Van der Pol oscillator under parametric and forced excitations
We study a van der Pol oscillator under parametric and forced excitations. The case where a system contains a small parameter and is quasilinear and the general case (without the assumption of the smallness of nonlinear terms and perturbations) are studied. In the first case, equations of the first...
Збережено в:
| Дата: | 2007 |
|---|---|
| Автори: | , , , , , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2007
|
| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/3304 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509369150472192 |
|---|---|
| author | Nguyen, Van Dao Nguyen, Van Dinh Tran, Kim Chi Нгуєн, Ван Дао Нгуєн, Ван Дінх Тран, Кім Чі |
| author_facet | Nguyen, Van Dao Nguyen, Van Dinh Tran, Kim Chi Нгуєн, Ван Дао Нгуєн, Ван Дінх Тран, Кім Чі |
| author_sort | Nguyen, Van Dao |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:50:43Z |
| description | We study a van der Pol oscillator under parametric and forced excitations. The case where a system contains a small parameter and is quasilinear and the general case (without the assumption of the smallness of nonlinear terms and perturbations) are studied. In the first case, equations of the first approximation are obtained by the Krylov-Bogolyubov-Mitropol’skii technique, their averaging is performed, frequency-amplitude and resonance curves are studied, and the stability of the given system is considered. In the second case, the possibility of chaotic behavior in a deterministic system of oscillator type is shown. |
| first_indexed | 2026-03-24T02:40:00Z |
| format | Article |
| fulltext |
UDC 517.9
Nguyen Van Dao (Vietnam Nat. Univ., Hanoi),
Nguyen Van Dinh, Tran Kim Chi* (Vietnam Acad. Sci., Hanoi),
VAN DER POL’S OSCILLATOR
UNDER THE PARAMETRIC AND FORCED EXCITATIONS
*
VPLYV NA SYSTEMU VAN DER POLQ
PARAMETRYÇNOHO TA ZOVNIÍN|OHO ZBUREN|
Van Der Pol’s oscillator under parametric and forced excitations is studied. The case where the system
contains a small parameter being quasilinear and the general case (without assumption on the smallness
of nonlinear terms and perturbations) are studied. In the first case, equations of the first approximation
are obtained by means of the Krylov – Bogoliubov – Mitropolskii technique, their averaging is
performed, frequency-amplitude and resonance curves are studied, on the stability of the given system is
considered. In the second case, the possibility of chaotic behavior in a deterministic system of oscillator
type is shown.
DoslidΩeno vplyv parametryçnoho ta zovnißn\oho zburen\ na systemu Van der Polq. Rozhlqnu-
to vypadky, koly dana systema mistyt\ malyj parametr i [ kvazilinijnog ta zahal\nyj vypadok
(bez prypuwennq pro malyznu nelinijnyx dodankiv ta zburen\). U perßomu vypadku za dopomo-
hog metodu Krylova – Boholgbova – Mytropol\s\koho otrymano rivnqnnq perßoho nablyΩen-
nq, provedeno ]x userednennq i vyvçeno çastotno-amplitudni ta rezonansni kryvi, doslidΩeno
stijkist\ reΩymiv dano] systemy. U druhomu vypadku pokazano moΩlyvist\ xaotyçno] povedinky
v determinovanyx systemax kolyvnoho typu.
Introduction. It is well-known that there always exists an iteraction of some kind
between nonlinear oscillating systems. N. Minorsky stated that “Perhaps the whole
theory of nonlinear oscillations could be formed on the basis of interaction” [1].
Different interesting cases of interaction have been investigated by us and published in
the monograph [1], using the effective asymptotic method of nonlinear mechanics
created by N. M. Krylov, N. N. Bogoliubov and Yu. A. Mitropolskii.
The present paper introduces our research on the behaviour of a Van der Pol’s
oscillator under the parametric and forced excitations. The dynamic system under
consideration is described by an ordinary nonlinear differential equation of type (1.1).
Section 1 is devoted to the case of small parameters. The amplitudes of nonlinear
deterministic oscillations and their stability are studied. Analytical calculations in
combination with a computer are used to obtain amplitude curves, which show a very
complicated form in Figs. 1.2 – 1.4. In Section 2 we study the chaotic phenomenon
occurring in the system described by equation (1.1) without assumption on the
smallness of the parameters.
As known, the fundamental characteristic of a chaotic system is its sensitivity to the
initial conditions. The diagnostic tool used in this work is the Lyapunov exponents.
