Regimes with the dynamic chaos in analisis of linear systems
The examples are shown which demonstrate that in the course of analysis of linear systems the modes with dynamic chaos can occur. This can happen when for analysis of linear systems the change of variables is used resulting in the need to investigate either non-linear equations or non-linear functio...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2014
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nasplib_isofts_kiev_ua-123456789-812092025-02-05T20:29:07Z Regimes with the dynamic chaos in analisis of linear systems Режимы с динамическим хаосом при анализе динамики линейных систем Режими з динамічним хаосом при аналізі динаміки лінійних систем Antipov, V.S. Buts, V.A. Плазменная электроника The examples are shown which demonstrate that in the course of analysis of linear systems the modes with dynamic chaos can occur. This can happen when for analysis of linear systems the change of variables is used resulting in the need to investigate either non-linear equations or non-linear functions. In the plasma physics field such situation may arise, e.g., when diagnosing some plasma parameters. Приведены примеры, которые показывают, что при анализе динамики линейных систем могут возникать режимы с динамическим хаосом. Это случается, когда для анализа линейных систем используются замены переменных, которые приводят к необходимости исследовать либо нелинейные уравнения, либо нелинейные функции. Для физики плазмы такая ситуация может возникнуть, например, при диагностике параметров плазмы. Наведено приклади, які показують, що при аналізі динаміки лінійних систем можуть виникати режими з динамічним хаосом. Це трапляється, коли для аналізу лінійних систем використовуються заміни змінних, які призводять до необхідності досліджувати, або нелінійні рівняння, або нелінійні функції. Для фізики плазми така ситуація може виникнути, наприклад, при діагностиці параметрів плазми. 2014 Article Regimes with the dynamic chaos in analisis of linear systems / V.S. Antipov, V.А. Buts // Вопросы атомной науки и техники. — 2014. — № 6. — С. 108-111. — Бібліогр.: 5 назв. — англ. 1562-6016 PACS: 05.45.-a https://nasplib.isofts.kiev.ua/handle/123456789/81209 en Вопросы атомной науки и техники application/pdf Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Плазменная электроника Плазменная электроника |
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Плазменная электроника Плазменная электроника Antipov, V.S. Buts, V.A. Regimes with the dynamic chaos in analisis of linear systems Вопросы атомной науки и техники |
| description |
The examples are shown which demonstrate that in the course of analysis of linear systems the modes with dynamic chaos can occur. This can happen when for analysis of linear systems the change of variables is used resulting in the need to investigate either non-linear equations or non-linear functions. In the plasma physics field such situation may arise, e.g., when diagnosing some plasma parameters. |
| format |
Article |
| author |
Antipov, V.S. Buts, V.A. |
| author_facet |
Antipov, V.S. Buts, V.A. |
| author_sort |
Antipov, V.S. |
| title |
Regimes with the dynamic chaos in analisis of linear systems |
| title_short |
Regimes with the dynamic chaos in analisis of linear systems |
| title_full |
Regimes with the dynamic chaos in analisis of linear systems |
| title_fullStr |
Regimes with the dynamic chaos in analisis of linear systems |
| title_full_unstemmed |
Regimes with the dynamic chaos in analisis of linear systems |
| title_sort |
regimes with the dynamic chaos in analisis of linear systems |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2014 |
| topic_facet |
Плазменная электроника |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/81209 |
| citation_txt |
Regimes with the dynamic chaos in analisis of linear systems / V.S. Antipov, V.А. Buts // Вопросы атомной науки и техники. — 2014. — № 6. — С. 108-111. — Бібліогр.: 5 назв. — англ. |
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Вопросы атомной науки и техники |
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ISSN 1562-6016. ВАНТ. 2014. №6(94)
108 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2014, №6. Series: Plasma Physics (20), p. 108-111.
REGIMES WITH THE DYNAMIC CHAOS IN ANALISIS OF LINEAR
SYSTEMS
V.S. Antipov, V.А. Buts
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov,
Ukraine
E-mail: vbuts@kipt.kharkov.ua
The examples are shown which demonstrate that in the course of analysis of linear systems the modes with
dynamic chaos can occur. This can happen when for analysis of linear systems the change of variables is used
resulting in the need to investigate either non-linear equations or non-linear functions. In the plasma physics field
such situation may arise, e.g., when diagnosing some plasma parameters.
PACS: 05.45.-a
INTRODUCTION
At present time it is generally accepted that the
regimes with dynamic chaos appear only in nonlinear
systems. In general, this is correct. However, in some
cases, when studying the linear systems one can run into
the dynamics similar to the dynamics of modes with
dynamic chaos. It may seem that this dynamic is not
possible. In reality, in many cases such dynamic is quite
natural. It can occur, for example, when for the analysis
of linear systems the change of variables is necessary,
and thus the initial linear system transforms to the one
described by a system of nonlinear equations. The well-
known examples are the equations of classical
mechanics and the equations of geometrical optics.
