Routes to chaos through the interaction of high- and low-frequency oscillations
Roads to chaos for systems with the interaction of high- and low-frequency oscillations are considered. Results are presented for quisi-periodically forced Duffing and Van-der-Pole oscillators and a two-mode system driven by a harmonic force. The focus is made on the conditions for chaos under weak...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
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| Zitieren: | Routes to chaos through the interaction of high- and low-frequency oscillations / D.V. Shygimaga, D.M. Vavriv // Вопросы атомной науки и техники. — 2001. — № 6. — С. 223-225. — Бібліогр.: 6 назв. — англ. |
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| author | Shygimaga, D.V. Vavriv, D.M. |
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| citation_txt | Routes to chaos through the interaction of high- and low-frequency oscillations / D.V. Shygimaga, D.M. Vavriv // Вопросы атомной науки и техники. — 2001. — № 6. — С. 223-225. — Бібліогр.: 6 назв. — англ. |
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| description | Roads to chaos for systems with the interaction of high- and low-frequency oscillations are considered. Results are presented for quisi-periodically forced Duffing and Van-der-Pole oscillators and a two-mode system driven by a harmonic force. The focus is made on the conditions for chaos under weak nonlinearity of the system.
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ROUTES TO CHAOS THROUGH THE INTERACTION
OF HIGH- AND LOW-FREQUENCY OSCILLATIONS
D.V. Shygimaga, D.M. Vavriv
Institute of Radio Astronomy of NAS of Ukraine, Kharkov, Ukraine
e-mail: shig@rian.kharkov.ua
Roads to chaos for systems with the interaction of high- and low-frequency oscillations are considered. Results
are presented for quisi-periodically forced Duffing and Van-der-Pole oscillators and a two-mode system driven by a
harmonic force. The focus is made on the conditions for chaos under weak nonlinearity of the system.
PACS: 29.27.Hj, 29.90.+r
1. INTRODUCTION
It is well known that interaction of two or more os-
cillations or modes may lead to chaotic oscillations. Up
to now, the most extensively studied systems were ones
with the resonant interaction of oscillations, when the
following conditions were met [1-4]:
mn≈21 ωω , or 21 ωωω ±= , (1)
where 1ω and 2ω are external frequencies or the natu-
ral frequencies of interacting modes, n and m are com-
paratively small integers, and ω stands for the frequen-
cy of an external force or the third interacting mode.
The most recent results [5] show that even the interac-
tion of oscillations with substantially different natural
frequencies, i.e., when the following condition is met:
21 ωω > > , (2)
may considerably change the system dynamics.
In this paper we present results of recent investiga-
tions of different dynamical systems, like quisiperiodic-
ally forced Duffing and Van-der-Pole oscillators and a
two-mode system driven by a harmonic force. We show
that interaction of low- and high-frequency oscillations
in such systems leads to the chaos onset even in the lim-
it of weak nonlinearity. Both, numerical and analytical
methods are used for the study of chaotic states.
2. CHAOS IN DUFFING OSCILLATOR
WITH HIGH- AND LOW-FREQUENCY EX-
TERNAL FORCING
Let us consider a two-frequency forced Duffing os-
cillator:
tFtkF
xxxxx
ω
γβωµ
coscos0
322
0
′+Ω′
=′′+′′−+′+
, (3)
under the following condition: Ω> >≈ 0ωω , where
2,1=k , ω and Ω are the frequencies of external exci-
tation, and 0ω is the natural frequency of oscillator.
This equation has been studied already for the case
when the following condition is met: 0ωω mn = , where
,,2,1, =mn or 210 ωωω += .
Our study has demonstrated that the oscillator (3)
demonstrates chaotic behavior as a result of interaction
of high- and low-frequency oscillations even in the
weakly nonlinear limit. For this case the equation (3)
can be simplified by applying the standard averaging
technique. This leads to the following averaged equa-
tions:
( )[ ]
( )[ ] .2cos2cos
,2cos2cos
222
222
PuBBvu
vv
vBBvu
uu
−++++∆
+−=
++++∆
−−=
ε τγε τβγ
µ
ε τγε τβγ
µ
(4)
for k=1, and
( ) ( )[
( ) ]
( ) ( )[
( ) ] .cos4cos22
2
,cos4cos22
2
22
22
22
22
uBPB
vuPPvv
vBPB
vuPPuu
ε τγε τγβ
γγβµ
ε τγε τγβ
γγβµ
++
+++++∆+−=
+−
+++−−∆−−=
(5)
for k=2.
