The influence of the dynamics features of the trapped particles on a spectrum of their oscillations
Some features of dynamics of ensembles of coupled linear and nonlinear oscillators are investigated. It is shown that spectral characteristics of this dynamics essentially depend not only on quantity of the oscillators making the ensemble, but also on features of motion of each oscillator. In partic...
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irk-123456789-1121562017-01-23T22:52:06Z The influence of the dynamics features of the trapped particles on a spectrum of their oscillations Buts, V.A. Vavriv, D.M. Tarasov, D.V. Нелинейные процессы в плазменных средах Some features of dynamics of ensembles of coupled linear and nonlinear oscillators are investigated. It is shown that spectral characteristics of this dynamics essentially depend not only on quantity of the oscillators making the ensemble, but also on features of motion of each oscillator. In particular, at chaotic motion – on the moments of their chaotic dynamics. Special attention is turned on appearance of LF dynamics of ensemble. In this case there is a possibility of an effective exchange of energy between HF and LF oscillations. Досліджені особливості динаміки ансамблів зв'язаних лінійних і нелінійних осциляторів. Показано, що спектральні характеристики цієї динаміки суттєво залежать не тільки від кількості осциляторів, складових ансамблю, але й від особливостей руху кожного осцилятора. Зокрема, при хаотичному русі – від моментів їх хаотичної динаміки. Особливе значення звернено на виникнення НЧ-динаміки ансамблю. У цьому випадку з'являється можливість ефективного обміну енергією між ВЧ- і НЧ-коливаннями. Исследованы особенности динамики ансамблей связанных линейных и нелинейных осцилляторов. Показано, что спектральные характеристики этой динамики существенно зависят не только от количества осцилляторов, составляющих ансамбль, но и от особенностей движения каждого осциллятора. В частности, при хаотическом движении – от моментов их хаотической динамики. Особое значение обращено на возникновение НЧ-динамики ансамбля. В этом случае появляется возможность эффективного обмена энергией между ВЧ- и НЧ-колебаниями. 2013 Article The influence of the dynamics features of the trapped particles on a spectrum of their oscillations / V.A. Buts, D.M. Vavriv, D.V. Tarasov // Вопросы атомной науки и техники. — 2013. — № 4. — С. 289-292. — Бібліогр.: 5 назв. — англ. 1562-6016 PACS: 05.45.Xt; 05.45.Ac http://dspace.nbuv.gov.ua/handle/123456789/112156 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Нелинейные процессы в плазменных средах Нелинейные процессы в плазменных средах |
| spellingShingle |
Нелинейные процессы в плазменных средах Нелинейные процессы в плазменных средах Buts, V.A. Vavriv, D.M. Tarasov, D.V. The influence of the dynamics features of the trapped particles on a spectrum of their oscillations Вопросы атомной науки и техники |
| description |
Some features of dynamics of ensembles of coupled linear and nonlinear oscillators are investigated. It is shown that spectral characteristics of this dynamics essentially depend not only on quantity of the oscillators making the ensemble, but also on features of motion of each oscillator. In particular, at chaotic motion – on the moments of their chaotic dynamics. Special attention is turned on appearance of LF dynamics of ensemble. In this case there is a possibility of an effective exchange of energy between HF and LF oscillations. |
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Article |
| author |
Buts, V.A. Vavriv, D.M. Tarasov, D.V. |
| author_facet |
Buts, V.A. Vavriv, D.M. Tarasov, D.V. |
| author_sort |
Buts, V.A. |
| title |
The influence of the dynamics features of the trapped particles on a spectrum of their oscillations |
| title_short |
The influence of the dynamics features of the trapped particles on a spectrum of their oscillations |
| title_full |
The influence of the dynamics features of the trapped particles on a spectrum of their oscillations |
| title_fullStr |
The influence of the dynamics features of the trapped particles on a spectrum of their oscillations |
| title_full_unstemmed |
The influence of the dynamics features of the trapped particles on a spectrum of their oscillations |
| title_sort |
influence of the dynamics features of the trapped particles on a spectrum of their oscillations |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2013 |
| topic_facet |
Нелинейные процессы в плазменных средах |