The positiveness of the largest Lyapunov exponent will help us to define the values of
parameters with which the chaotic motions are occurred. Chaotic attractors and
associated power spectra will be presented.
1. The case of small parameters. In this section, let us consider the case where
the parameters are small. The opposite case will be investigated in the next section.
The smallness of parameters is characterized by introducing a small positive parameter
ε. For this case the asymptotic method of Krylov – Bogoliubov – Mitropolskii (K - B -
M) [2, 3] is used to seek the approximate solutions and to study their stability.
1.1. The differential equation of oscillation and its stationary solution. The
system under consideration is described by the equation
˙̇x + ω
2
x = ε γ ω ω σ∆ x x h k x x px t e t− + − + + +{ }3 21 2 2( ) ˙ cos cos( ) , (1.1)
where h > 0 and k > 0 are coefficients characterizing the self-excitation of a pure
* This work is completed with the financial support of the Council for Natural Science of Vietnam.
© NGUYEN VAN DAO , NGUYEN VAN DINH, TRAN KIM CHI, 2007
206 ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 2
VAN DER POL’S OSCILLATOR UNDER THE PARAMETRIC AND FORCED … 207
Van der Pol’s oscillator, 2 p > 0 is the intensity of the parametric excitation, e > 0 is
the intensity of the forced excitation and σ, 0 ≤ σ ≤ 2 π, in the phase shift between the
parametric and forced excitations. Bellow, two subcases will be investigated separately
for a weak parametric excitation when p
2 < h2 and for a strong parametric excitation
when p
2 > h2.
The solution of (1.1) is found in the form
x = a cos ψ, ẋ = – a ω sin ψ, ψ = ω t + θ, (1.2)
where a and θ are new variables, which satisfy the following equations in the
standard form:
ȧ = –
ε
ω
γ ω ω σ ψ∆ x x h k x x px t e t− + − + + +{ }3 21 2 2( ) ˙ cos cos( ) sin ,
(1.3)
θ̇ = –
ε
ω
γ ω ω σ ψ
a
x x h k x x px t e t∆ − + − + + +{ }3 21 2 2( ) ˙ cos cos( ) cos .
Following the K - B - M method, in the first approximation these equations can be
replaced by averaged ones
ȧ = –
ε
ω2 0f = –
ε
ω
ω θ θ σ
2 4
1 2
2
h
ka
a pa e−
+ + −
sin sin( ) ,
(1.4)
a θ̇ = –
ε
ω2 0g = –
ε
ω
γ θ θ σ
2
3
4
23∆a a pa e− + + −{ }sin sin( ) .
The amplitude a and phase θ of stationary oscillations are determined from the
equations ȧ = θ̇ = 0:
f0 = h
ka
aω
2
4
1−
+ p a sin 2 θ + e sin ( θ – σ ) = 0,
(1.5)
g0 = ∆ a –
3
4
3γa + p a cos 2 θ + e cos ( θ – σ ) = 0.
These equations are equivalent to
f = f0 cos θ – g0 sin θ =
= p a a− +
∆ 3
4
2γ θsin + h
ka
aω θ
2
4
1−
cos – e sin σ = 0,
(1.6)
g = f0 sin θ + g0 cos θ =
= h
ka
aω θ
2
4
1−
sin + p a a+ −
∆ 3
4
2γ θcos + e cos σ = 0,
or
f = A sin θ + B cos θ – E = 0,
(1.7)
g = G sin θ + H cos θ – K = 0,
where
A = p a a− +
∆ 3
4
2γ , B = h
ka
aω
2
4
1−
, E = e sin σ,
(1.8)
G = h
ka
aω
2
4
1−
, H = p a a+ −
∆ 3
4
2γ , K = – e cos σ.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 2
208 NGUYEN VAN DAO , NGUYEN VAN DINH, TRAN KIM CHI
1.2. The amplitude-frequency relationship and resonance curve. The
characteristic determinants of equations (1.7) are
D =
A B
G H
=
p a a h
ka
a
h
ka
a p a a
− +
−
−
+ −
∆
∆
3
4 4
1
4
1
3
4
2
2
2
2
γ ω
ω γ
=
= a p a p a h
ka2 2 2 2 2
2 2
3
4
3
4 4
1+ −
− +
− −
∆ ∆γ γ ω ,
D1 =
E B
K H
=
e h
ka
a
e p a a
sin
cos
σ ω
σ γ
2
2
4
1
3
4
−
− + −
∆
=
= ae p a h
ka+ −
+ −
∆ 3
4 4
12
2
γ σ ω σsin cos , (1.9)
D2 =
A E
G K
=
p a a e
h
ka
a e
− +
−
−
∆ 3
4
4
1
2
2
γ σ
ω σ
sin
cos
=
= – ae p a h
ka− +
+ −
∆ 3
4 4
12
2
γ σ ω σcos sin .