Here the point means derivative with respect to a
slow time τ , 1< <ε is a normalized frequency of low-
frequency external forcing, P is a normalized amplitude
of high-frequency external forcing, B is amplitude of
the low-frequency external forcing, β and γ are non-
linearity coefficients, and ∆ is frequency detuning para-
meter.
We have studied this system by using two methods:
Melnikov technique and the method of second averag-
ing. The first one allows us to find conditions for the
chaos onset in the system, and the second one allows
finding conditions for the period-doubling bifurcation
and for the tangential bifurcation. The results obtained
by using the both techniques show good correspondence
with the results of numerical experiments (see Fig. 1).
The criterion obtained in accordance with the Melnikov
technique, is shown by solid line. Crosses show regions
of chaos, obtained in numerical experiment. The border
of the first period-doubling bifurcation, obtained by the
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 6, p. 223-225. 223
-5 -4 -3 -2 -1 0
0
1
2
3
4
5 B
∆
∆ = - 0.2
Fig. 1. Chaos regions on the ),( B∆ parameters
plane for k=1, 1.0=β , 1.0=γ , 1.0=P , 08.0=ε ,
and 01.0=µ
second averaging method, is shown by dashed line, and
the border of the tangential bifurcation is shown by dot-
ted line. With respect to the initial equation (3) the
transition to chaos is realised via destruction of two- di-
mensional tori.
3. CHAOS IN THE VAN-DER-POLE OSCIL-
LATOR WITH LOW-FREQUENCY ANODE
VOLTAGE MODULATION
The next system under consideration is the Van-der-
Pole oscillator with a low- and high-frequency external
forcing. This system is described by the following equa-
tion:
( )
( ) )cos('sin'
2 322
10
τν+τθ−
=γ+β−+µ−µ−
BA
qqqqqq
(6)
Here 0µ and 1µ are the damping coefficients, β
and γ are coefficients of nonlinearity, 'A and θ are the
amplitude and phase of low-frequency modulation, cor-
respondingly, 'B and ν are the amplitude and phase of
the synchronizing force ( 1≈ν , i.e., ν is close to the
natural frequency of the oscillator).
After the application of the averaging technique to
(6) we obtain the following equations:
( )
.sinsincos
,sinsin1
22
22
θ τγ+θ τ−ϕ−γ+∆=
τ
ϕ
θ τ−ϕ−−=
τ
AC
a
Ba
d
d
AaBaa
d
da
(7)
Here C is related to A and β .
Conditions for the stability of the synchronous mode
are as following:
2
12 >a , (8)
( )( ) ( )( ) 03311 2222 >+∆+∆+−− aaaa γγ . (9)
The application of method of the second averaging has
also allowed us to obtain conditions for the period-
doubling bifurcation in this system.
-20 -18 -16 -14 -12 -10 -8 -6
0,0
0,2
0,4
0,6
0,8
1,0
1,2
1,4
A=0.4
C=1.7
γ =10
θ =1
B
∆
Fig. 2. Bifurcation diagram on the ( )∆,B plane for
given values of parameters
Results of numerical experiments as well as condi-
tions for the period-doubling bifurcation and for the sta-
bility of the synchronous mode are shown in Fig. 2.
Crosses here represent chaotic regions obtained from
numeric simulations; points are period-2 oscillations;
the border of synchronous oscillations is shown by solid
line, and the boundary for the period-doubling bifurca-
tion obtained analytically is shown by dashed line.
4. CHAOS IN A NONLINEAR TWO-MODE
HARMONICALLY FORCED SYSTEM
In this section we review results of the investigation
of a harmonically forced system of two coupled passive
oscillators, which natural frequencies differ essentially.
