The regular and chaotic dynamics at weak-nonlinear interaction of waves
It is shown that different regimes at weak-nonlinear interaction of waves are possible: regular regimes with the growth of degree of coherence of some waves which take part in interaction and also chaotic regimes. It is shown that in some cases the increasing number of waves which take part in inter...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2010
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| Cite this: | The regular and chaotic dynamics at weak-nonlinear interaction of waves / V.A. Buts, I.K. Kovalchuk, D.V. Tarasov, A.P. Tolstoluzhsky // Вопросы атомной науки и техники. — 2010. — № 6. — С. 132-134. — Бібліогр.: 12 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860249934466383872 |
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| author | Buts, V.A. Kovalchuk, I.K. Tarasov, D.V. Tolstoluzhsky, A.P. |
| author_facet | Buts, V.A. Kovalchuk, I.K. Tarasov, D.V. Tolstoluzhsky, A.P. |
| citation_txt | The regular and chaotic dynamics at weak-nonlinear interaction of waves / V.A. Buts, I.K. Kovalchuk, D.V. Tarasov, A.P. Tolstoluzhsky // Вопросы атомной науки и техники. — 2010. — № 6. — С. 132-134. — Бібліогр.: 12 назв. — англ. |
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| description | It is shown that different regimes at weak-nonlinear interaction of waves are possible: regular regimes with the growth of degree of coherence of some waves which take part in interaction and also chaotic regimes. It is shown that in some cases the increasing number of waves which take part in interaction leads to significant simplification of dynamics of interacting waves and not to complication. Possible electrodynamics structures in which the regimes with dynamical chaos can realize are discussed in this paper. In particular the properties of plasma waveguides filled with rare plasma are investigated. It is shown that in such waveguides the regimes with chaotic dynamics may be realized.
Показано, что при слабонелинейном взаимодействии волн возможны разнообразные режимы: регулярные режимы с ростом степени когерентности отдельных, участвующих во взаимодействии волн, а также хаотические режимы. Показано, что в некоторых случаях рост участвующих во взаимодействии волн (числа степеней свободы) приводит не к усложнению динамики взаимодействия волн, а к существенному упрощению этой динамики. Обсуждаются возможные электродинамические структуры, в которых могут реализоваться режимы с динамическим хаосом. В частности, изучены дисперсионные свойства плазменных волноводов с редкой плазмой. Показано, что в таких волноводах легко реализовать режимы с динамическим хаосом.
Показано, що при слабконелінійній взаємодії хвиль можливі різноманітні режими: регулярні режими з ростом ступеня когерентності окремих, що беруть участь у взаємодії, хвиль, а також хаотичні режими. Показано, що в деяких випадках ріст хвиль, що беруть участь у взаємодії (числа ступенів волі), приводить не до ускладнення динаміки взаємодії хвиль, а до істотного спрощення цієї динаміки. Обговорюються можливі електродинамічні структури, у яких можуть реалізуватися режими з динамічним хаосом. Зокрема, вивчені дисперсійні властивості плазмових хвилеводів з рідкою плазмою. Показано, що в таких хвилеводах легко реалізувати режими з динамічним хаосом.
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THE REGULAR AND CHAOTIC DYNAMICS AT WEAK-NONLINEAR
INTERACTION OF WAVES
V.A. Buts, I.K. Kovalchuk, D.V. Tarasov, A.P. Tolstoluzhsky
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
E-mail: vbuts@kipt.kharkov.ua
It is shown that different regimes at weak-nonlinear interaction of waves are possible: regular regimes with the
growth of degree of coherence of some waves which take part in interaction and also chaotic regimes. It is shown that in
some cases the increasing number of waves which take part in interaction leads to significant simplification of dynamics
of interacting waves and not to complication. Possible electrodynamics structures in which the regimes with dynamical
chaos can realize are discussed in this paper. In particular the properties of plasma waveguides filled with rare plasma
are investigated. It is shown that in such waveguides the regimes with chaotic dynamics may be realized.