Below, in the ( ∆, a )-plane we identify the regular region in which the characteristic
determinant D is nonzero and the critical region in which D is identically zero.
By solving equations (1.7) relatively to sin θ and cos θ and eliminating the phase
θ we obtain the amplitude-frequency relationship
W ( ∆, a ) = D1
2 + D2
2 – D
2 =
= a e p a h
ka2 2 2
2 2
3
4 4
1+ −
+ −
∆ γ σ ω σsin cos +
+ a e p a h
ka2 2 2
2 2
3
4 4
1− +
+ −
∆ γ σ ω σcos sin –
– a p a p a h
ka4 2 2 2 2
2 2 2
3
4
3
4 4
1− +
+ −
− −
∆ ∆γ γ ω = 0. (1.10)
The regular part C1 of the resonance curve satisfies (1.10) and lies in the regular
region, where D ≠ 0.
The critical part C2 of the resonance curve lies in the critical region, where
D = 0 or p
2 – ∆ –
3
4
2
2
γa
– h
ka2 2
2 2
4
1ω −
= 0,
and satisfies:
the compatibility conditions
D1 = 0 or p a+ −
∆ 3
4
2γ σsin + h
kaω σ
2
4
1−
cos = 0,
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 2
VAN DER POL’S OSCILLATOR UNDER THE PARAMETRIC AND FORCED … 209
D2 = 0 or p a− +
∆ 3
4
2γ σcos + h
kaω σ
2
4
1−
sin = 0;
the trigonometrical conditions
A
2 + B
2 ≥ E
2 or a p a h
ka2 2
2
2 2
2 2
3
4 4
1− +
+ −
∆ γ ω ≥ e
2
sin
2
σ,
(1.11)
G
2 + H
2 ≥ K
2 or a p a h
ka2 2
2
2 2
2 2
3
4 4
1+ −
+ −
∆ γ ω ≥ e
2
cos
2
σ.
It is easy to see that the critical region is the resonance curve of Van der Pol’s
oscillator under the action of the parametric excitation without the forced excitation
( e = 0 ). For a weak parametric excitation ( p
2 < h
2
), the resonance curve is an oval
encircling the point A a a a
k0
2 2
0
23
4
4∆ = = =
γ , which is the representative point of
the self-oscillation of Van der Pol’s oscillator. This oval lies completely above the
abscissa axis ∆. When the parametric excitation is strong enough ( p
2 > h
2
), the
critical oval enlarges and cuts the abscissa axis.
From the compatibility conditions it follows that
∆ = p cos 2 σ +
3
4
2γa , h
kaω
2
4
1−
= – p sin 2 σ. (1.12)
Hence, the compatibility point has coordinates
∆ = ∆* = p cos 2 σ +
3
4
2γa
*
, a
2 = a*
2 = a
p
h p0
2 1
2
1 2
−
+
sin
cos
σ
σ
.
The existence condition of the compatibility point is
a*
2 > 0 or
p
h
sin
cos
2
1 2
σ
σ+
< 1. (1.13)
Obviously, this condition is satisfied if
sin 2 σ < 0, i.e.,
π
2
< σ < π or
3
2
π
< σ < 2 π.
In the case
sin 2 σ ≥ 0, i.e., 0 ≤ σ ≤
π
2
or π ≤ σ ≤
3
2
π
(1.14)
the condition (1.13) can be transformed into
Λ ( cos 2 σ ) = p
2
cos
2
2 σ + p h
2
cos 2 σ + h
2 – p
2 ≥ 0. (1.15)
The left-hand side of (1.15) is a trinomial of cos 2 σ ∈ [ – 1, 1 ] with the discriminant
Γ = p
2
( h
4 – 4 h
2 + 4 p
4
).
If h > 2 – 2 1 − p (the case h
2 > 2 + 2 1 − p is not considered here) then Γ <
< 0 and the trinomial Λ ( cos 2 σ ) always has the same positive sign as its first
coefficients, and the condition (1.15) is satisfied with all values of σ in the interval
(1.14).
