Chaotic dynamics at cyclotron resonances
Mechanisms and criteria for the transition to chaotic dynamics of particles and fields under conditions of electron cyclotron resonances (ECR) are considered. It is shown that the known conditions for the onset of dynamic chaos of charged particles in external electromagnetic fields require careful...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
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| Цитувати: | Chaotic dynamics at cyclotron resonances / V.A. Buts, V.V. Kuzmin, A.P. Tolstoluzhsky // Вопросы атомной науки и техники. — 2017. — № 6. — С. 66-70. — Бібліогр.: 9 назв. — англ. |
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Buts, V.A. Kuzmin, V.V. Tolstoluzhsky, A.P. 2018-06-16T06:57:45Z 2018-06-16T06:57:45Z 2017 Chaotic dynamics at cyclotron resonances / V.A. Buts, V.V. Kuzmin, A.P. Tolstoluzhsky // Вопросы атомной науки и техники. — 2017. — № 6. — С. 66-70. — Бібліогр.: 9 назв. — англ. 1562-6016 PACS: 41.75.-i; 05.45.-a https://nasplib.isofts.kiev.ua/handle/123456789/136192 Mechanisms and criteria for the transition to chaotic dynamics of particles and fields under conditions of electron cyclotron resonances (ECR) are considered. It is shown that the known conditions for the onset of dynamic chaos of charged particles in external electromagnetic fields require careful use. The mechanism of the appearance of regimes with dynamic chaos has been discovered and described, even under conditions of isolated cyclotron resonance. Anomalous sensitivity of particle dynamics to external fluctuations is described. It is shown that the higher moments of particle dynamics can play a significant role. In this case, the usual diffusion equations require a revision. Розглянуто механізми та критерії переходу до хаотичної динаміки частинок і полів в умовах електронних циклотронних резонансів (ЕЦР). Показано, що відомі умови виникнення динамічного хаосу заряджених частинок у зовнішніх електромагнітних полях вимагають обережного використання. Виявлено та описано механізм виникнення режимів з динамічним хаосом навіть в умовах ізольованого циклотронного резонансу. Описана аномальна чутливість динаміки частинок на зовнішні флуктуації. Показано, що значну роль можуть грати вищі моменти динаміки частинок. В цьому випадку звичні дифузійні рівняння вимагають перегляду. Рассмотрены механизмы и критерии перехода к хаотической динамике частиц и полей в условиях электронных циклотронных резонансов (ЭЦР). Показано, что известные условия возникновения динамического хаоса заряженных частиц во внешних электромагнитных полях требуют осторожного использования. Обнаружен и описан механизм возникновения режимов с динамическим хаосом даже в условиях изолированного циклотронного резонанса. Описана аномальная чувствительность динамики частиц на внешние флуктуации. Показано, что значительную роль могут играть высшие моменты динамики частиц. В этом случае привычные диффузионные уравнения требуют пересмотра. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Динамика пучков Chaotic dynamics at cyclotron resonances Хаотична динаміка при циклотронних резонансах Хаотическая динамика при циклотронных резонансах Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Chaotic dynamics at cyclotron resonances |
| spellingShingle |
Chaotic dynamics at cyclotron resonances Buts, V.A. Kuzmin, V.V. Tolstoluzhsky, A.P. Динамика пучков |
| title_short |
Chaotic dynamics at cyclotron resonances |
| title_full |
Chaotic dynamics at cyclotron resonances |
| title_fullStr |
Chaotic dynamics at cyclotron resonances |
| title_full_unstemmed |
Chaotic dynamics at cyclotron resonances |
| title_sort |
chaotic dynamics at cyclotron resonances |
| author |
Buts, V.A. Kuzmin, V.V. Tolstoluzhsky, A.P. |
| author_facet |
Buts, V.A. Kuzmin, V.V. Tolstoluzhsky, A.P. |
| topic |
Динамика пучков |
| topic_facet |
Динамика пучков |
| publishDate |
2017 |
| language |
English |
| container_title |
Вопросы атомной науки и техники |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| format |
Article |
| title_alt |
Хаотична динаміка при циклотронних резонансах Хаотическая динамика при циклотронных резонансах |
| description |
Mechanisms and criteria for the transition to chaotic dynamics of particles and fields under conditions of electron cyclotron resonances (ECR) are considered. It is shown that the known conditions for the onset of dynamic chaos of charged particles in external electromagnetic fields require careful use. The mechanism of the appearance of regimes with dynamic chaos has been discovered and described, even under conditions of isolated cyclotron resonance. Anomalous sensitivity of particle dynamics to external fluctuations is described. It is shown that the higher moments of particle dynamics can play a significant role. In this case, the usual diffusion equations require a revision.