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http://dspace.nbuv.gov.ua/handle/123456789/112156 |
| citation_txt |
The influence of the dynamics features of the trapped particles on a spectrum of their oscillations / V.A. Buts, D.M. Vavriv, D.V. Tarasov // Вопросы атомной науки и техники. — 2013. — № 4. — С. 289-292. — Бібліогр.: 5 назв. — англ. |
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Вопросы атомной науки и техники |
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ISSN 1562-6016. ВАНТ. 2013. №4(86) 289
THE INFLUENCE OF THE DYNAMICS FEATURES OF THE TRAPPED
PARTICLES ON A SPECTRUM OF THEIR OSCILLATIONS
V.A. Buts1, D.M. Vavriv2, D.V. Tarasov1
1National Science Center «Kharkov Institute of Physics and Technology», Kharkov, Ukraine;
2Department of Microwave Electronics, Radio Astronomy Institute NASU, Kharkov, Ukraine
E-mail: vbuts@kipt.kharkov.ua
Some features of dynamics of ensembles of coupled linear and nonlinear oscillators are investigated. It is shown
that spectral characteristics of this dynamics essentially depend not only on quantity of the oscillators making the
ensemble, but also on features of motion of each oscillator. In particular, at chaotic motion – on the moments of
their chaotic dynamics. Special attention is turned on appearance of LF dynamics of ensemble. In this case there is a
possibility of an effective exchange of energy between HF and LF oscillations.
PACS: 05.45.Xt; 05.45.Ac
INTRODUCTION
It is known that spectral characteristics of ensemble
of oscillators can essentially differ from spectral charac-
teristics of separate oscillator. In particular, the fact of
appearance of normal frequencies and normal modes at
system of the coupled linear oscillator is widely known.
Moreover, the problem of finding of normal modes of the
various complex distributed systems is one of the most
important and complex problems which exists in the the-
ory of oscillations – especially in radiophysics and in
hydrodynamics. Importance of finding of normal modes
and normal frequencies is caused, first of all, with possi-
bility to describe linear dynamics of complex distributed
systems in space and in time of systems with their help.
Moreover, they also allow to describe nonlinear dynamics
of such systems in many cases. Besides that, exactly
normal frequencies are those frequencies which the con-
sidered system answers resonant responses.
The special role in a spectrum of ensemble of oscilla-
tor is played by low-frequency components. Such appear-
ance can lead to linear and nonlinear interaction between
low-frequency (LF) and high-frequency (HF) modes.
Such interaction allows to organize an effective exchange
between these oscillations. In particular, such interaction
allows to transform energy of LF oscillations to energy of
HF-oscillations. Besides, such coupling leads to appear-
ance of regimes with chaotic dynamics [1 - 4]. It can lead
to saturation of level of oscillations exited by the elec-
tronic beam [5]. In particular, to saturation of level of
exited oscillations at plasma-beam instability.
The spectral characteristics of ensemble of the
charged particles can be useful to diagnostics of function
of distribution of the charged particles. Such information
is useful, for example, for analysis the bunches of the
charged particles on an accelerators output, and also on
an output of generators, for example, after transmission
of the charged particles through undulator in LSE.
In the paper the results of investigation of dynamics
of system of the coupled linear and nonlinear oscillator
are presented. The dynamics of ensemble of nonlinear
oscillator is stochastic. Stochasticity is caused by exis-
tence of coupling between oscillators, or by influence of
external forces.
In the second section it is shown that the system of a
large number of weakly-coupled linear oscillators can
have normal frequencies which are hundreds times less,
than partial frequencies of separate oscillator. If to im-
pact on such system of oscillator with the external low-
frequency signal which frequency coincides with this
normal frequency, energy of this low-frequency signal
will be transformed to energy of oscillator. If to break
coupling between oscillator after obtaining sufficient
energy from an external source, they will oscillate on
the partial frequencies. Amplitudes of these partial fre-
quencies will be almost considerably large, than their
initial amplitudes. Such scenario allows to transform the
energy of LF oscillations in to the energy of HF-
oscillations.