If h
2 ≤ 2 – 2 1 − p , then Γ ≥ 0 and the trinomial Λ ( cos 2 σ ) has either two
simple roots or a double root. The simple roots cos 2 σ1, 2 are
cos 2 σ1, 2 =
1
2 2
2
p
ph− ±( )Γ . (1.16)
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 2
210 NGUYEN VAN DAO , NGUYEN VAN DINH, TRAN KIM CHI
It is noted that Λ ( 1 ) = h
2
( 1 + p ) > 0, Λ ( – 1 ) = h
2
( 1 – p ), ( p = 0 ( ε ) ) and the
numerical average of two roots:
S
2
= –
ph
p
2
22
satisfy – 1 <
S
2
< 1. Hence, two roots
(1.16) lie in the interval [ – 1, 1 ]. Condition (1.15) leads to
cos 2 σ ≤ cos 2 σ2 or cos 2 σ ≥ cos 2 σ1
. (1.17)
Combining (1.17) with (1.14) we obtain
0 ≤ σ ≤ σ1 or σ2 ≤ σ ≤
π
2
, or π ≤ σ ≤ π + σ1 or π + σ2 ≤ σ ≤
3
2
π
.
In summary, we have
if h
2 > 2 – 2 1 2− p , then the compatibility point I* exists for every σ;
if h
2 ≤ 2 – 2 1 2− p , then the compatibility point I* exists only for
0 ≤ σ ≤ σ1 or σ2 ≤ σ ≤ π + σ1 or π + σ2 ≤ σ ≤ 2 π. (1.18)
In Fig. 1.1 the heavy arcs give the values σ with which the compatibility point I*
exists when h
2 ≤ 2 – 2 1 2− p . Since 2 – 2 1 2− p is approximately equal to p
2,
then
if p
2 < h
2, i.e., when the parametric excitation is weak in comparison with self-
excitation, the critical oval D = 0 lies completely above the abscissa axis ∆. The
critical point I* always exists;
if p
2 ≥ h
2, i.e., when the parametric excitation is strong enough, the critical oval
D = 0 cuts the abscissa axis ∆. The critical point I* exists only with the values of σ
lying in the interval (1.18).
Fig. 1.1. The heavy arcs give the values σ with which the compatibility point I* exists.
Verifying the trigonometrical conditions by substituting (1.12) into (1.11) we obtain
4 2 2 2p a
*
sin σ ≥ e
2
sin
2
σ, 4 2 2 2p a
*
cos σ ≥ e
2
cos
2
σ. (1.19)
Because the right-hand sides of (1.19) are not equal to zero simultaneously, from (1.19)
we find
4 2 2p a
*
≥ e
2 or a
*
2 ≥
e
p
2
24
. (1.20)
Hence, the compatibility point I* is only a critical point when the amplitude a is
large enough, i.e., when the forced excitation is still not too strong in comparison with
the parametric one.
1.3. Forms of resonance curves. To identify the forms of resonance curves we
give in advance th values h, k, then for each chosen value p we change e and σ to
have the resonance curves. For example, with h = 0.1, k = 4, ω = 1, the self-excited
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 2
VAN DER POL’S OSCILLATOR UNDER THE PARAMETRIC AND FORCED … 211
oscillation of Van der Pol’s original system has an amplitude a0
2 = 1 and is
represented by the point A a0 0
23
4
1∆ = =( )γ , .
The case of weak parametric excitation ( p
2 < h
2 ). As it is known, in this case the
critical oval D = 0, i. e., the resonance curve of Van der Pol’s oscillator under the
parametric excitation, runs around the point A0 and lies entirely above the abscissa
axis ∆. We take p = 0.05 and σ = 0. For a weak forced excitation, i.e., when e is
small enough, the condition (1.20) is satisfied and the critical point I* with
coordinates ∆
* *
,= + = =( )p a a
3
4
12
0
2γ exists. For enough strong forced excitation,
i.e., when e is large enough, point I* is only a trivial compatibility point which does
not belong to the resonance curve.
In Fig.1.2 the resonance curves 0 ,1 , 2, 3 , 4 , 5 correspond to the linear case
γ = 0, for e = 0; 0.015; 0.0177; 0.050; 0.100; 0.120, respectively. The curve 0 is
a critical oval. The curve 1 has two branches: branch C ′ lies near abscissa axis,
branch C ′′ lies higher and consists of two cycles, one of them ′′C1 is outside and the
other ′′C2 is inside the critical oval. These cycles are connected to one another at the
critical node I* on the critical oval.