Розглянуто механізми та критерії переходу до хаотичної динаміки частинок і полів в умовах електронних циклотронних резонансів (ЕЦР). Показано, що відомі умови виникнення динамічного хаосу заряджених частинок у зовнішніх електромагнітних полях вимагають обережного використання. Виявлено та описано механізм виникнення режимів з динамічним хаосом навіть в умовах ізольованого циклотронного резонансу. Описана аномальна чутливість динаміки частинок на зовнішні флуктуації. Показано, що значну роль можуть грати вищі моменти динаміки частинок. В цьому випадку звичні дифузійні рівняння вимагають перегляду.
Рассмотрены механизмы и критерии перехода к хаотической динамике частиц и полей в условиях электронных циклотронных резонансов (ЭЦР). Показано, что известные условия возникновения динамического хаоса заряженных частиц во внешних электромагнитных полях требуют осторожного использования. Обнаружен и описан механизм возникновения режимов с динамическим хаосом даже в условиях изолированного циклотронного резонанса. Описана аномальная чувствительность динамики частиц на внешние флуктуации. Показано, что значительную роль могут играть высшие моменты динамики частиц. В этом случае привычные диффузионные уравнения требуют пересмотра.
|
| issn |
1562-6016 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/136192 |
| citation_txt |
Chaotic dynamics at cyclotron resonances / V.A. Buts, V.V. Kuzmin, A.P. Tolstoluzhsky // Вопросы атомной науки и техники. — 2017. — № 6. — С. 66-70. — Бібліогр.: 9 назв. — англ. |
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| fulltext |
ISSN 1562-6016. ВАНТ. 2017. №6(112) 66
CHAOTIC DYNAMICS AT CYCLOTRON RESONANCES
V.A. Buts
1,2,3
, V.V. Kuzmin
1,2
, A.P. Tolstoluzhsky
1
1
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine;
2
V.N. Karazin Kharkiv National University, Kharkov, Ukraine;
3
Institute of Radio Astronomy of the National Academy of Sciences of Ukraine, Kharkov, Ukraine
Mechanisms and criteria for the transition to chaotic dynamics of particles and fields under conditions of elec-
tron cyclotron resonances (ECR) are considered. It is shown that the known conditions for the onset of dynamic
chaos of charged particles in external electromagnetic fields require careful use. The mechanism of the appearance
of regimes with dynamic chaos has been discovered and described, even under conditions of isolated cyclotron res-
onance. Anomalous sensitivity of particle dynamics to external fluctuations is described. It is shown that the higher
moments of particle dynamics can play a significant role. In this case, the usual diffusion equations require a revi-
sion.
PACS: 41.75.-i; 05.45.-a
INTRODUCTION
It would seem that in the dynamics of particles and
fields in ECR conditions everything is quite understand-
able and investigated. However, as will follow from the
results of our work, many important questions of the
dynamics of particles and fields in ECR have important
features that were not previously studied. Really in sec-
tion 2 it is shown, that the transition of regular dynamics
to a regime of chaotic dynamics requires an additional
analysis of transition conditions. In the 3rd section it is
shown that the dynamics of particles at ECR is anoma-
lously sensitive to external fluctuations. Ad-ditive fluc-
tuations can give rise to superdiffusion. Multiplicative
fluctuations give rise to a fluctuation instability. In the
4th section it is shown that a regime with dynamic chaos
in the excitation of electromagnetic waves by a stream
of charged particles can arise even under conditions of
one isolated nonlinear cyclotron resonance.
In Section 5 it is shown that the higher moments can
play a more significant role than the lower moments.
This means that for the kinetic description of these re-
gimes the known diffusion equations can not be used.
Equations are needed in which these higher moments
are taken into account
1. CONDITION OF ARISING REGIMES
WITH DYNAMICAL CHAOS
Let us consider the motion of a charged particle in a
constant magnetic field 0H and in the field of a plane
electromagnetic wave of arbitrary polarization:
0Re , Re /
i t
e c
kr
E E H kE ,
0 0 , ,x y zE i E . (1)
Here
0E is amplitude of electric field strength,
, ,x y zi α is vector polarization of wave.
