Nonrelativistic dynamics of the charged particles at cyclotron resonances
It is shown that transition to the chaotic dynamics of the charged particles at cyclotron resonances can take place on an unusual scenario - not through overlapping of nonlinear resonances, but through qualitative periodic tuning of topology of phase space. It is shown that such scenario can be real...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2008
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| Cite this: | Nonrelativistic dynamics of the charged particles at cyclotron resonances / V.A. Buts, A.P. Tolstoluzhsky // Вопросы атомной науки и техники. — 2008. — № 6. — С. 117-119. — Бібліогр.: 3 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859993507035348992 |
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| author | Buts, V.A. Tolstoluzhsky, A.P. |
| author_facet | Buts, V.A. Tolstoluzhsky, A.P. |
| citation_txt | Nonrelativistic dynamics of the charged particles at cyclotron resonances / V.A. Buts, A.P. Tolstoluzhsky // Вопросы атомной науки и техники. — 2008. — № 6. — С. 117-119. — Бібліогр.: 3 назв. — англ. |
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| description | It is shown that transition to the chaotic dynamics of the charged particles at cyclotron resonances can take place on an unusual scenario - not through overlapping of nonlinear resonances, but through qualitative periodic tuning of topology of phase space. It is shown that such scenario can be realized at excitation of low-frequency oscillations by dense flows of the charged particles in the strong magnetic field.
Показано, що перехід до хаотичної динаміки заряджених частинок при циклотронних резонансах може відбуватися за незвичайним сценарієм - не через перекриття нелінійних резонансів, а через якісну періодичну перебудову топології фазового простору. Показано, що такий сценарій може реалізуватися при збудженні низькочастотних коливань щільними потоками заряджених частинок в сильному магнітному полі.
Показано, что переход к хаотической динамике заряженных частиц при циклотронных резонансах может происходить по необычному сценарию - не через перекрытие нелинейных резонансов, а через качественную периодическую перестройку топологии фазового пространства. Показано, что такой сценарий может реализоваться при возбуждении низкочастотных колебаний плотными потоками заряженных частиц в сильном магнитном поле.
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| first_indexed | 2025-12-07T16:33:15Z |
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NONRELATIVISTIC DYNAMICS OF THE CHARGED PARTICLES
AT CYCLOTRON RESONANCES
V.A. Buts, A.P. Tolstoluzhsky
National Science Center “Kharkov Institute of Physics and Technology”,
Kharkov, Ukraine, e-mail: tolstoluzhsky@kipt.kharkov.ua
It is shown that transition to the chaotic dynamics of the charged particles at cyclotron resonances can take place on an
unusual scenario - not through overlapping of nonlinear resonances, but through qualitative periodic tuning of topology
of phase space. It is shown that such scenario can be realized at excitation of low-frequency oscillations by dense flows
of the charged particles in the strong magnetic field.
PACS: 52.20.-j; 05.45.-a
1. INTRODUCTION
Now conditions of transition from regular dynamics
to chaotic are received for all known resonances of
interaction a wave-particle [1-3]. Thus a source of chaotic
dynamics is crossing homoclinic or heteroclinic
trajectories. In physical language this fact is more
convenient for expressing as overlapping of nonlinear
resonances. The received results are correct at interaction
of a particle with a separate electromagnetic wave with
constant amplitude. In real physical problems of waves
amplitude vary, except for that waves can be several.
Generally such complication of physical system does not
lead to any qualitative changes of processes chaotic.
Simply there are additional resonances appear or width of
existing resonances varies. Therefore the studied scenario
of transition to chaos is rather universal. Besides its
universality is caused by that fact, that interaction of the
charged particles with electromagnetic waves can be
considered as perturbation in Hamilton formalism. Thus,
as is known, the truncated equations in a vicinity of
resonances are described by the equation of a
mathematical pendulum. Therefore it seemed natural what
exactly the scenario of overlapping of nonlinear
resonances should describe practically all variety of
conditions of interaction wave-particle. However, as we
shall see below, at cyclotron resonances, for the
description of dynamics nonrelativistic particles, the
truncated equations are described not by the equation of a
mathematical pendulum, and system of the equations
topological similar to Duffing oscillator. Duffing
oscillator unlike the mathematical pendulum has two free
parameters. This fact leads to that at change of amplitude
of the wave in which the particle moves, its phase portrait
(the phase portrait Duffing oscillator) can qualitatively
vary. Presence of such qualitative change of the phase
portrait, as we shall see below, is the reason of occurrence
of chaotic dynamics. Change of amplitude of the wave in
which particles move, can be caused by external factors.