Fig. 1.2. Resonances curves for γ = 0, σ = 0, e = 0 (curve 0), e = 0.015 (curve 1), e = 0.0177
(curve 2), e = 0.050 (curve 3), e = 0.100 (curve 4), e = 0.120 (curve 5).
Increasing the forced excitation ( e ), the lower branch C ′ moves up. The inner
loop ′′C2 of the upper branch is pressed while the outer loop ′′C1 is expanded, but
both loops are tied at the node I*
. For e ≈ 0.0177, the lower branch C ′ is connected
with the outer loop ′′C1 at the node J and we have the curve 2, where J is a
singular point belonging to the regular region D ≠ 0. For e larger than 0.0177, the
singular point J disappears. Then the lower branch and the outer loop are unified into
one branch which lies outside the critical oval. We have the resonance curve 3.
Increasing e further, the inner loop ′′C2 continues to be pressed into I* and
disappears when e = 0.1 (see the resonance curve 4). At this moment I* is a
returning point. The curve 5 corresponds to a very large value of forced excitation;
the point I* is a trivial compatibility point which lies outside the resonance curve and
does not belong to it.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 2
212 NGUYEN VAN DAO , NGUYEN VAN DINH, TRAN KIM CHI
Figures 1.3 show the resonances curves in the nonlinear case γ x
3, γ ≠ 0 with γ =
= – 0.1 (a) and γ = 0.1 ( b). The curves 1, 3, 5 in these figures have the same
values of parameters (except γ ) of the curves 1, 3, 5 in Fig.1.2.
a b
Fig. 1.3. Resonance curves for γ = – 0.1 (a), γ = 0.1 (b), σ = 0, e = 0 (curve 0), e = 0.015 (curve
1), e = 0.050 (curve 3), e = 0.12 (curve 5).
With the negative value of γ (see Fig. 1.3, a) resonance curves lean toward the
left in comparison with the case γ = 0 (Fig. 1.2). Otherwise, resonance curves lean
toward the right for the positive value of γ (see Fig. 1.3, b). This situation is
common for nonlinear Duffing’s systems.
The case of strong parametric excitation ( p
2 > h
2 ). As before we take h = 0.1,
k = 4 but p = 0.12. In this case the critical oval 0 is enlarged and cuts the abscissa
axis ∆ . In Fig. 1.4, a ( γ = 0 ), and 1.4, b ( γ = 0.1 ), the resonance curves 1
correspond to σ = 0, e = 0.06. The curve 1 has a cycle lying inside the critical oval
and is connected with the outside branch by the critical point I*
. If only e increases,
the inside cycle is tied and then disappears. The critical point I* first becomes a
returning point and then an isolated trivial compatibility point. The resonance curve is
the only outside branch which is moving up.
a b
Fig. 1.4. Resonance curves for γ = 0 (a), γ = 0.1 (b), and σ = 0 (curve 1), σ = π /12 (curve 2),
σ = π / 6 (curve 3), σ = π / 4 (curve 4).
Changing σ, the critical point moves along the critical oval 0. In Fig. 1.4, a, b,
the resonance curves 2, 3, 4 correspond to the values γ = 0 (a), γ = 0.1 (b) and σ =
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 2
VAN DER POL’S OSCILLATOR UNDER THE PARAMETRIC AND FORCED … 213
=
π
12
;
π
6
;
π
4
respectively. We see that, when σ increases the critical point moves
down, the critical point I* becomes a returning one and then disappears. Then the
resonance curves separate into a cycle lying inside the oval 0 and a branch lying
outside this oval (curves 3, 4).
1.4. Stability conditions. To have the stability condition we use the variational
equations by putting in (1.4) a = a0 + δ a, θ = θ0 + δ θ and neglecting the terms of
higher degrees with respect to δ a and δ θ
d
dt
a( )δ = –
ε
ω
δ
2
0
0
∂
∂
f
a
a –
ε
ω θ
δθ
2
0
0
∂
∂
f
,
(1.21)
a
d
dt0 ( )δθ = –
ε
ω
δ
2
0
0
∂
∂
g
a
a –
ε
ω θ
δθ
2
0
0
∂
∂
g
,
where a0
, θ0 are stationary values of the amplitude a and phase θ – the roots of
equations (1.5). The characteristic equation of the system (1.21) is
a0
ρ
2 +
ε
ω θ
ρ
2 0
0
0
0
0
a
f
a
g∂
∂
+ ∂
∂
+
+
ε
ω θ θ
2
2
0
0
0
0
0
0
0
04
∂
∂
∂
∂
− ∂
∂
∂
∂
f
a
g f g
a
= 0.