The equations of motion of a charged particle in this
case has the form
0
d d e e
m e
dt dt mc mc
p
v E vH vH , (2)
/d
dt
r p .
For simplicity of writing the formulas, we shall con-
sider the simplest structure of the electromagnetic-wave
field
Re 0, ,0yiEE , 0,0, zHH , ,0,0xkk .
In this case, the criterion for the overlap of two
neighboring nonlinear cyclotron resonances, which was
obtained in [1, 2], can be written in the form
2
0
216H s xW k , (3)
where ( ), /s s x HW p J k p
, cosxp p ,
sinyp p and dimensionless variables are intro-
duced t , / mcp p , kr r , /c k k ,
0 0 /eE mc , 0 0/h H H ,
0 /H eH mc .
From the condition (3) follows that at 0sW the
wave amplitude necessary for the arising of regime with
dynamic chaos tends to infinity. However, our numeri-
cal calculations show that this does not occur. We will
remind that the criterion (3) has been obtained as a con-
dition for overlapping two neighboring nonlinear cyclo-
tron resonances (Chirikov criterion). Our analytical and
numerical studies show that in this case (
0 ) for
the formation of chaotic dynamics, the main role is
played by other resonances that have not been taken into
account in obtaining the criterion (3).
We note that the influence of a large number of non-
linear cyclotron resonances with which the particle in-
teracts weakly can be modeled by the presence of an
external noise influence. Indeed, as we shall see in the
next section, the role of even small external fluctuations
can radically change the dynamics of charged particles
at cyclotron resonances. Qualitatively, these results can
explain the arisen contradictions.
2. INFLUENCE FLUCTUATIONS
ON PARTICLES DYNAMICS
It was shown in [3, 4] that under conditions close to
autoresonance conditions, particle dynamics can be
anomalously sensitive to external fluctuations. Below,
we consider this question in more detail for the simplest
structure of the field of an electromagnetic wave propa-
gating along the direction of the field 0 || zH :
Re ,0,0xEE , 0, ,0yHH , 0,0, 1zk k .
We will analyze the influence of additive and multi-
plicative fluctuations in the most interesting case, under
conditions close to autoresonance:
/ 1 0s z z HR k s .
ISSN 1562-6016. ВАНТ. 2017. №6(112) 67
2.1. ADDITIVE FLUCTUATION
At the beginning, we will estimate the role of addi-
tive fluctuations. For this, taking into account the small
value of the field amplitude
0 1 , the system of
equations (2) can be linearized [4]:
/ , / ,d d B d d f (4)
where 0 1 0 0/ 2 sinB W ,
1 , 1 ,
1 ,
0
0 /nR
, ,
Hf is additive
fluctuation force. At the analytical study, we assume
that ( )f Gaussian, delta correlated random pro-
cess with zero mean:
2f f D , 0f ,
where D is diffusion coefficient.
In works [3, 4] it has been shown anomalous sensi-
tivity of the particles dynamics to such fluctuations at
approach to an autoresonance ( 0 ). However, the
energy gain of the particle remained diffusive:
2 /DB . (5)
It is of interest to find out under what conditions the
law of ordinary diffusion is replaced by the law of su-
perdiffusion:
2 32 / 3 D B (6)
Numerical studies were carried out for this purpose.
For numerical calculations, the parameter value
~ 0.033B was chosen. The value of the parameter
varied from 110 to 710 . To find the mean
values of the square of the energy increment, averaging
over the ensemble of forty realizations was carried out.
As fluctuations, a random variable with a uniform dis-
tribution law in the interval ,H H was chosen.
The value 0.1H . Initial conditions for the addition
of energy and phase: (0) 0, (0) / 60 .
Studies have shown that at change of the parameter
right up to 410 the dependence of the mean
square of energy on time corresponds to the diffusion
law (5) Figs. 1,a-b.
Fig. 1. The dependence of the mean square
of the particle energy on time: a) 110 ; b) 410 ;
c) 510 ; d) 710
Decrease the parameter leads to a qualitative
change in the dependence of the mean square of the
particle energy on the time Figs. 1,c-d. The dependence
of the mean square of energy on time instead of linear
becomes quadratic with 510 and increases to cubic
710 .