Besides as is known, at beams instabilities in a nonlinear
mode of amplitude of excited waves also periodically
vary. Depth of amplitude modulation essentially depends
on density of the electron beam. Below it is shown, that at
enough big density of the beam this modulation appears
sufficient for realization of the chaotic regime in the
isolated cyclotron resonance. The mechanism responsible
for a randomness in this case is qualitative change of
topology of phase space.
2. STATEMENT OF THE PROBLEM.
THE BASIC EQUATIONS
We shall consider a problem about excitation of an
electromagnetic field by a monoenergetic beam of
oscillators with function of distribution:
0 0( ) (
2
bN
|| )f p p p
p
δ δ
π ⊥ ⊥
⊥
= − (1)
where - perpendicular and parallel axes z
components of pulse , - equilibrium beam density.
||,p p⊥
bN
The beam moves in the constant magnetic field directed
along z axis. We shall consider excitation of wave which
propagates perpendicularly to the magnetic field. Considered
statement of the problem does not differ from that which has
been formulated in works [1-3]. Moreover, we shall take
advantage of many results received in these works. The self-
consistent system of the equations which describes dynamics
of excited fields and dynamics of the charged particles
consists of Maxwell equations and the equations of motion
of separate particles.
The full system of the equations is written out in
[2,3]. Below we shall write out the truncated system of
the equations describing dynamics of particles and fields
in the isolated cyclotron resonance with number s :
( ) si
s
dp
iJ e
d
θμ ε
τ
⊥ ′= ,
2
2
11 1 ( ) sis H
s
H
d s s J e
d
θθ ω
μ ε
τ γ ω μ
⎛ ⎞
= − + −⎜ ⎟
⎝ ⎠
, (2)
22
0
0
( )
2
sib
s s
pd i d J e
d
π
θωε θ μ
τ π γ
−⊥ ′= ∫
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2008. № 6. 117
Series: Plasma Physics (14), p. 117-119.
where - /p p mc⊥ ⊥= , 2 2/ , 1H Hpμ ω γ μ ω⊥= = + ,
/H oeH mcω ω= , 2 24 e /b bn mω π= e /eE mcω= . ε
The system of the equations (2) differs from what have
been analyzed in works [2,3], presence of last equation
term in the right part for phase. For relativistic particles
this term can be neglected. In this case first two equations
of system (2) at constant intensity of the wave's field
( constε = ) represent the equation of a mathematical
pendulum. These equations have been used in works [2,3]
for the finding of conditions of local instability
occurrence. For not relativistic particles this term can be
essential and as we shall see below, it leads to the
scenario of stochasticity occurrence distinct from the
scenario of overlapping of nonlinear resonances.
mailto:vbuts@kipt.kharkov.ua
3. DYNAMICS OF NONRELATIVISTIC
PARTICLE MOTION IN THE FIELD
WITH CONSTANT AMPLITUDE
If the amplitude of the field does not vary, the third
equation in system (2) can be not considered. Dynamics
of particles is described by first two equations. Such
system has Hamiltonian
( )( , ) 2 ( 2 ) cos( )s s
H H
s dH I I I J I
dI
ε
sθ γ θ
ω ω
= − + , (3)
where . 2 / 2I μ=
The phase portrait of system with Hamiltonian (3)
topologically is similar to the phase portrait Duffing
oscillator. Really, on a phase plane ( , sp θ⊥ ), in general
case, there are three critical points: ( 0sθ = , 1p⊥ = Π ),
( 0sθ = , ), (2 1 /p⊥ = −Π
118
2 sθ π= , ).
Here , - initial impulse of particle.
Where, two of these critical points (the second and the
third) represent points of type "center", and one (first) –
saddle point. Such kind of phase space is realized at small
amplitude of an external wave ( ). If the amplitude
is large enough ( ), that two critical points, namely
saddle point and the point of type "center" (the first and
second critical points) merge and disappear. There is only
one special point - a point of type "center". All these
features of phase space are similar to features of phase
space Duffing oscillator.