Hence, the stability conditions for stationary solutions a0
, θ0 is
S1 = a0
∂
∂
f
a
0
0
+
∂
∂
g0
0θ
= a h
ka hk
a p0
0
2
0
2
04
1
4
2ω ω θ−
+ +
sin –
– 2 20 0 0pa esin sin( )θ θ σ+ −{ } > 0. (1.22)
From the first equation of (1.5) we find sin 2
θ0
, then by substituting into (1.22) we
get
S1 = hωa ka0 0
2 2−( ) > 0 or a0
2 >
2
k
.
This condition means that only the oscillations with large amplitudes may be stable.
The second stability condition still has the abbreviated form [1, 4]
S2 =
a
D
W a
a
0 0
2
0
2
∂ ( )∆,
∂
> 0.
The curves D a∆, 0
2( ) = 0 and W a∆, 0
2( ) = 0 divide the plane P a∆, 0
2( ) into
regions. In each region the functions D a∆, 0
2( ) and W a∆, 0
2( ) have a determined
sign. Moving upwards along a straight line, parallel to the ordinate axis a0
2 and
cutting the resonance curve at a point M, if we go from the region D W < 0 (> 0) to
the region D W > 0 (< 0) then point M corresponds to the stable (unstable)
oscillation. Therefore, basing on the sign distribution of the functions D and W in
the P-plane, we can identify the stable and unstable branches of the resonance curve.
2. The case of arbitrary parameters. Regular and chaotic solutions. Let us go
back to the equation (1.1), ignoring the assumption on the smallness of the parameters,
i. e., we will consider the following differential equation:
˙̇x + ω
2 x = ∆ x – γ x
3 + h k x x( ) ˙1 2− + 2 p x cos 2 ω t + e cos ( ω t + σ ). (2.1)
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 2
214 NGUYEN VAN DAO , NGUYEN VAN DINH, TRAN KIM CHI
We fix the parameters: ω = 0. 83, ∆ = 0. 01, γ = 1, h = 1, k = 0.6, p = 0.001, σ =
= 0 and use e as a control parameter. With different values of e, solutions of the
equation (2.1) can be regular or chaotic. To identify the regular or chaotic character of
a solution, we can use various methods, such as consideration of the sign of the largest
Lyapunov exponent, or building the Poincaré sections [5 – 11]. To construct a
Poincaré section of an orbit, we use the period T =
2π
ω
of the external excitation force.
Then, the Poincaré section acts like a stroboscope, freezing the components of the
motion commensurate with the period T. If we have a collection of n discrete points
on the Poincaré section, the corresponding motion is periodic with the period n T. For
example, for e = 5.09, the Poincaré section consists of three points (Fig. 2.1, a), the
motion is periodic with the period 3 T; for e = 5.116, the Poincaré section consists of
six points (Fig. 2.1, b), the motion is periodic with the period 6 T. When the Poincaré
section does not consist of finite number of discrete points, the motion is aperiodic, it
may be chaotic (Fig. 2.2).
a b
Fig. 2.1. Poincaré section: e = 5.09 (a), e = 5.116 (b).
Fig.2.2. Poincaré section realized at e = 5,15.
The periodic attractors and the corresponding power spectrums realized at e = 5.09
and e = 5.116 are illustrated in Fig. 2.3, a, b. The aperiodic attractor and its power
spectrum realized at e = 5.15 are illustrated in Fig. 2.4. In this case the power
spectrum has a continuous broadband character. The Poincaré section has a distinctive
form shown in Fig. 2.2, it consists of about 8000 points after the transition decays (the
first 500 periods).
To verify that the motion realized at e = 5.15 is chaotic, we need to show the
sensitivity to initial conditions on this attractor. We choose two points separated by
d0 = 10
– 7 close to the attractor and examine evolutions initiated from them. Figure
2.5 illustrates the variation of the separation d with time t. The exponential growth of
separation for 20 < t < 300 is clearly noticeable. The separation saturates at the size
of the attractor for t > 300. Therefore, there is a positive Lyapunov exponent
associated with the chaotic orbit at e = 5.15. The evaluated largest Lyapunov exponent
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 2
VAN DER POL’S OSCILLATOR UNDER THE PARAMETRIC AND FORCED … 215
is λ ≈ 0.062 > 0 (its calculation will be mentioned below).