Original for these equations are linear equations of
quantum mechanics and Maxwell equations,
respectively.
In such a case, these new variables can lead to
appearance of the modes with dynamic chaos. The
change of variables modes to those with dynamic chaos
can occur in linear systems if the processing of the
results of the linear system dynamics is produced by
non-linear characteristics. In our previous works (see,
for example, [1-3]), we have illustrated the possibility
for such regimes to be realized.
This paper presents the results of further studies in
this direction. Attention is drawn to the fact that there
are two causes of dynamic chaos: (i) an immediate
change of variables and (ii) the features of these non-
linear equations. The first reason apparently is
nonphysical. As an example of the role of the second
reason, the dynamics of waves propagating in layered
dielectric was considered. The recurrence relations for
the coefficients of reflection from and transmission
through the layered media weкe found. Thus a non-
linear characteristic of the field was inserted. The
conditions when this recurrent sequence becomes
chaotic have been found too.
1. REPLACEMENT LEADING TO THE
CHAOTIC DYNAMICS
A convenient characteristic allowing to determine
such changes is a measure of "volume" x in the phase
space: ( )ip x x . Here ( )ip x is a probability
density for the studied dynamical system to get to point
ix , that belongs to "volume" x . Let’s assume that
substitution ( )iz f x
was done in such a way that z is
an image and
ix is an inverse image of the point z .
Principally, there can be many inverse images. The set
of vectors ( )iz f x do form a new phase space.
We consider in this space the "volume" z
( / 2; / 2z z z z ). By definition, a measure of
the magnitude of this volume will be:
( ) ( )z i i
i
g z z p x x . Here the sum is carried
out along the number of inverse images. Then the
density of probability of the new phase space is
determined by the formula:
( )
( ) ( ) i i
i
i i
x p x
g z p x
z J
, (1)
where det( / )J f x is the Jacobian of
transformation.
Equation (1) is practically the Perron - Frobenius
formula for transformation of probability density when
converting the functions. Let’s consider, as an example,
the high-usage and most important replacement.
Suppose, that we have ,k k kx A A , where
k k kA A iA . If one carries out the replacement:
expk k k k kA A iA a i , then it is easy to verify
that 1/J a . If in initial variables the motion was
regular, then ( ) ~ ( ( ))i i ip x x x t . The density of
probability for new variables become indeterminate
( ( ) ?g z ) when 0a . It is useful to note that this
transformation describes the transition from quantum
consideration to classical consideration. In this case the
condition 0a may mean that the velocity of particles
are going to zero. This fact corresponds to a well-known
result that the transition from quantum to classical
consideration for particles with zero speed is not
correct. Using this simplest example, we will examine
where this might lead to. Let the complex variables
satisfy the following equations:
x y , y x . (2)
Then there are two possible replacements. First
R Ix x i x ,
R Iy y i y . In this case, the new
mailto:vbuts@kipt.kharkov.ua
ISSN 1562-6016. ВАНТ. 2014. №6(94) 109
dynamics does not arise. But the new dynamics occurs
when another replacement is used, namely:
0 2expx x i x ,
1 3expy x i x .
Here ( ), 0,1,2,3,kx t k are real functions. After
substitution these replacement into (2) we will get the
system of equations:
0 1 cosx x
1 0 cosx x
0 1
1 0
sin
x x
x x
(3)
with
3 2x x
Fig. 1. Time dependence of
0x
Fig. 2. Time dependence of phase
2x . One can
see jumps of phase at the moments when
amplitude don’t change sign when passing
zero point
In addition, it follows from first two equations of the
system (6) the existence of the integral: 2 2
0 1x x const .
Taking into account this integral, the system of
equations (3) has only one degree of freedom. As soon
as it gets to the point, for example
0 0x , then the
corresponding phase (
2x ) can be undefined. These
features of the dynamics of the amplitude
0x and phase
2x are shown in Figs. 1-3.
Fig. 3. Autocorrelation function for
0x variable
From Figs. 1, 2 we see that at random time moments
the value of
0x does not change its sign and the value of
phase undergoes a jump. Similar random phase jumps
for
3x are observed when
1 0x .
