17)
x = a0 cosω0t, ẋ = −a0ω0 sinω0t, ẍ = −a0ω2
0 cosω0t,
and equating the coefficients of the harmonics sinω0t and cosω0t separately to zero,
a20 =
4
β
, ω2
0 = 1 +
3
4
γa20 = 1 +
3γ
β
.
We see that the frequency ω0 depends on the amplitude a0 of self-oscillation. This results from
the presence of the nonlinear restoring terms.
In the present article, the case we will be dealing with is the oscillation in a vicinity of the
main resonance, so that ν is close to ω0. The solution of (17) characterizing the forced oscillation
can be approximated by the form
x = a cosψ, ẋ = −aν sinψ,
(18)
ẍ = −aν2 cosψ, ψ = νt− ϕ,
where a is the amplitude of the forced oscillation and ϕ is the angle of the phase displacement
relative to the forced excitation.
518 NGUYEN VAN DAO, NGUYEN VAN DINH, TRAN KIM CHI
Fig. 29. The resonance curves for the case Q = 1, γ = 0.005, β = 0.2,
and for various values of δ.
To find the equations for determining the amplitude a and phase ϕ, we use the method of
harmonic balance which consists of substituting the assumed solution (18) into the equation of
motion (17) and comparing the terms of the harmonic components cosψ and sinψ. This gives
a system of nonlinear algebraic equations with respect to a and ϕ,
a(1− ν2) +
3
4
γa3 = Qν2 cosϕ,
(19)
δ
(
1− 1
4
βa2
)
aν = −Qν2 sinϕ.
Eliminating the phase ϕ from these equations we obtain the following formula for the am-
plitude a:
W (a2, ν) = a2
[(
1− ν2 +
3
4
γa2)2 + δ2ν2(1− 1
4
βa2
)2]
−Q2ν4 = 0, (20)
and the phase ϕ:
tgϕ =
−δν
(
1− 1
4
βa2
)
1− ν2 +
3
4
γa2
. (21)
The amplitude curves, giving the dependence of a on ν, are presented in Fig. 29 for Q =
1, γ = 0.005, β = 0.2, and for different values of δ, δ = 0.05 (curve 1), δ = 0.45 (curve 2), δ =
0.60 (curve 3) and δ = 0.75 (curve 4) and in Fig. 30 for Q = 1, γ = 0.005, β = 0.04 and for
δ = 0.05 (curve 1), δ = 0.23 (curve 2), δ = 0.276 (curve 3) and δ = 0.65 (curve 4).
A VAN DER POL OSCILLATOR SUBJECTED TO COMPLICATED EXCITATIONS . . . 519
Fig. 30. The resonance curves for the case Q = 1, γ = 0.005, β = 0.04,
and for various values of δ.
The value δ∗ at which the resonance curve crosses itself can be calculated approximately as
follows. The resonance curve (20) cuts the skeleton line
ν2 = 1 +
3
4
γa2
at the points with ordinate a∗ satisfying the relationship
δ2a2∗ν
2
(
1− 1
4
βa2∗
)2
= Q2ν4
or approximately (ν2 ' 1),
u = (1− u)2 = F, (22)
where
u =
1
4
βa2∗, F =
βQ2
4δ2
.
Equation (22) has a double root relative to u, which corresponds to the crossing resonance
curve on the skeleton line, when F is equal to 4/27. So, the corresponding value δ∗ is approxi-
mately,
δ∗ =
Q
4
√
27β.
From Figs 29 and 30, one observes that withQ = 1 and with a small value of δ, the resonance
curve has only one branch (see curve 1) which reaches asymptotically the straight line a = a1 =
520 NGUYEN VAN DAO, NGUYEN VAN DINH, TRAN KIM CHI
Fig. 31a. The resonance curves for the case Q = 1, γ = 0.005, β = 0.2,
and very large values of δ.
Fig. 31b. The resonance curves for the case Q = 1, γ = 0.005, β = 0.2,
and very large values of δ (compressed form of Fig. 31a).