PACS: 52.35.Mw, 52.35.Qz
1. THE INCREASING OF THE DEGREE
OF COHERENCE AT THREE-WAVE
INTERACTION
132 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2010. № 6.
Series: Plasma Physics (16), p. 132-134.
Let us consider the process of decay of a high-
frequency wave on high-frequency and low-frequency
one. The regular process of such decay is well enough
investigated [1, 2]. We shall consider that during the
initial moment of time in nonlinear medium the
nonmonochromatic high-frequency wave is propagated.
The spectrum width of this wave is equal δω . Besides for
simplicity of the further analysis, we shall consider that
the source of a nonregularity of this wave is random
diffusion of a phase. In such model this fact is important
and the equation for random phase can be written down in
the form ( )tϕ ξ=& , where ( )tξ - delta-correlated function
with a zero average: 0ξ = , ( )( ) ( )t t t tξ ξ δω δ′ ′⋅ = ⋅ − .
Function ( )tξ is the instantaneous frequency of a
signal. The diffusion coefficient δω is the width of the
spectrum of signal. Besides high-frequency wave in
nonlinear medium there are small coherent high-
frequency perturbations and also small low-frequency
perturbations. We shall consider also that the amplitude of
the basic signal is great enough, so the increment of decay
instability is more than the width of a spectrum of this
signal ( δωΓ > ). In this case the equations that describe
dynamics of complex amplitudes of interacting waves can
be presented in such form:
1 2 3A A A= − ⋅&
2 1 3A A A∗= ⋅&
23 1A A A∗= ⋅& . (1)
When we obtain these equations the laws of
conservation of energy and an impulse have been
considered:
3
1
i i
i
N constω
=
=∑ , . (2)
3
1
i i
i
k N const
=
=∑
r
Besides that the normalization has been used:
2
k kA N= - number of quanta in -wave; k /A dA dτ=& ,
V tτ = ⋅ , V - matrix element of waves interaction.
At the initial stage of process of decay it is possible to
consider that the amplitude of decaying wave does not
vary. In this case it is necessary to analyze the last two
equations of system (1). Thus it is convenient to introduce
new function 1 3B A A∗= ⋅ . Then the last two equations of
system (1) can be rewrite in the form of:
2A B=& , 2
1B B Aξ 2A= ⋅ + ⋅& . (3)
From system (3) using Furutsu-Novikov equation [3] it
is possible to obtain the following equations for the first
moments (average quantity):
2A B=& , 2
1 2 / 2B A A Bδω= ⋅ + ⋅& . (4)
It is seen from (4), that the process of decay for
average quantities goes the same way (practically with the
same increment ( )2 2
1 1/16 /16A Aδω δωΓ = + + ≈ ) as
well as at decay of the regular wave ( 2
1A δω>> ). Thus
from the function ( )B τ it is possible to make the
inference, that the fluctuations of the phase of a decaying
wave are compensated by the fluctuations of a phase of a
low-frequency wave. As a result, the fluctuations of
complex amplitude 2A and function ( )B τ are small. For
the analysis of nonlinear dynamics it is useful to rewrite
the set of equations (1) in such form:
( )
2
2
1
2
2 2
2 1 1
,
2 (0) .
d A
2 ,A B A B A B
d
B A A A
τ
∗ ∗⎡ ⎤= − + =⎣ ⎦
⎡ ⎤= −⎣ ⎦
&
&
(5)
Considering that the fluctuations of amplitude 2A and
the function ( )B τ are small, it is easy to make averaging
of the set of equations (5). One can see, that in this case
these average equations will coincide with the equations
(5). In turn, the equations (5) do not differ from the
equations which describe the regular dynamics of decay
process. Thus, as it is known, during the order of
reciprocal increment (T 1~ −Γ ) practically all the energy
of the decaying wave transfers into the energy of high-
frequency coherent components. The insignificant part of
the energy, according to Manley-Rowe's relation,
transfers to a low-frequency wave
( ( )1~ /E Eδ ω EΩ << ).