Fig. 2.3. Periodic attractors and associated power spectra: e = 5.09 (a), e = 5.116 (b).
Fig. 2.4. Chaotic attractor and associated power spectra at e = 5.15.
Fig. 2.5. Sensitivity to initial conditions at e = 5.15.
The evaluation of the largest Lyapunov exponent. To evaluate the largest
Lyapunov exponent in the case ω = 0.83, ∆ = 0.01, γ = 1, h = 1, k = 0.6, p = 0.001,
σ = 0 and e = 5.15, we represent the equation (2.1) in the form
ẋ1 = x2
,
ẋ2 = – 0.6889x1 + 0.01 x1 – x1
3 + 1 0 6 1
2
2−( ). x x +
+ 0.002 x1 cos 2z + 5.15 cos z, (2.2)
ż = 0.83.
Let u = ( x1 , x2 , z ) is a three dimensional vector and u* = u*(t, u0
) is a reference
trajectory of the system (2.2), where u0 is the initial condition. The variational
equation corresponding to this reference trajectory is
η̇ = A η,
where η = u – u* and the matrix A depends on u*
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 2
216 NGUYEN VAN DAO , NGUYEN VAN DINH, TRAN KIM CHI
A =
0 1
0 6789 3
1 2 0 002 2
1 0 6
0
0 004 2
5 15 2
0 0 0
1
2
1 2
1
2 1− − −
− +
− ( )
− −
−
.
. . cos
.
. sin
. sin
*
* * *
*
* *
*
x
x x z
x
x z
z
.
This time evolution of the Lyapunov exponent is presented in Fig. 2.6. The largest
Lyapunov exponent is a positive number λ ≈ 0.062, which shows the chaotic
character of the motion of the system (2.2). This means that two trajectories starting
closely one to another in the phase space will move exponentially away from each
other for small times on the average
d ( t ) = d t
0 2λ ,
where d0 is the initial distance between two adjacent starting points at t = t0 and d is
the distance between two these points at the moment t. We return again to Fig 2.5, it
shows how the distance d between evolutions initiated from two points separated by
d0 = 10
–
7 varies with time. The separation growth exponentially in the range 20 < t <
< 300 before leveling off at the size of the attractor.
Fig. 2.6. Time evolution of the largest Lyapunov exponent (one cycle = 2π ω/ , ω = 0.83).
3. Conclusion. The first section of the paper shows how the asymptotic method
created by N. M. Krylov, N. N. Bogoliubov and Yu. A. Mitropolskii is effective in
solving a complicated nonlinear problem. Figure 1.2 presents different resonance
curves in “linear” case γ = 0. There exists a special returning point I* on the
resonance curves. In the “nonlinear case” γ ≠ 0, the resonance curve leans toward to
the right for γ > 0 (Fig.1.3, b), and to the left for γ < 0 (Fig.1.3, a), which is
common for nonlinear Duffing’s systems.
In the second section it is seen that chaotic phenomenon occurs in a deterministic
system described by (2.1). The Poincaré section, chaotic attractor and associated
power spectra of the nonlinear oscillator (2.1) have been found. The Fortran and
Matlab softwares were used for calculating data and building the graphs.
1. Nguyen Van Dao, Nguyen Van Dinh. Interaction between nonlinear oscillating systems. – Hanoi:
Vietnam Nat. Univ. Publ. House, 1999. – 358 p.
2. Mitropolskii Yu. A., Nguyen Van Dao. Applied asymptotic methods in nonlinear oscillations. –
Kluwer Publ., 1997. – 341 p.
3. Mitropolskii Yu. A., Nguyen Van Dao. Lectures on asymptotic methods of nonlinear dynamics. –
Hanoi: Vietnam Nat. Univ. Publ. House, 2003. – 503 p.
4. Nguyen Van Dao. Stability of dynamics systems. – Hanoi: Vietnam Nat. Univ. Publ. House, 1998.
5. Nayfeh A., Balakuma Balachandran. Applied nonlinear dynamics. – John Wiley&Sons, 1995. –
685 p.
6. Ueda Y. The road to chaos. – Aerial Press, 1992.
7. Kapitaniak T., Steeb W. H. Transition to chaos in a generalized Van der Pol’s equation // J. Sound
and Vibration. – 1990. – 143, # 1. – P. 167 – 170.