One of important results is the proof (section 3) that
with increasing of number of nonlinear oscillators the
dynamics of their centre shift becomes more and more
regular, and the spectrum is displaced in low-frequency
area. And the shift size is defined by the moments of
distribution function.
In conclusion the obtained results are discussed.
1. REGULAR DYNAMICS OF THE ENSEMBLE
OF THE COUPLED LINEAR OSCILLATORS
Essential changing of dynamics of ensemble of os-
cillator in comparing with dynamics of a separate oscil-
lator can be seen even for ensemble of coupled linear
oscillators. Really, it is known that the system of the
coupled linear oscillator has a set of normal frequencies
which differ from partial frequencies of separate oscilla-
tor. The important feature of normal frequencies is that
fact that the minimum of them is much less then all par-
tial frequencies, and the maximum – is much more then
all partial frequencies. And, the minimum normal fre-
quency can be very small. In this case, influencing such
ensemble of the coupled oscillator with the external
low-frequency signal which frequency is close to the
minimum normal frequency, it is possible to strengthen
oscillations of such ensemble resonantly. However it is
necessary to keep in mind that all coupled oscillator at
such excitement will oscillate too on frequency consid-
erably smaller then their partial frequencies. However,
if at some point of time we will break the coupling, the
energy of oscillator received from external excitation,
will pass to the energy of oscillations of separate oscil-
lator on their partial frequencies. These frequencies
(partial) can be much higher than frequency of external
excitation. Thus, in such scheme of excitation the
strengthening of oscillations of high-frequency oscilla-
tor due to the energy of an external low-frequency sig-
ISSN 1562-6016. ВАНТ. 2013. №4(86) 290
nal is possible. Let's illustrate such possibility on an
example of a large number of the coupled, identical,
linear oscillator.
We have a system with a Hamiltonian:
2 2
2
0 0
0 12 2
N N
i i
j
i j
p q
H q qω μ
= =
⎛ ⎞
= + + ⋅ ⋅⎜ ⎟
⎝ ⎠
∑ ∑ , (1)
i
i
Hp
q
∂
= −
∂
& ,
i
i
Hq
p
∂
=
∂
& , constμ = ,
0H
t
∂
=
∂
, H const= . (2)
This system represents the N coupled linear oscilla-
tors. As the coefficient of coupling and natural frequen-
cies (partial) do not depend on time, the Hamiltonian is
continuous function (the energy of system should be
constant). From (1) it is easy to obtain system of the
equations for oscillator:
2
0 0
2
0 0 0
1
2
0 0
2
0 0 0
1
, ,
,
,
.
i i i i
N
i
i
i i
N
i
i
q p p q q
p q q
p p p
p p p
ω μ
ω μ
ω μ
ω μ
=
=
= = − −
= − −
+ = − ⋅
+ = − ⋅
∑
∑
& &
&
&&
&&
(3)
The system (3) describes the N coupled linear os-
cillator. And for simplicity we consider system in which
all oscillators are coupled with each other only across
the zero oscillator. It is easy to find the normal frequen-
cies of such system. For this purpose we will solve sys-
tem (3) in such form:
( )exp ,i i ip a i t a constω= ⋅ ⋅ = .
Substituting this solution in (3), it is easy to receive
the dispersive equation:
( )22 2 2
0 .Nω ω μ− + = (4)
The equation (4) is solved elementary:
0 2
0
1 Nω ω μ
ω
= ± ± ⋅ . (5)
Signs + and – in equation (5) before the root and un-
der the root are independent. It is seen, what even at
very small coupling coefficient, but at a large number of
the oscillator, one of normal modes can be very small
(for the case with minus sign under a root).
Fig. 1. The dynamics of oscillators with external
periodic force
If now the system of oscillator (3) is exited by exter-
nal periodic force with frequency which is equal to this
minimum normal frequency of ensemble, then oscilla-
tions of ensemble will increase under the linear law in
time. As an example we have taken 100 oscillator. As a
result, the minimum normal frequency was 100 times
less, than partial frequencies. In the Figs. 1, 2 given be-
low the increasing of amplitudes of oscillations of sepa-
rate oscillator is seen. Oscillations only two of them are
presented. They differ only with initial conditions.