Thus, we have shown that despite that fact, that the
phase of the decaying wave undergoes random
fluctuations, this wave can practically transmit all the
energy to a monochromatic coherent high-frequency wave
and the part of the energy will transfer to a low-frequency
wave. Now if we interrupt nonlinear interaction of waves
after the complete transmitting total energy from pumping
wave pass into new high-frequency one and in the field of
low-frequency wave. The process appear as transformation
of the energy of incoherent wave into the energy of
coherent wave. It can seem that such process proceeds with
breaking of the second law of thermodynamics. However,
as it is shown in paper [4], such processes proceed with the
growth of entropy in the complete accordance with the
second law of thermodynamics. The matter is that all the
entropy from the high-frequency wave transfers into the
entropy of low-frequency wave.
133
2. DYNAMICS OF CASCADES ALLOW
FOR PROCESSES OF FUSION
In the theory of weak nonlinear interactions of waves
it is known the model which is widely used at the analysis
of the processes which arise at the action of powerful
laser radiation on plasma. In this model we shall consider,
that the number of high-frequency waves can be infinitely
large and interaction between them is carried out through
one low-frequency wave. The mathematical model of
such processes, apparently, for the first time has been
obtained in papers [5,6] and analyzed by many authors:
*
1 1 ,i t i tn
n n
dai ba e b a e
dt
δ δ−
− += +
2
1
*
1 .
n
i t
n n
n n
dbi a a e
d t
δ−
−
= −
= ∑ (6)
Here , - amplitude of a HF and LF wave, na b
1n nδ ω ω −= − −Ω - detuning, nω - frequency of a HF
wave, -frequency of a LF wave. Ω
In real systems the number of high-frequency
interacting waves is great enough but finite. Therefore, we
shall assume – the amplitudes of waves if is
outside the range of values ( n -number of
"red" satellites, - number of "blue" satellites). It is
possible to show, that from (6) except the conservation
laws of type (2) at there is an integral of a motion
and the equation of system (6)
becomes free. As it is seen from (6) excitation of more
and more high numbers of harmonics eventually occurs
because of interaction with LF wave. Analytically (at
) it is possible to show [5] that if the quantity
is restricted, the maximal
number of excited waves on an order of magnitude is
equal
0na = n
1 2n n n− ≤ ≤ 1
2
t′
2n
1n n=
2
1
*
1
n
n n
n n
a a const−
=−
=∑
n → ∞
0
( ) ( ) exp( )
t
t b t i t dδ′ ′Β = −∫
max maxn B . The amplitudes of higher harmonics
will be negligibly small. In case when , the
number of excited waves is defined from the requirement
. Depending on the initial state the various
modes of the dynamics of wave interaction can be
realized: with constant amplitude of a LF-wave, with
periodic changes of this amplitude, with frequency
detuning
max 1 2,n n> n
21n n n− ≤ ≤
δ and at 0δ → the amplitude of a LF wave
linearly grows in time [5, 7].
For the case with finite values the set of equations
(6) is solved numerically for various initial values of the
amplitudes of HF and LF fields, and also for the
parameter of detuning
n
δ [7]. Dynamics of LF and HF
fields is in strict correspondence with analytical
consideration up to the moment of time *t t= of
excitation of boundary harmonics (harmonic with
numbers or ). At dynamics of amplitudes
essentially differs from the case though equality
1n 2n *t t>
n →∞
ia a− = i
2
is carried out. In the symmetrical case
( 1n n= ) dynamics of waves remains enough regular. In
the asymmetrical case 1n n2≠ dynamics of waves gets the
nonregular character. The spectrum of power of HF
waves is wide with slow diminution of the field of high
frequencies. The correlation function promptly decreases
up to zero and has the nonregular oscillations close to
zero level what is characteristic for chaotic processes.