Q. When increasing δ, the resonance curve is deformed (curve 2) then intersected (curve 3).
As δ increases further, the resonance curve separates into two branches (curves 4); the upper
branch has an oval form. The center of the oval has an ordinate which is equal to the amplitude
a0 of self-excitation, a0 = 2/
√
β. The ratio γ/β in Fig. 30 is 5 times larger than in Fig. 29. The
maximum of amplitudes in Fig. 30 is bigger than in Fig. 29 and the resonance curves bend to the
right.
Figs 31a and 31b show resonance curves for Q = 1 for large values of δ. In these cases
the lower branch goes down, approaching the straight line a = a1 = Q either from above (see
A VAN DER POL OSCILLATOR SUBJECTED TO COMPLICATED EXCITATIONS . . . 521
Fig. 32. The resonance curves for the case γ = 0.005, β = 0.2, very large
values of the external force Q, Q = 5, and for various values of δ.
Fig. 33. The resonance curves for the case γ = 0.005, β = 0.2, very large
values of the external force Q, Q = 10, and for various values of δ.
curve 1 for δ = 1) or from below (see curve 2 for δ = 2 and curve 3 for δ = 5). The upper
branch is an oval which is compressed as δ increases (see oval 1 for δ = 1, oval 2 for δ = 2, and
oval 3 for δ = 5).
Figure 32 represents resonance curves for large values of external force (Q = 5). In this
case the oval no longer exists, the curves in Fig. 32 correspond to different values of the friction
coefficient δ : δ = 0.05 (curve 1), δ = 0.55 (curve 2) and δ = 2.9 (curve 3). These curves
approach the straight line a1 = Q = 5 from above.
Figure 33 gives resonance curves for an external force twice as large as that of Fig. 32 (Q =
522 NGUYEN VAN DAO, NGUYEN VAN DINH, TRAN KIM CHI
10). The resonance curve has only one branch which approaches the straight line a1 = Q = 10
either from above (curve 1 for δ = 0.05) or from below (curve 2 for δ = 0.55 and curve 3 for
δ = 0.65).
Stability of Stationary Oscillations. The stability of solution (18) can be determined by
means of an equation in variation which is obtained by substituting x = x0 + ξ, ẋ = ẋ0 + ξ̇
into equation (17), where x0 and ẋ0 are of the form (18) with a and ϕ satisfying (19). We have
the following equation in variation (the high degree terms relative to ξ are neglected):
ξ̈ + F (t)ξ̇ +G(t)ξ = 0,
where
F (t) = −δ(1− βx20) = −δ
[
1− a2
2
β(1 + cos 2ψ)
]
,
G(t) = 1 + 2δβx0ẋ0 + 3γx20 = 1− δβa2 sin 2ψ +
3γa2
2
(1 + cos 2ψ).
Using the transformation to the new variable η by the formula
ξ =
{
exp
[
−1
2
∫
F (τ)dτ
]}
η,
we get
η̈ + (θ0 + 2θ1c cos 2ψ + 2θ1s sin 2ψ + 2θ2c cos 4ψ)η = 0, (23)
where
θ0 = 1 +
3
2
γa2 − δ2
4
[(
1− 1
2
βa2
)2
+
1
8
β2a4
]
,
θ1c =
3
4
γa2 +
1
8
δ2βa2
(
1− 1
2
βa2
)
,
θ1s = −1
4
δβνa2,
θ2c = − δ
2
64
β2a4.
Equation (23) belongs to Hill’s differential equation. As is already well known [3], the
stability conditions for periodic solution (18) of equation (17) are
1) 0 < h = 1/2〈F (t)〉 = −δ(1− βa2/2)/2, (24)
where 〈F 〉 is the averaged value of the function F (t),
2) (θ0 − n2ν2)2 + 2(θ0 + n2ν2)h2 + h4 > θ2n = θ2nc + θ2ns, n = 1, 2, (25)
A VAN DER POL OSCILLATOR SUBJECTED TO COMPLICATED EXCITATIONS . . . 523
or in an open form,
{
1− n2ν2 +
3
2
γa2 − δ2
4
[(
1− 1
2
βa2
)2
+
1
8
β2a4
]}2
+
1
2
δ2
{
1 + n2ν2 +
3
2
γa2 − δ2
4
[
(1− 1
2
βa2)2 +
1
8
β2a4
]}
(1− β
2
a2)2
+
δ4
16
(
1− β
2
a2
)4
> θ2n, n = 1, 2.