In the general case such system can be described by the
following equations:
.2
,)cos2(2
22
2
2
2
2
1
2
2
HF
LF
LF
LF
LFHF
HF
HF
HF
x
d
dx
x
d
xd
Sxx
d
dx
x
d
xd
γε−
τ
ε µ−=ε+
τ
ν τ−γε−
τ
ε µ−
=+
τ
(10)
Here HFx and LFx are variables describing high-
and low-frequency oscillators, correspondingly, 1µ and
2µ represent damping in high- and low-frequency os-
cillators, correspondingly; γ is the coefficient of non-
linearity; τ is a slow time; S is the amplitude of the
external forcing; ν is relevant to the natural frequency
of high-frequency oscillator.
A possible physical realization of the above system
is shown in Fig. 3. Here 1L , 1R , and 1C represent reso-
nantly driven by an external harmonic force high-fre-
quency circuit I. 2L , 2R , and 2C represent low-fre-
quency circuit II. It is generally believed that if the con-
ditions ,2121 , CCLL εε ≈≈ are met, the influence of the
low-frequency circuit on the dynamics of the whole sys-
tem can be neglected. Our studies [6] have shown that
224
such interaction can exert a considerable influence on
the system dynamics.
The first equation of (10) can be considered as the
motion equation of a quasilinear oscillator. So one can
apply an averaging technique to it. After performing
corresponding transformations one can come to the fol-
lowing system of averaged equations:
.
2
12
,
,cos
,sin
2
2
1
au
u
a
Su
Saa
γµ
ϕγϕ
ϕµ
−−−=
=
−+∆−=
−−=
vv
v
(11)
L2L1
C1(u)
R1 R2C2
e(t)
Fig. 3. Two-mode externally forced system
-9 -6 -3 0 3 60
2
4
6
8
S
∆
Fig. 4. Obtained from numerical simulations bifur-
cation diagram on the parameter plane (S, ∆) at γ=1.0;
µ1=0.7 and µ2=0.01
∆
S
0
2
4
6
8
-10 -8 -6 -4 -2 0 2 4 6
Fig. 5. Experimental bifurcation diagram on the pa-
rameter plane (S, ∆) with the same notations as in
Fig. 4. The fine structure of the chaos region is not indi-
cated
The overdot here denotes differentiation with respect
to the slow time ετ, LFx≡v , LFxu ≡ are independent
variables which define the state of the low-frequency
oscillator, and ( ) ( )ε νν 212 −=∆ is the parameter of the
frequency mismatch.
Bifurcation diagram of the system (1) obtained nu-
merically is shown in Fig. 4. It should be noted that the
system demonstrates chaotic behaviour in a wide range
of variation of control parameters, and that the threshold
for chaos onset is going down with a decrease of the
damping coefficients. Results of experimental investiga-
tions of the circuit in Fig. 3 are shown in Fig. 5 on the
same parameter plane. A good qualitative agreement
between these results should be mentioned.
CONCLUSION
The results of the presented investigations allow us
to make the following conclusions:
(i) periodically excited systems with the interaction of
high- and low-frequency oscillations are susceptible to
chaotic instabilities to a great extent,
(ii) the chaotic oscillations can arise under weakly non-
linear conditions of excitation,
(iii) chaotic instabilities due to the interaction of low-
and high-frequency oscillations can exert a strong influ-
ence on the dynamics of many practical systems.
ACKNOWLEDGMENT
The authors are indebted to V.V. Vinogradov for his
contributions to this work. The work was partially sup-
ported by EC under Contract IC15CT980509.
REFERENCES
1. C. Holmes and P. Holmes. Second order averaging
and bifurcations to period two in DuAEng’s equa-
tion // J. Sound Vib. 1981, v. 78, p. 161-174.
2. J. Miles. Chaotic motion of a weakly nonlinear,
modulated oscillator // Appl. Phys. and Math. Sci.
1984, v. 81, №6, p. 3919-3923.