If now at some point of time (it is defined by exis-
tence of damping, and, respectively, saturation of
strengthened oscillations) we break coupling, frequency
of oscillator becomes partial frequency. Amplitudes of
oscillator, of course, considerably will fall, but they will
oscillate on much higher frequency, than frequency of
external excitation. The amplitude of these partial fre-
quencies will essentially exceed initial amplitudes. The
illustration of this fact is on figure given below.
Fig. 2. The dynamics of oscillators after breaking coupling
The important feature of dynamics of such ensemble
of oscillators is that fact that after braking the coupling
all of them oscillate in one phase. And it is possible to
organize coherent radiation.
2. COMPLEX, NONREGULAR DYNAMICS
OF THE SYSTEM OF COUPLED
NONLINEAR OSCILLATORS
Above we have considered rather simple system –
the system consisting of linear coupled oscillators. Al-
ready this simple example has shown the essential dif-
ference in dynamics of separate oscillator from dynam-
ics of the whole ensemble of such oscillators. The real
dynamics of the charged particles which are trapped, for
example, by the field of electromagnetic wave, will be
nonlinear. Moreover, as a result of action on this dy-
namics with the external regular or random forces or as
a result of interaction between these oscillators, will be
chaotic in most cases. There is a question. What features
of dynamics of such ensemble can be observed in this
case? Below we will pay attention to the case of chaotic
dynamics of separate nonlinear oscillator. Let's show
that such dynamics of ensemble of noninteracting oscil-
lator with chaotic behavior has spectral characteristics
which can essentially differ from spectral characteristics
of separate oscillator.
Let's consider the system representing ensemble of
coupled N nonlinear oscillator with external regular
periodic force. The Hamiltonian of such system can be
presented in following form:
( ) ( ) ( )
2
1
,
2
N
i
i i j i
j i
x
H x G x x xε τ
=
⎡ ⎤
= +Φ + − ⋅⎢ ⎥
⎣ ⎦
∑∑
& . (6)
For Hamiltonian (6) corresponds the following sys-
tem of the equations of second order:
ISSN 1562-6016. ВАНТ. 2013. №4(86) 291
( ) ( ) ( )0 1 ,i i i jx F x F x x ε τ= + +&& . (7)
If the last two members of the right part of system
(7) are absent, this system describes dynamics of en-
semble of nonlinear oscillator independent from each
other. Thus the behavior of nonlinearity is defined by
function ( )0 /i iF x x= −∂Φ ∂ . The second member
1 /
N
i
i
F G x= − ∂ ∂∑ describes interaction between nonlinear
oscillators. The third one – describes external periodic
force. Let, for the definiteness, each of considered
nonlinear oscillators represents the charged particle
which moves in some nonlinear potential. Below we
will consider that as a result of interaction between os-
cillators or as a result of influence on them with external
regular force the dynamics is chaotic. In this case it is
possible to present the shift of each of these oscillator in
following form:
i ix x δ= + , (8)
where
1
lim ( ) /iN i
x x N
∞
→∞
=
= ∑ − average coordinate of the
shift of nonlinear oscillator; iδ − random deviation. And
0iδ = . In this case average sizes of ensemble for
functions 0F and 1F are convenient for moments series
expansion:
( ) ( )0 0 0 0
1
|
! i
n
n
i
n i
M dF x F x F
n dx δδ
∞
=
=
⎛ ⎞
+ = + ⋅⎜ ⎟
⎝ ⎠
∑ , (9)
where ( )n
n δΜ = − the moments.
For example we will consider ensemble of oscilla-
tor, each of which represents the mathematical pendu-
lum. In this case ( ) ( )0 sini iF x x= − . The average size
from this function is:
( ) ( ) 2
0
1
sin 1 sin
(2 !)
m
i i
m
F x x x
m
∞
=
Μ⎡ ⎤
= − = − −⎢ ⎥
⎣ ⎦
∑ . (10)
For equation (10), it is possible to receive at once the
one of the most important results. This result is that
ensemble characteristics even noninteracting mathe-
matical pendulums can essentially differ from character-
istics of separate oscillator. In order to illustrate this fact
from system (7) we will find the equation which de-
scribes dynamics of average deviation. The external
force and the coupling between oscillator can be ne-
glected for the simplicity.