8. Lakshmanan M., Rajasekar S. Nonlinear dynamics. – Springer, 2003.
9. Baker G. L., Gollub J. P. Chaotic dynamics. An introduction. – Cambridge Univ. Press, 1990.
10. Francis C Moon. Chaotic vibration. – John Wiley, 1996.
11. Kathleen T. Alligood, Tom D. Sauer, Jame A. Yorke. Chaos. An introduction to dynamical systems.
– New York: Springer, 1997.
Received 14.08.2006
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 2
|
| id | umjimathkievua-article-3304 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:40:00Z |
| publishDate | 2007 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/a9/ac3ee0d0e59dc1afd2bc40ca2a6950a9.pdf |
| spelling | umjimathkievua-article-33042020-03-18T19:50:43Z Van der Pol oscillator under parametric and forced excitations Вплив нa систему Ван дер Поля параметричного та зовнішнього збурень Nguyen, Van Dao Nguyen, Van Dinh Tran, Kim Chi Нгуєн, Ван Дао Нгуєн, Ван Дінх Тран, Кім Чі We study a van der Pol oscillator under parametric and forced excitations. The case where a system contains a small parameter and is quasilinear and the general case (without the assumption of the smallness of nonlinear terms and perturbations) are studied. In the first case, equations of the first approximation are obtained by the Krylov-Bogolyubov-Mitropol’skii technique, their averaging is performed, frequency-amplitude and resonance curves are studied, and the stability of the given system is considered. In the second case, the possibility of chaotic behavior in a deterministic system of oscillator type is shown. Досліджено вплив параметричного та зовнішнього збурень на систему Ван дер Поля. Розглянуто випадки, коли дана система містить малий параметр і є квазілінійною та загальний випадок (без припущення про мализну нелінійних доданків та збурень). У першому випадку за допомогою методу Крилова - Боголюбова - Митропольського отримано рівняння першого наближення, проведено їх усереднення і вивчено частотно-амплітудні та резонансні криві, досліджено стійкість режимів даної системи. У другому випадку показано можливість хаотичної поведінки в детермінованих системах коливного типу. Institute of Mathematics, NAS of Ukraine 2007-02-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3304 Ukrains’kyi Matematychnyi Zhurnal; Vol. 59 No. 2 (2007); 206–216 Український математичний журнал; Том 59 № 2 (2007); 206–216 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3304/3353 https://umj.imath.kiev.ua/index.php/umj/article/view/3304/3354 Copyright (c) 2007 Nguyen Van Dao; Nguyen Van Dinh; Tran Kim Chi |
| spellingShingle | Nguyen, Van Dao Nguyen, Van Dinh Tran, Kim Chi Нгуєн, Ван Дао Нгуєн, Ван Дінх Тран, Кім Чі Van der Pol oscillator under parametric and forced excitations |
| title | Van der Pol oscillator under parametric and forced excitations |
| title_alt | Вплив нa систему Ван дер Поля параметричного та зовнішнього збурень |
| title_full | Van der Pol oscillator under parametric and forced excitations |
| title_fullStr | Van der Pol oscillator under parametric and forced excitations |
| title_full_unstemmed | Van der Pol oscillator under parametric and forced excitations |
| title_short | Van der Pol oscillator under parametric and forced excitations |
| title_sort | van der pol oscillator under parametric and forced excitations |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3304 |
| work_keys_str_mv | AT nguyenvandao vanderpoloscillatorunderparametricandforcedexcitations AT nguyenvandinh vanderpoloscillatorunderparametricandforcedexcitations AT trankimchi vanderpoloscillatorunderparametricandforcedexcitations AT nguênvandao vanderpoloscillatorunderparametricandforcedexcitations AT nguênvandính vanderpoloscillatorunderparametricandforcedexcitations AT trankímčí vanderpoloscillatorunderparametricandforcedexcitations AT nguyenvandao vplivnasistemuvanderpolâparametričnogotazovníšnʹogozburenʹ AT nguyenvandinh vplivnasistemuvanderpolâparametričnogotazovníšnʹogozburenʹ AT trankimchi vplivnasistemuvanderpolâparametričnogotazovníšnʹogozburenʹ AT nguênvandao vplivnasistemuvanderpolâparametričnogotazovníšnʹogozburenʹ AT nguênvandính vplivnasistemuvanderpolâparametričnogotazovníšnʹogozburenʹ AT trankímčí vplivnasistemuvanderpolâparametričnogotazovníšnʹogozburenʹ |