N ote.
1. Condition (24) gives
a >
√
2
β
. (26)
2. When the parameters γ, δ are sufficiently small for the terms with smallness higher
than two relative to γ, δ to be neglected, the stability condition (25) is satisfied for n = 2. This
condition, in the case ν ' 1 and n = 1, can be presented in the form
(
1− ν2 +
3
2
γa2
)2
+ δ2ν2
[(
1− 1
2
βa2
)2
− 1
16
β2a4
]
− 9
16
γ2a4 > 0,
or
∂W
∂a2
> 0, or the same,
∂W
∂a
> 0, because a > 0,
where W = 0 is the equation for the resonance curve (20). Using the rule stated in [1] we
can easily identify stable branches of the resonance curve. In the figures presented, the stable
branches are shown by heavy solid lines, while the unstable branches are shown by dash lines.
The hatching in the figures shows the unstable region in which the condition (26) is not satisfied.
Because the problem is interesting for large values of δ and a, the stability condition (25)
should therefore be considered for concrete values of the parameters.
Transitional Oscillation. In the oscillatory system described by equation (17) the phenome-
non of synchronization of harmonic oscillation occurs when the initial point is given inside the
region containing the stable branches of the resonance curve. The oscillation starting from the
unstable zone of the resonance curve will be in a transitional regime. After some instants, this
oscillation reaches the periodic stationary and stable regime.
524 NGUYEN VAN DAO, NGUYEN VAN DINH, TRAN KIM CHI
Fig. 34. The resonance curve for Q = 1,
γ = 0.005, β = 0.04, δ = 0.05.
Let us consider some cases of oscillations originating from the unstable branch of the reso-
nance curve. Figure 34 represents the resonance curve for the case Q = 1, γ = 0.005, β =
0.04, δ = 0.05. The point A(ν0 = 1.37, a0 = 14.30) lies on the unstable branch. Now, we
construct the phase trajectory, which departs from A, of the equations
dx
dt
= ẋ,
(27)
dẋ
dt
= δ(1− βx2)ẋ− x− γx3 +Qν2 cos νt,
which are equivalent to equation (17). The behaviour of the system under consideration is
described by the movement of a representative point (x(t), ẋ(t)) along the solution curves of
Eqs. (27) in the (x, ẋ) plane.
The phase ϕ of oscillation is determined by formulas (21). The corresponding initial condi-
tions t0 = 0, x0 = x(0), ẋ0 = ẋ(0) are found and the Cauchy problem relative to the nonlinear
differential equation (17) is solved by using the Runge – Kutta method. The Maple and Fortran
power station software have been used.
Figure 35 gives the phase trajectory of oscillation with the initial amplitude and frequency
which correspond to point A of Fig. 34. When t → ∞ this trajectory asymptotically approaches
the limit cycle with a stable amplitude a = 15.86. This limit cycle is called an attractor. Point
S in the Fig. 36 is the stroboscopic point at the instants t = nT (n = 1, 2 . . . , T is the period of
the external force). All stroboscopic points coincide at S because the motion is periodic. The
point S is a sink (by Ueda’s concept [4]). This means that there exists a neighbourhood of the
point S, around which all stroboscopic points should be attracted.
A VAN DER POL OSCILLATOR SUBJECTED TO COMPLICATED EXCITATIONS . . . 525
Fig. 35. The phase trajectory corresponding
to the point A of Fig. 24.
Fig. 36. The limit cycle and the stroboscopic
point S at the instants nT .
Chaotic motion. In the case considered above with the chosen initial conditions, we have
regular motions. The largest Lyapunov’s exponent is negative.
Now, let us study the motion with the initial conditions t0 = 0, x0 = 1, ẋ0 = 0, keeping
the same parameters of Fig. 35. The largest Lyapunov’s exponent is positive (λ w 0.008) and
therefore we have a chaotic motion [5].
Figs 37 and 38 give phase trajectories in the first seventy periods and the first one hundred
and fifty periods, respectively.