2
1
1 sin 0
(2 !)
m
m
x x
m
∞
=
Μ⎡ ⎤
+ − =⎢ ⎥
⎣ ⎦
∑&& . (11)
The equation (11) describes the dynamics of a
mathematical pendulum. However the potential of this
mathematical pendulum, and respectively, and oscilla-
tory characteristics of this pendulum essentially depend
on statistical characteristics of the separate oscillators
making considered ensemble. Thus, oscillatory proper-
ties of ensemble even of independent nonlinear oscilla-
tor appear essentially dependent on chaotic dynamics of
separate oscillator. It is necessary to keep in mind that
chaotic dynamics of separate oscillator appears only as
result or interactions between these oscillators, or, as a
result of external influence on these oscillators. In equa-
tion (11) this fact in not represented explicitly. Let's pay
attention that the second addend in square brackets in a
equation (11) has the minus sign. It means that the ran-
dom nature of the dynamics of separate oscillator al-
ways leads to reduction of effective potential where the
ensemble moves. In particular, frequency of small linear
oscillations of such ensemble decreases also.
Above we have considered the case when nonlinear
oscillators are not coupled and external force does not
impact on them. However we assumed that dynamics of
each oscillator is random. Random dynamics can be
caused either interaction between oscillator or influence
of external force. The analysis of the general case (when
there is a interaction and external forces) is possible only
with numerical methods. Such investigation has been
carried out. Some results of such investigation for ensem-
ble of mathematical pendulums are presented below.
For numerical investigations the interaction between
oscillators which corresponds to one-dimensional cou-
lomb interaction between oscillator has been chosen:
( ) ( )
( )1 2
1
,
N
i j
i j
j i j
sign x x
F x x
x x a
μ
=
−
= −
− −
∑ . (12)
In a equation (12) size a characterizes the minimum
distance between oscillator. External force we choose
as: ( ) cosAε τ ωτ= ⋅ . The set of the equations (7) with
such external force and with coupling (12) describes
ensemble of N the charged particles which moves in
external periodic potential and which affects by external
periodic force.
Fig. 3. Dependence of the shift of separate oscillator on time
Fig. 4. Dependence of the mean position
of oscillators on time
Fig. 5. Characteristic spectrum of the separate oscillator
Fig. 6. Spectrum of the mean position of oscillators
The main results of numerical calculations are in a
good qualitative agreement with the situation described
ISSN 1562-6016. ВАНТ. 2013. №4(86) 292
above, i.e. the existence of interaction and external force
leads to chaotic dynamics of each separate oscillator,
and the dynamics of the whole ensemble essentially
depends on statistical characteristics of the dynamics of
separate oscillator. Below in Figs. 3-6 some of character-
istic results are presented.
So, in Figs. 3, 4 the dependence of shift of separate os-
cillator (see Fig. 3) and the dependence of shift of mean
position of ensemble of oscillator (see Fig. 4) on time is
presented. First of all, it is seen, that the dynamics of the
separate oscillator is chaotic, and the dynamics of ensem-
ble is much more regular. It is seen also that characteristic
frequencies of oscillations of ensemble are more lower,
than characteristic frequencies of separate oscillator. This
fact is proved in Figs. 5 and 6. In Fig. 5 − the spectrum of
oscillation of a separate oscillator, and in Fig. 6 – the spec-
trum of oscillations of ensemble is presented.
Besides the presented Figs. 3-6 correlation functions
and Lyapunov's main exponents have been obtained.
The correlation functions of oscillations of a separate
oscillator quickly fall down, and correlation functions of
oscillations of all ensembles oscillate without damping.
Lyapunov's main exponents for separate oscillator prac-
tically in all phase plane are positive, and similar values
for ensemble practically do not differ from zero.