3 1 /p⊥ = +Π 2
3
,0/ pε ⊥Π = ,0p⊥
1Π <<
1Π >>
However it is necessary to pay attention to that fact,
that oscillations of Duffing oscillator are potential, and for
the equations considered by us it is not possible to find
potential. The important feature of phase space topology
of considered system is that fact, that the closed
trajectories in a vicinity of a critical point of type "center"
can identify the trapped particles. Unclosed trajectories
which surround the closed trajectories, it is possible to
identify passing particles. We have got used, that the
trapped and passing particles are divided by separatrix,
i.e. homoclinic or heteroclinic trajectories. In this case
such trajectories are absent. It is necessary to tell, that
absence of such trajectories leads to different dynamics of
particles which pass through the region dividing region of
trapped and passing particles.
4. NUMERICAL ANALYSIS OF DYNAMICS
OF PARTICLES AND FIELDS
By numerical methods, first of all, had been investigated
dynamics of particles in the field of external
electromagnetic wave with constant and with periodically
varying amplitude. Besides self-consistent dynamics of
particles and fields excited by these particles has been
investigated. As a whole the received results will well be
agreed with available representations about studied
processes. So in Fig. 1 the phase portraits are presented
for case (Fig.1, a) and for (Fig.2, b).
Straight lines correspond to initial coordinates of testing
particles. Apparently from Fig.1(a) on the phase plane, in
full conformity with the results obtained above, is
available three critical points: two type of the center and
saddle point.
1Π << 1Π >
a b
Fig.1. Phase trajectories at a) ε =0.08; b) ε =0.12
At increasing of the wave amplitude two points ("saddle"
and "center" at 0sθ = ) come together and disappear
(Fig.1,b). If the amplitude of the external field
periodically varies in the vicinity of the merge point the
dynamics of particles turns out chaotic: the spectra of
particles motion is wide, correlation functions quickly
enough fall down (Fig. 2).
Fig.2. Spectrum of the impulse near saddle point (left);
correlation function near saddle point (right)
If the amplitude of field varies in region where there is no
qualitative change of the phase portrait the dynamics of
particles undergoes small changes (see Fig. 3).
Fig.3. Spectrum of the impulse far from saddle point (left);
correlation function far from saddle point (right)
The received results confirm the above formulated
assumption about chaotization mechanism which is
caused by periodic qualitative change of the phase
portrait.
By numerical methods had been investigated also
self-coordinated dynamics of particles and fields, which
are excited by these particles (system of the equations
(2)). Dynamics of studied processes thus completely
corresponds to the qualitative picture described above.
Really, if density of the beam particles is not too large, so
the amplitude of excited field satisfies to an inequality
1Π << , the dynamics of particles and fields is similar to
dynamics of particles and fields at beam-plasma
interaction (Fig.4, a).
Fig.4. Amplitude of field versus time at: a) small beam
density: 2
bω =0.02; b) at large beam density: 2
bω =0.1
119
45.
If the density of particles becomes such that the
amplitude of the wave excited by beam and depth of its
modulation become such, that occur qualitative change of
the phase portrait described above then irregular
dynamics as fields, and particles is appeared much earlier,
than for the case of relatively small amplitudes (Fig.4, b).
5. CONCLUSIONS
Thus, transition from regular dynamics to chaotic one
at interactions of the wave-particle type can be realized
not only as result of overlapping nonlinear resonances,
but also as the result of phase portrait topology changing.
Above we have considered such mechanism of the local
instability occurrence for cases of nonrelativistic charged
particles interaction with electromagnetic wave in
conditions of the isolated cyclotron resonance. Clearly,
that similar mechanisms can be realized in the large
number of other physical systems. It will be those systems
which motions can be described by motion of Duffing
oscillator. These are numerous systems. Really, as is
known, to the analysis of dynamics of the mathematical
pendulum it is possible to reduce studying of the
Hamiltonian systems under periodic disturbance.
Hamiltonian of the mathematical pendulum in this
connection name the first fundamental Hamiltonian. To
represent the fact, that to the analysis of the Duffing
oscillator dynamics is reduced the investigation of many
physical systems, Hamiltonian of the Duffing oscillator
name the second universal Hamiltonian.
REFERENCES
1. V.A..Buts, A.N.Lebedev, V.I.Kurilko. The Theory of
Coherent Radiation by Intense Electron Beams. Berlin:
“Springer”, New York: “Heidelberg”, 2006, p. 263.
2. V.A. Balakirev, V.A. Buts, A.P. Tolstoluzhsky,
Yu.A. Turkin. Chaotization of motion of phased
oscillator beam // JETP. 1983, v.84, N 4, p. 741-7
3. V.A. Balakirev, V.A.. Buts, A.P. Tolstoluzhsky,
Yu.A. Turkin. Dynamic of charged particles motion in
the field of two electromagnetic waves // JETP. 1989,
v. 95, N 4, p. 710-717.