Presence of parameter of detuning δ conducts to
additional complication of dynamics of interaction of
waves. In case with the broken symmetry ( 1n n2≠ ) and
values of detuning 0.05δ = the dynamics of waves
becomes chaotic.
3. THE DISPERSION PROPERTIES
OF THE CYLINDRICAL WAVEGUIDE
FILLED WITH RARE PLASMA
It is known, that for excitation of broadband noise
radiation making special type of generators is necessary.
This problem is complex enough and not well
investigated at present time. On the other hand, the
generators of regular intensive oscillations are well
investigated. It is enough to mention such devices as
klystrons, magnetrons, TWT and the others; however on
output they give narrow spectral lines. In many cases (for
example, for the purposes of broadband detection) wide
spectrums are necessary to us. Therefore it is possible to
imagine, that radiation with the narrow spectral line
obtained from magnetron, for example, gets in nonlinear
medium in which the regimes with dynamic chaos are
possible. On an output from such medium the spectrum of
radiation can have the necessary form. However, as
numerical estimations show, in usual requirements the
necessary intensity for embodying chaotic regimes
becomes significantly greater (see [8-10]). So, for
example, for the unbounded plasma this intensity of fields
make is more then 20 kV/cm. Is of interest to find such
electrodynamic structures with nonlinear devices in which
the regime with dynamic chaos would develop at much
smaller intensities of the fields. We know (from the
previous investigations), that the less the distance on
frequency between eigen waves of electrodynamic
structure are, the smaller the intensities of the fields are,
witch are necessary for transition into regimes with
dynamic chaos. Therefore our problem is to find such
mediums and structures in which the distance on
frequency between eigen waves could be minimal. The
possible candidate for such structure is the cylindrical
134
metal waveguide filled with rare plasma and held in
external magnetic field.
Such electrodynamic structures were investigated but
consideration in them was restricted by studying slow
waves [11]. It is related with the fact that such structures,
first of all, were considered for the purposes of
acceleration of charged particles - especially heavy
particles. The dispersion of fast waves remained
insufficiently investigated [12].
We investigated the dispersion of cylindrical
waveguide partially filled with cylindrical magnetoactive
plasma, coaxial with a wave guide. For this system the
dispersion equation which was explored analytically and
numerically was obtained. It is shown analytically and
numerically, that in the field of frequencies between
electronic cyclotron and upper hybrid there is the infinite
number of branches of eigenmodes. In this area such
frequencies can be easily chosen and the distance
between them will correspond to the necessary
requirements for realization of the regimes with dynamic
chaos.
REFERENSES
1. B.B. Kadomtsev. Collectrive phenomena in plasma.
M: “Nauka”, 1988 (in Russian).
2. H. Wilhelmsson, J. Weiland. Coherent non-linear
interaction of waves in plasmas. M: “Energoizdat”,
1981 (in Russian).
3. V.I. Klyatskin. Statistical description of dynamical
systems with fluctuation parameters. In book: Modern
Problem of Physics. M.: “Nauka”, 1975 (in Russian).
4. V.A. Buts. Some features of the law of degradation of
energy // Uspekhi sovremennoj radioelectroniki. 2008,
N 7, p. 42-53 (in Russian).
5. A.S. Bakaj. Interaction of high frequency and low
frequency waves in a plasma // Nuclear Fusion. 1970,
v. 10, p. 53-67.
6. A.A. Vodyanitskij, N.S. Repalov. Nonlinear
interaction of longitudinal waves in nonisothermal
plasma // ZhTF. 1970, v. 40, N 1, p. 32 (in Russian).
7. V.A. Buts, A.P Tolstoluzhsky. Regular and chaotic
dynamics of decaying cascade of waves in plasma //
Problems of Atomic Science and Technology. Series
“Plasma Physics” (7). 2002, N 4, p. 106-108.