These results show that the presence of additive
fluctuations, even of very small amplitudes, actually
leads to the appearance of superdiffusion. However, this
occurs only in an exceptionally small neighborhood of
the exact fulfillment of the autoresonance conditions.
Under conditions of real experiments, it is practically
impossible to satisfy the conditions of autoresonance
with the required accuracy. Therefore, it is necessary to
focus on the formula (5), and not on the formula (6).
2.2. MULTIPLICATIVE FLUCTUATION
Let us now consider what the presence of multi-
plicative fluctuations will lead to. Such fluctuations
arise, for example, in the presence of fluctuations in the
amplitude of the wave in which the particle moves. In
this case, the dynamics of a particle located not in the
vicinity of a singular point of the "saddle" type, but in
the neighborhood of the "center" is of the greatest inter-
est. This is due to the fact that from the vicinity of the
saddle point the particles exponentially move away
from each other even under the action of regular forces.
Equations for finding time dynamics and particle,
which are close to points of the "center" type of a math-
ematical pendulum, in this case it is convenient to repre-
sent in the form [4]:
(1 ( )) , .
du d
f u
d d
(7)
Here t B . The relationship between par-
ticle energy and angle takes the form:
/B .
The numerical analysis of equations (7) has been
carried out for the initial conditions (0) 0u ,
(0) / 60 , amplitude of fluctuations 0.1H .
Fig. 2 shows the dependence of the mean square of the
energy: the solid line is the result of numerical calcula-
tion, the dots indicate the approximation by the curve
exp ( ) exp( )multF D . The exponential dependence
of the mean square of energy on time is clearly visible
from the graphs of Fig. 2.
Fig. 2. The dependence of the mean square
of the particle energy on time
3. EXCITATION OF ELECTROMAGNETIC
WAVES BY A BUNDLE OF OSCILLATORS
The dynamics of charged particles in external pre-
scribed electromagnetic fields becomes chaotic only in
the presence of at least two nonlinear resonances. How-
ever, if an electromagnetic wave is excited by the parti-
cles themselves, then regimes with dynamic chaos can
appear even in the presence of only one nonlinear reso-
nance. Let us show this result. To this end, we consider
ISSN 1562-6016. ВАНТ. 2017. №6(112) 68
the problem of excitation of an electromagnetic field by
a monoenergetic beam of oscillators with a distribution
function
0 0 ||( ) ( )
2
bN
f p p p
p
, (8)
where
||,p p
perpendicular and parallel to the axis z
impulse component,
bN equilibrium beam density.
We shall consider the excitation of a wave propagat-
ing perpendicular to the external magnetic field. A
complete nonlinear self-consistent system of equations
that describes the dynamics of particles and fields con-
sists of the Maxwell equations and the equations of par-
ticle motion. Such a system is given in [1, 2]. Below, we
write out truncated system of equations describing the
dynamics of particles and fields in an isolated cyclotron
resonance with number s :
( ) si
s
dp
iJ e
d
,
2
2
1
1 1 Re ( ) sis H
s
H
d s s
J e
d
, (9)
22
0
0
( )
2
sib
s s
pd
i d J e
d
,
where: /p p mc , 2 2/ , 1H Hp ,
/H oeH mc , 2 24 e /b b en m , /eE mc .
From the results of numerical calculations shown in
Fig. 3, we can mark out the following features of the
dynamics of particles and fields:
with an increase in the density of active particles
(within 20.002 0.04b ), the level of the excited
field increases. The dynamics of particles and the excit-
ed field is regular;
with the beam density greater than 2 0.04b a
chaotic component appears in the dynamics of the excit-
ed field;
beginning approximately from the beam density of
0.5, the asymptotic value of the field does not exceed
0.15.
Thus, just as in overlapping cyclotron resonances
(see, for example, [5]), the onset of local instability
leads to a limitation of the level of the field excited by
the beam (see Fig. 3).