3. D.M. Vavriv, V.B. Ryabov, S.A. Sharapov, and
H.M. Ito. Chaotic states of weakly and strongly non-
linear oscillators with quasiperiodic excitation //
Physical Review E. 1996, v. 53, №1, p. 103-114.
4. D.M. Vavriv, Yu.A. Tsarin, and I.Yu. Chernyshov.
Forced Oscillations of Two Coupled Passive Oscil-
lators // Radiotekh. Electron. 1991, v. 36, №10,
p. 2015-2023 (in Russian).
5. A.H. Nayfeh, S.A. Nayfeh, and B. Balachandran.
Transfer of Energy from High-Frequency to Low-
Frequency Modes, in Nonlinearity and Chaos in En-
gineering Dynamics, J.M.T. Thompson and S.R.
Bishop, Eds. John Wiley & Sons Ltd., 1994, p. 39-
58.
6. D.V. Shygimaga, D.M. Vavriv, V.V. Vinogradov.
Chaos due to the interaction of high-and low-fre-
quency modes // IEEE Tran. on Circuits and Sys-
tems. 1998, v. 45, № 12, p. 1255-1259.
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2000, №2.
Серия: Ядерно-физические исследования (36), с. 3-6.
225
D.V. Shygimaga, D.M. Vavriv
Institute of Radio Astronomy of NAS of Ukraine, Kharkov, Ukraine
PACS: 29.27.Hj, 29.90.+r
1. INTRODUCTION
2. CHAOS IN DUFFING OSCILLATOR WITH HIGH- AND LOW-FREQUENCY EXTERNAL FORCING
3. CHAOS IN THE VAN-DER-POLE OSCILLATOR WITH LOW-FREQUENCY ANODE VOLTAGE MODULATION
4. CHAOS IN A NONLINEAR TWO-MODE HARMONICALLY FORCED SYSTEM
CONCLUSION
ACKNOWLEDGMENT
REFERENCES
|
| id | nasplib_isofts_kiev_ua-123456789-79893 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T18:54:39Z |
| publishDate | 2001 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Shygimaga, D.V. Vavriv, D.M. 2015-04-06T16:26:39Z 2015-04-06T16:26:39Z 2001 Routes to chaos through the interaction of high- and low-frequency oscillations / D.V. Shygimaga, D.M. Vavriv // Вопросы атомной науки и техники. — 2001. — № 6. — С. 223-225. — Бібліогр.: 6 назв. — англ. 1562-6016 PACS: 29.27.Hj, 29.90.+r https://nasplib.isofts.kiev.ua/handle/123456789/79893 Roads to chaos for systems with the interaction of high- and low-frequency oscillations are considered. Results are presented for quisi-periodically forced Duffing and Van-der-Pole oscillators and a two-mode system driven by a harmonic force. The focus is made on the conditions for chaos under weak nonlinearity of the system. The authors are indebted to V.V. Vinogradov for his contributions to this work. The work was partially supported by EC under Contract IC15CT980509. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Anomalous diffusion, fractals, and chaos Routes to chaos through the interaction of high- and low-frequency oscillations Пути к хаосу через взаимодействие высоко- и низкочастотных колебаний Article published earlier |
| spellingShingle | Routes to chaos through the interaction of high- and low-frequency oscillations Shygimaga, D.V. Vavriv, D.M. Anomalous diffusion, fractals, and chaos |
| title | Routes to chaos through the interaction of high- and low-frequency oscillations |
| title_alt | Пути к хаосу через взаимодействие высоко- и низкочастотных колебаний |
| title_full | Routes to chaos through the interaction of high- and low-frequency oscillations |
| title_fullStr | Routes to chaos through the interaction of high- and low-frequency oscillations |
| title_full_unstemmed | Routes to chaos through the interaction of high- and low-frequency oscillations |
| title_short | Routes to chaos through the interaction of high- and low-frequency oscillations |
| title_sort | routes to chaos through the interaction of high- and low-frequency oscillations |
| topic | Anomalous diffusion, fractals, and chaos |
| topic_facet | Anomalous diffusion, fractals, and chaos |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/79893 |
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