CONCLUSIONS
Thus, the ensemble of linear identical coupled oscilla-
tors can has in the spectrum of normal frequencies, fre-
quency which can be many times smaller (hundreds
times) then partial frequencies of separate oscillator. This
feature is necessary to take into account in the analysis of
dynamics of such ensemble. Besides, this feature can be
used for transformation of energy of low-frequency oscil-
lations to energy of high-frequency oscillations. The
scheme of such transformation can be such, for example.
Let we have a large number of the coupled high-
frequency vibrators. As it is seen above, the ensemble of
such vibrators can have a low-frequency normal mode. If
to influence now on such ensemble with external signal
which frequency is equal to this low-frequency normal
mode, such ensemble will gain energy from an external
low-frequency source. If, as it is shown in the second
section, we will break coupling between vibrators, they
will oscillate on the high-frequency partial frequencies.
Amplitude of such oscillations will be essential more than
their initial amplitudes. It is important that phases of os-
cillations of all these separate vibrators will be identical
similar irrespectively of initial phases of these vibrators.
Such synchronisation of phases allows to organize coher-
ent radiation of this ensemble of vibrators on their high-
frequency partial frequencies.
The important was found that dynamics of ensemble
of nonlinear oscillator can essentially depend on statis-
tical characteristics of dynamics of separate oscillator.
This feature is shown even in that case when oscillator
are independent and do not interact with each other. The
influence of the statistical moments of separate oscilla-
tor on size of effective potential in which makes oscilla-
tions considered ensemble is also important. This fea-
ture of dynamics of nonlinear oscillator can be function
of distribution of the charged particles grasped in vari-
ous potentials useful for definition.
It is necessary to note that these results could be ex-
pected a priori. Really, if we have large number of oscil-
lators, and their dynamics is described by stationary
casual process, then after averaging, the first summand
in (11) will become zero (as derivative of a constant).
And with inevitability will become zero a square
bracket in second summand in this equation. In this ex-
treme case the frequency of oscillation of averaged dis-
placement becomes zero too.
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and high-frequency oscillations in a nonlinear RLC-
circuit // IEEE Trans. Circuits & Syst. 1994, v. 41,
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№ 12, p. 1255-1259.
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O.F. Kovpik, E.A. Kornilov, V.G. Svichensky,
D.V. Tarasov. Regular and stochastic decays of
waves in a plasma cavity // Plasma Physics Reports.
2012, v. 38, № 8, p. 636-650.
Article received 13.05.2013.
ВЛИЯНИЕ ОСОБЕННОСТЕЙ ДИНАМИКИ ЗАХВАЧЕННЫХ ЧАСТИЦ НА СПЕКТР ИХ КОЛЕБАНИЙ
В.А. Буц, Д.М. Ваврив, Д.В. Тарасов
Исследованы особенности динамики ансамблей связанных линейных и нелинейных осцилляторов. Показано, что
спектральные характеристики этой динамики существенно зависят не только от количества осцилляторов, составляю-
щих ансамбль, но и от особенностей движения каждого осциллятора. В частности, при хаотическом движении – от мо-
ментов их хаотической динамики. Особое значение обращено на возникновение НЧ-динамики ансамбля. В этом случае
появляется возможность эффективного обмена энергией между ВЧ- и НЧ-колебаниями.
ВПЛИВ ОСОБЛИВОСТЕЙ ДИНАМІКИ ЗАХОПЛЕНИХ ЧАСТИНОК НА СПЕКТР ЇХ КОЛИВАНЬ
В.О. Буц, Д.М. Ваврів, Д.В. Тарасов
Досліджені особливості динаміки ансамблів зв'язаних лінійних і нелінійних осциляторів. Показано, що спектральні
характеристики цієї динаміки суттєво залежать не тільки від кількості осциляторів, складових ансамблю, але й від особ-
ливостей руху кожного осцилятора. Зокрема, при хаотичному русі – від моментів їх хаотичної динаміки. Особливе зна-
чення звернено на виникнення НЧ-динаміки ансамблю. У цьому випадку з'являється можливість ефективного обміну
енергією між ВЧ- і НЧ-коливаннями.
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