Article received 22.09.08.
НЕРЕЛЯТИВИСТСКАЯ ДИНАМИКА ЗАРЯЖЕННЫХ ЧАСТИЦ
ПРИ ЦИКЛОТРОННЫХ РЕЗОНАНСАХ
В.А. Буц, А.П. Толстолужский
Показано, что переход к хаотической динамике заряженных частиц при циклотронных резонансах может
происходить по необычному сценарию - не через перекрытие нелинейных резонансов, а через качественную
периодическую перестройку топологии фазового пространства. Показано, что такой сценарий может
реализоваться при возбуждении низкочастотных колебаний плотными потоками заряженных частиц в сильном
магнитном поле.
НЕРЕЛЯТИВІСТСЬКА ДИНАМІКА ЗАРЯДЖЕНИХ ЧАСТИНОК
ПРИ ЦИКЛОТРОННИХ РЕЗОНАНСАХ
В.О. Буц, О.П. Толстолужський
Показано, що перехід до хаотичної динаміки заряджених частинок при циклотронних резонансах може
відбуватися за незвичайним сценарієм - не через перекриття нелінійних резонансів, а через якісну періодичну
перебудову топології фазового простору. Показано, що такий сценарій може реалізуватися при збудженні
низькочастотних коливань щільними потоками заряджених частинок в сильному магнітному полі.
|
| id | nasplib_isofts_kiev_ua-123456789-110773 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T16:33:15Z |
| publishDate | 2008 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Buts, V.A. Tolstoluzhsky, A.P. 2017-01-06T11:49:00Z 2017-01-06T11:49:00Z 2008 Nonrelativistic dynamics of the charged particles at cyclotron resonances / V.A. Buts, A.P. Tolstoluzhsky // Вопросы атомной науки и техники. — 2008. — № 6. — С. 117-119. — Бібліогр.: 3 назв. — англ. 1562-6016 PACS: 52.20.-j; 05.45.-a https://nasplib.isofts.kiev.ua/handle/123456789/110773 It is shown that transition to the chaotic dynamics of the charged particles at cyclotron resonances can take place on an unusual scenario - not through overlapping of nonlinear resonances, but through qualitative periodic tuning of topology of phase space. It is shown that such scenario can be realized at excitation of low-frequency oscillations by dense flows of the charged particles in the strong magnetic field. Показано, що перехід до хаотичної динаміки заряджених частинок при циклотронних резонансах може відбуватися за незвичайним сценарієм - не через перекриття нелінійних резонансів, а через якісну періодичну перебудову топології фазового простору. Показано, що такий сценарій може реалізуватися при збудженні низькочастотних коливань щільними потоками заряджених частинок в сильному магнітному полі. Показано, что переход к хаотической динамике заряженных частиц при циклотронных резонансах может происходить по необычному сценарию - не через перекрытие нелинейных резонансов, а через качественную периодическую перестройку топологии фазового пространства. Показано, что такой сценарий может реализоваться при возбуждении низкочастотных колебаний плотными потоками заряженных частиц в сильном магнитном поле. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Plasma electronics Nonrelativistic dynamics of the charged particles at cyclotron resonances Нерелятивістська динаміка заряджених частинок при циклотронних резонансах Нерелятивистская динамика заряженных частиц при циклотронных резонансах Article published earlier |
| spellingShingle | Nonrelativistic dynamics of the charged particles at cyclotron resonances Buts, V.A. Tolstoluzhsky, A.P. Plasma electronics |
| title | Nonrelativistic dynamics of the charged particles at cyclotron resonances |
| title_alt | Нерелятивістська динаміка заряджених частинок при циклотронних резонансах Нерелятивистская динамика заряженных частиц при циклотронных резонансах |
| title_full | Nonrelativistic dynamics of the charged particles at cyclotron resonances |
| title_fullStr | Nonrelativistic dynamics of the charged particles at cyclotron resonances |
| title_full_unstemmed | Nonrelativistic dynamics of the charged particles at cyclotron resonances |
| title_short | Nonrelativistic dynamics of the charged particles at cyclotron resonances |
| title_sort | nonrelativistic dynamics of the charged particles at cyclotron resonances |
| topic | Plasma electronics |
| topic_facet | Plasma electronics |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/110773 |
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