Fig. 3. The amplitude of the field versus time
at a beam density: a) 2
b =0.04; b) 2
b =0.1;
c) 2
b =0.5; d) 2
b =4
We note that the same stabilization process is also
characteristic of plasma-beam instability [6]. Such dy-
namics of the field with increasing particle density re-
mains fairly familiar as long as the particle density satis-
fies inequality 2 1b . With further increasing of the
particle density, when 2 1b it was possible to assume
that excitation of oscillations at the selected frequencies
(
H ) will be absent. Indeed, if inequality 2 1b
oscillations at frequencies
H are not eigenmodes
in such medium. When excited, these oscillations are
damped. Indeed, due to the nonequilibrium nature of the
beam system at frequencies
H , there is excitation
of oscillations at these frequencies. However, these os-
cillations decay rapidly enough. The regime of relaxa-
tion oscillations appears in Fig. 3,d. It exists on a suffi-
ciently large time interval. However, over time, this
regime goes into a regime of chaotic oscillations, and
the process of excitation of oscillations at these frequen-
cies is stopped. With increasing particle density, the
amplitudes of the excited oscillations decrease. As far as
we know, the excitation of such relaxation oscillations
has not yet been described. Such oscillations may, ap-
parently, arise in the ionospheric plasma.
In the above model (see formulas (9)), one cyclotron
resonance is isolated, and the dynamics of the interac-
tion of particles and fields in the isolated cyclotron res-
onance model is studied. In this case, the chaotization
mechanism due to the overlap of the cyclotron reso-
nances is absent. An additional analysis was made of the
dynamics of particles in an isolated cyclotron resonance.
We assume that the amplitude of the wave is constant.
In this case, the system of equations that describes the
dynamics of a particle coincides with the system of
equation (9), in which the third equation can be neglect-
ed. The dynamics of the particles is described by the
first two equations. Such a system has the Hamiltonian:
( , ) 2 ( 2 ) cos( )s s s
H H
s d
H I I I J I
dI
, (10)
where 2 / 2I .
It is easy to show that the phase portrait of the sys-
tem with the Hamiltonian (10) is topologically similar to
the phase portrait of the Duffing oscillator. For a small
value of the parameter 3
,0/G p ( ,0p initial parti-
cle momentum) ( 1G ) on the phase plane there are
three singular points. Two of them are points of the
"center" type, one is the "saddle" type. If the amplitude
is sufficiently large ( 1G ), then two singular points,
namely the saddle point and the point of the "center"
type merge and disappear. There remains only one sin-
gular point a point of the "center" type. It is necessary
to pay attention to that fact that oscillations of the Duff-
ing oscillator are potential, and for the equations con-
sidered by us, it isn't possible to find potential. Typical
types of phase portrait at small ( 1G ) and at high
( 1G ) field strengths of the external waves are pre-
sented in Fig. 4. There are selected regions for trapped
particles and a region for transiting particles. As can be
seen from Fig. 4,a, on the phase plane, in full accord-
ance with the results given above, there are three singu-
ISSN 1562-6016. ВАНТ. 2017. №6(112) 69
lar points: two types of center and a saddle point. As
amplitude of the wave increases, two points ("saddle"
and "center" with 0s ) approach and disappear
(see Fig. 4,b).
Fig. 4. Phase trajectories: а) =0.08; b) =0.105
Such a process of qualitative change in dynamics
can cause a regime with dynamic chaos. Moreover, it
can be seen that even the quantitative characteristics of
the appearance of such a qualitative change in dynam-
ics, given in Figs. 3,a-b, confirm this possibility. Indeed,
it can be seen from this figure that as soon as the ampli-
tude of the excited wave exceeds 0.105, the dynamics of
the particles acquire an irregular character. With a fur-
ther increase in the particle density, and, corresponding-
ly, with increasing intensity of the excited wave, this
irregularity becomes more noticeable. Already the in-
tensity of the field being excited for short times may
exceed 0.2. However, the dynamics of the particle turns
out to be such that, irregularly oscillating, the field am-
plitude reaches a level of the order of 0.15. This value
of the field strength agrees qualitatively with the intensi-
ty of the wave field at which a qualitative change in the
phase dynamics of the particles occurs
4. ROLE OF THE MOMENTS IN DYNAMICS
OF PARTICLES
Often, particle dynamics in regimes with dynamic
chaos are described in the framework of a diffusion
equation of the Einstein-Fokker-Planck type equations.