8. V.A. Buts, O.V. Manuylenko, K.N. Stepanov,
A.P. Tolstoluzhsky. Chaotic dynamics of the charged
particles at interaction of type wave-particle and
chaotic dynamics of waves at weak-nonlinear
interaction of wave-wave type // Physic of Plasma.
1994, v. 20, N 9, p. 794-801 (in Russian).
9. V.A. Buts, O.V. Manuylenko, A.P. Tolstoluzhsky.
Instability and dynamicall chaos in a weak nonlinear
interaction of waves // EPAC’94 European Particle
Accelerator Conference. London, June 27 - July 1,
1994, p. 145-146.
10. V.A. Buts, O.V. Manuylenko, A.P. Tolstoluzhsky.
Development of Dynamical Chaos under Nonlinear
Interaction of Waves in Bounded Magnetized Plasma
System // 5th European Particle Accelerator
Conference. Barcelona, 10-14 June,1996, p. 111-112.
11. A.N. Kondratenko. Plasma waveguide. М.:
“Аtоmizdat”, 1976 (in Russian).
12. A.N. Antonov, V.A. Buts, O.F. Kovpik,
E.A. Kornilov, V.G. Svichensky, D.V. Tarasov.
Chaotic Decayes in Resonators filled with rare Plasma
// Problems of Atomic Science and Technology. Series
“Plasma Electronic and New Methods of
Acceleration” (6). 2008, N4, p. 245-249 (in Russian).
Article received 17.09.10
РЕГУЛЯРНАЯ И ХАОТИЧЕСКАЯ ДИНАМИКА ПРИ СЛАБОНЕЛИНЕЙНОМ ВЗАИМОДЕЙСТВИИ ВОЛН
В.А. Буц, И.К. Ковальчук, Д.В. Тарасов, А.П. Толстолужский
Показано, что при слабонелинейном взаимодействии волн возможны разнообразные режимы: регулярные
режимы с ростом степени когерентности отдельных, участвующих во взаимодействии волн, а также
хаотические режимы. Показано, что в некоторых случаях рост участвующих во взаимодействии волн (числа
степеней свободы) приводит не к усложнению динамики взаимодействия волн, а к существенному упрощению
этой динамики. Обсуждаются возможные электродинамические структуры, в которых могут реализоваться
режимы с динамическим хаосом. В частности, изучены дисперсионные свойства плазменных волноводов с
редкой плазмой. Показано, что в таких волноводах легко реализовать режимы с динамическим хаосом.
РЕГУЛЯРНА Й ХАОТИЧНА ДИНАМІКА ПРИ СЛАБКОНЕЛІНІЙНІЙ ВЗАЄМОДІЇ ХВИЛЬ
В.О. Буц, І.К. Ковальчук, Д.В. Тарасов, О.П. Толстолужський
Показано, що при слабконелінійній взаємодії хвиль можливі різноманітні режими: регулярні режими з
ростом ступеня когерентності окремих, що беруть участь у взаємодії, хвиль, а також хаотичні режими.
Показано, що в деяких випадках ріст хвиль, що беруть участь у взаємодії (числа ступенів волі), приводить не
до ускладнення динаміки взаємодії хвиль, а до істотного спрощення цієї динаміки. Обговорюються можливі
електродинамічні структури, у яких можуть реалізуватися режими з динамічним хаосом. Зокрема, вивчені
дисперсійні властивості плазмових хвилеводів з рідкою плазмою. Показано, що в таких хвилеводах легко
реалізувати режими з динамічним хаосом.