The chaotic attractor is shown in Fig. 39. It is the set of stroboscopic points at the instants
t = nT (T = 2π/1.37). The enlargement of a part of this chaotic attractor can be observed in
Fig. 40.
Figs 41 and 42 are time dependents of the motion corresponding to the phase trajectories
37 and 38.
Fig. 37. Phase trajectories in the first
seventy periods.
Fig. 38. Phase trajectories in the first
one hundred and fifty periods.
526 NGUYEN VAN DAO, NGUYEN VAN DINH, TRAN KIM CHI
Fig. 39. The chaotic attractor. Fig. 40. The enlargement of
a small region of the
chaotic attractor.
Fig. 41. The transitional process.
Fig. 42. The steady state after transition.
Fig. 43. The average power spectrum of the chaotic motion.
Figure 43 shows the average power spectrum in which there are continuous power spectra
representing the chaotic nature of the motion.
A VAN DER POL OSCILLATOR SUBJECTED TO COMPLICATED EXCITATIONS . . . 527
Fig. 44. The basins of atraction in the space of initial conditions.
Fig. 45. The diagram of districts in the plane of parameters δ and b1 = δβ with
the initial conditions t0 = 0, x0 = 1, ẋ0 = 0. The shaded area corres-
ponds to regular motions, the grey area to chaotic motions.
Figure 44 shows basins of attraction in the space of initial conditions, in which the set A is a
chaotic attractor. This set, together with the surrounding regionM , makes a basin of attraction,
corresponding to chaotic motions. The shaded region is the basin of attraction, corresponding
to regular motions. Note that Fig. 44 could not be called an ”attractor — basin phase por-
trait”[4] because we have not yet exhaustively studied all motions occuring in equations (27).
Here we simply described qualitatively the basins of attraction.
Figure 45 is the diagram of districts in the plane of parameters δ and b1 = δβ with the
initial conditions t0 = 0, x0 = 1, ẋ0 = 0. The shaded area (bold vertical lines) corresponds
to regular motions and the grey area, except for some isolated black points, corresponds to
the chaotic motion. Isolated black points in the grey area are doubt points, which correspond
to either regular or chaotic motions and should be investigated very carefully. The results
of investigation depend strongly on the exactness and the convergence of the computational
process.
528 NGUYEN VAN DAO, NGUYEN VAN DINH, TRAN KIM CHI
4. Concluding Remarks
The nonlinear phenomena observed through the resonance curves are more diverse and
interesting than those of a simple classical oscillating systems. These curves were obtained by
using analytical methods and numerical calculations.
The resonance curves for a strongly nonlinear Van der Pol’s oscillator in a forced harmonic
regime have been drawn for rather large values of the parameter δ, characterizing the nonlinear
friction. In comparison with the case of a small value of δ [6], the resonance curve for Q = 1
and for a large value of δ consists of two parts, one of which has an oval form with a center at
ordinate a0 of self-oscillation. For large values of the external force (Q) there no longer exists
an oval branch of the resonance curve. It has only one branch approaching the straight line
a1 = Q. The transitional motion from an unstable regime to a stable one is examined.
The chaotic phenomenon in the forced strongly nonlinear Van der Pol’s oscillator, described
by equation (17), has been studied for the parameters δ = 0.05, β = 0.04, ν = 1.37, Q = 1,
and γ = 0.005.
REFERENCES
1. Mitropolskii Yu.A. and Nguyen Van Dao. Applied Asymptotic Methods in Nonlinear Oscillations, Kluwer
Acad. Publ., Dordrecht (1997).
2. Nguyen Van Dao and Nguyen Van Dinh. Interaction Between Nonlinear Oscillating Systems, Vietnam Na-
tional University Publ., House, Hanoi (1999).
3. Hayashi C. Nonlinear Oscillations in Physical Systems, Mc Graw-Hill Book Company (1964).
4. Ueda Y. The Road to Chaos, Acrial Press, Inc. (1992).
5. Moon F.C. Chaotic Vibrations, John Wiley and Sons, New York (1987).
6. Tondl A. “On the interaction between self - excited and forced vibrations,” Nat. Res. Inst. Machine Design
Bechovice. Monographs and Memoranda, No. 20 (1976).
Received 17.08.2001
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