In particular, this approach is used to describe the dy-
namics of particles in ECR. See, for example, [7] and
the literature cited there. However, such diffusion equa-
tions are valid only when the higher moments rapidly
decrease and it is sufficient to take into account only the
second moment. Below we show that in the regimes
with dynamic chaos in ECR, in most cases the higher
moments can play a more significant role. They need to
be taken into account. In this case, the kinetic equation
must contain these moments. The results of numerical
studies of the dependence of the magnitude of the mo-
ments on their number and on the field strength are pre-
sented in Fig. 5.
a b
Fig. 5. Dependences of the magnitudes of the moments
divided by the factorial of their number !m
for the field amplitude: а)
0 0.1 ; b)
0 0.19
These figures show the dependence of the magnitude
of the moments on their number and the magnitude of
each moment is divided by the factorial of its number
(on !m ). It can be seen from these figures that, for a
sufficiently low external field strength
(
0 / 0.1eE mc ) the moments rapidly fall with
increasing number (see Fig. 5,a).
However, for higher strengths (for
0 0.19 ) the
higher moments turn out to be larger than the moments
with smaller numbers. Fig. 5,b shows that the moments
increase with the number up to the number 6m .
This feature of the moments requires the modifica-
tion of the equations for describing the particle kinetics.
To do this, we write down the relationship between the
particle density at the instant of time and the
particle density at time :
( , ) ( , ) f( )n p n p p p dp
. (11)
Expression (11) is a mathematical reflection of the
fact that the density of particles that have a momentum
p at a time , will be determined by all other
particles (with other energies). In this case, such parti-
cles with probability ( )f p , after an interval of time
, acquire momentum p . It is convenient to rewrite
equation (11) in the form:
( , ) ( , ) ( , ) ( , ) f( )n p n p n v p n p p dp
. (12)
If the moments are finite, then, decomposing the in-
tegrands (12) with respect to small displacements and
limiting ourselves to the second moments, we obtain the
usual diffusion equation for the particle density with the
diffusion coefficient 2 / 2D p . If the moments do
not decrease, then a more general equation:
!
m
m
m
m
pn n
m p
, 2 ; 1,2,3...m j j . (14)
For the case presented in Fig. 5,b, it is necessary to
take into account 4-5 terms in the sum (14).
CONCLUSIONS
Thus, the results obtained above show that the chaot-
ic dynamics of particles and fields in ECR is not fully
understood at the present time. Note the most important
result for the application. Great hopes were placed on
using the autoresonance condition to accelerate charged
particles and to excite high-frequency oscillations. 8
However, real attempts to construct such installations
have shown their insignificant efficiency (see, for ex-
ample, Ref. 9). Such insignificant efficiency of energy
exchange between particles and waves can be related to
the anomalous sensitivity of particle dynamics with re-
spect to fluctuations (see Section 3 above).
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Article received 30.09.2017
ХАОТИЧЕСКАЯ ДИНАМИКА ПРИ ЦИКЛОТРОННЫХ РЕЗОНАНСАХ
В.А. Буц, В.В. Кузьмин, А.П. Толстолужский
Рассмотрены механизмы и критерии перехода к хаотической динамике частиц и полей в условиях элек-
тронных циклотронных резонансов (ЭЦР). Показано, что известные условия возникновения динамического
хаоса заряженных частиц во внешних электромагнитных полях требуют осторожного использования. Обна-
ружен и описан механизм возникновения режимов с динамическим хаосом даже в условиях изолированного
циклотронного резонанса. Описана аномальная чувствительность динамики частиц на внешние флуктуации.
Показано, что значительную роль могут играть высшие моменты динамики частиц. В этом случае привыч-
ные диффузионные уравнения требуют пересмотра.
ХАОТИЧНА ДИНАМІКА ПРИ ЦИКЛОТРОННИХ РЕЗОНАНСАХ
В.О. Буц, В.В. Кузьмін, О.П. Толстолужський
Розглянуто механізми та критерії переходу до хаотичної динаміки частинок і полів в умовах електронних
циклотронних резонансів (ЕЦР). Показано, що відомі умови виникнення динамічного хаосу заряджених час-
тинок у зовнішніх електромагнітних полях вимагають обережного використання. Виявлено та описано ме-
ханізм виникнення режимів з динамічним хаосом навіть в умовах ізольованого циклотронного резонансу.
Описана аномальна чутливість динаміки частинок на зовнішні флуктуації. Показано, що значну роль мо-
жуть грати вищі моменти динаміки частинок. В цьому випадку звичні дифузійні рівняння вимагають перег-
ляду.
http://iopscience.iop.org/journal/0029-5515
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