1. THE INCREASING OF THE DEGREE OF COHERENCE AT THREE-WAVE INTERACTION
2. DYNAMICS OF CASCADES ALLOW FOR PROCESSES OF FUSION
3. THE DISPERSION PROPERTIES OF THE CYLINDRICAL WAVEGUIDE FILLED WITH RARE PLASMA
REFERENSES
|
| id | nasplib_isofts_kiev_ua-123456789-17481 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T18:42:14Z |
| publishDate | 2010 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Buts, V.A. Kovalchuk, I.K. Tarasov, D.V. Tolstoluzhsky, A.P. 2011-02-26T22:11:45Z 2011-02-26T22:11:45Z 2010 The regular and chaotic dynamics at weak-nonlinear interaction of waves / V.A. Buts, I.K. Kovalchuk, D.V. Tarasov, A.P. Tolstoluzhsky // Вопросы атомной науки и техники. — 2010. — № 6. — С. 132-134. — Бібліогр.: 12 назв. — англ. 1562-6016 https://nasplib.isofts.kiev.ua/handle/123456789/17481 It is shown that different regimes at weak-nonlinear interaction of waves are possible: regular regimes with the growth of degree of coherence of some waves which take part in interaction and also chaotic regimes. It is shown that in some cases the increasing number of waves which take part in interaction leads to significant simplification of dynamics of interacting waves and not to complication. Possible electrodynamics structures in which the regimes with dynamical chaos can realize are discussed in this paper. In particular the properties of plasma waveguides filled with rare plasma are investigated. It is shown that in such waveguides the regimes with chaotic dynamics may be realized. Показано, что при слабонелинейном взаимодействии волн возможны разнообразные режимы: регулярные режимы с ростом степени когерентности отдельных, участвующих во взаимодействии волн, а также хаотические режимы. Показано, что в некоторых случаях рост участвующих во взаимодействии волн (числа степеней свободы) приводит не к усложнению динамики взаимодействия волн, а к существенному упрощению этой динамики. Обсуждаются возможные электродинамические структуры, в которых могут реализоваться режимы с динамическим хаосом. В частности, изучены дисперсионные свойства плазменных волноводов с редкой плазмой. Показано, что в таких волноводах легко реализовать режимы с динамическим хаосом. Показано, що при слабконелінійній взаємодії хвиль можливі різноманітні режими: регулярні режими з ростом ступеня когерентності окремих, що беруть участь у взаємодії, хвиль, а також хаотичні режими. Показано, що в деяких випадках ріст хвиль, що беруть участь у взаємодії (числа ступенів волі), приводить не до ускладнення динаміки взаємодії хвиль, а до істотного спрощення цієї динаміки. Обговорюються можливі електродинамічні структури, у яких можуть реалізуватися режими з динамічним хаосом. Зокрема, вивчені дисперсійні властивості плазмових хвилеводів з рідкою плазмою. Показано, що в таких хвилеводах легко реалізувати режими з динамічним хаосом. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Плазменная электроника The regular and chaotic dynamics at weak-nonlinear interaction of waves Регулярная и хаотическая динамика при слабонелинейном взаимодействии волн Регулярна й хаотична динаміка при слабконелінійній взаємодії хвиль Article published earlier |
| spellingShingle | The regular and chaotic dynamics at weak-nonlinear interaction of waves Buts, V.A. Kovalchuk, I.K. Tarasov, D.V. Tolstoluzhsky, A.P. Плазменная электроника |
| title | The regular and chaotic dynamics at weak-nonlinear interaction of waves |
| title_alt | Регулярная и хаотическая динамика при слабонелинейном взаимодействии волн Регулярна й хаотична динаміка при слабконелінійній взаємодії хвиль |
| title_full | The regular and chaotic dynamics at weak-nonlinear interaction of waves |
| title_fullStr | The regular and chaotic dynamics at weak-nonlinear interaction of waves |
| title_full_unstemmed | The regular and chaotic dynamics at weak-nonlinear interaction of waves |
| title_short | The regular and chaotic dynamics at weak-nonlinear interaction of waves |
| title_sort | regular and chaotic dynamics at weak-nonlinear interaction of waves |
| topic | Плазменная электроника |
| topic_facet | Плазменная электроника |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/17481 |
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