Above we have considered the transformation
occuring most frequently in physics, especially in
radiophysics. In addition to this transformation, often is
used the transformation that transforms the linear
equation of second order to the Riccati equation, which
is a first order nonlinear equation. It may be expected
that the dynamics of such new variables can also be
chaotic. We will show that the exchange, which led to
the Riccati equation, is equivalent to the one used
above. Indeed, suppose we have the
equation 2 0x x . By the use of the
replacement
0 /x x x for the new variable this linear
equation transforms into nonlinear Riccati equation:
2 2
0 0x x , (4)
with the relation between initial variable and new
variable as: 0
0
( ) exp( ( ) )
t
x t a x t dt
If we present
0( )x t in the form
0 ( ) ( )x i t i t ,
0Re x ,
0( ) Imt x ,
then the expression for x(t) can be rewritten as:
( ) ( ) exp( ( ))x t A t i t i t , where
0
( ) exp
t
A t a dt , a const ,
0
( ) ( )
t
t t dt .
One can see that this expression is identical to the
previous expression.
2. TRANSMISSION OF
ELECTROMAGNETIC WAVES THROUGH
THE LAYER WITH LAYERED
INHOMOGENEITY
Now we shall consider the layered media, with
layers arranged perpendicular to the axis z with
inhomogeneous medium occupying the
region 0 z L . For simplicity, it is assumed that there
are only two different periodically alternating layers.
The dielectric constant of these layers and their
thicknesses are equal
0 0 1 1, ; ,d d , respectively.
We assume that from the left a homogeneous half-
space ( 0z ) a flat monochromatic electromagnetic
wave falls on a non-uniform layer. For simplicity, the
wave is incident at right angle to the interface. The
components of electric and magnetic fields of the wave
obey the following equations:
x
y
E
ikH
z
, ( )
y
x
H
ik z E
z
,
2
2
2
( ) 0x
x
E
k z E
z
. (5)
We are looking for reflectance and transmittance of
the wave through an inhomogeneous layer with the
fields in an arbitrary homogeneous layer having the
form:
( ) ( )x n n n nE A exp ik z B exp ik z , (6)
( ) ( )y n n n n nH A exp ik z B exp ik z ,
where k kn - the refractive index of the layer.
110 ISSN 1562-6016. ВАНТ. 2014. №6(94)
We use the matrix method (see, e.g., [4, 5]) for
determination of the transmission and reflection
coefficients. Under this method, the fields at the left and
right edges of the layer are connected with each other by
the following matrix:
x x
n
y y
E E
M
H H
,
11 12
21 22
n
a a
M
a a
, (7)
where
11 22cos n na k d a ; 12 sin /n n na i k d ;
21 sin n n na i k d ;
The connection between fields at the left and right
boundaries of the double layer will be determined by the
following relation:
1 0
x x
y y
E E
M M
H H
. (8)
The elements of matrix
11 12
0 1
21 22
A A
M M M
A A
are
easy to determine:
1
11 0 0 1 1 0 0 1 1
0
cos cos sin sinA k d k d k d k d ,
0
22 0 0 1 1 0 0 1 1
1
cos cos sin sinA k d k d k d k d ,
12 0 0 1 1 0 0 1 1
1 0
cos sin sin cos
i i
A k d k d k d k d ,
12 1 0 0 1 1 0 0 0 1 1cos sin sin cosA i k d k d i k d k d .
If there are N such double layers, the relation
between fields at the left ( 0z ) and right ( z L )
boundaries will be expressed by the formula:
x xN
y y
E E
M
H H
. (9)
In the general case the matrix NL M looks like a
quite complicated one, and its elements could be found
only by calculus of approximations. However, in some
special cases the elements of the matrix can be strongly
simplified, in particular, the elements become much
simpler if the layers with identical optical thickness
0 0 1 1k d k d kd , are under consideration.
We should pay attention that the matrix
nM is a 2х2
matrix, which diagonal elements are real functions and
nondiagonal elements are imaginary functions. The
matrix M is of a similar structure. It is easy to show that
similar structurewill also the transfer matrix, which
links the fields at the outer boundaries of the
inhomogeneous layer:
11 12
21 22
N
l il
L M
il l
. (10)
These elements, after some transformations, can be
expressed in terms of Chebyshev polynomials of the
second kind:
11 11 1 2( ) ( )N Nl A U s U s ,
22 22 1 2( ) ( )N Nl A U s U s ,
12 12 1( )Nl A U s .
Here
11 22 / 2s A A ; ( )NU s - Chebyshev
polynomials .
Using elements of matrix L it is easy to express
transmission and reflection coefficients:
11 22 12 21
2
( )N N
T
l l i l l
, (11)
11 22 12 21
11 22 12 21
( )
( )
N N
N N
l l i l l
R
l l i l l
. (12)
Expressions (11) and (12) can be analyzed by
numerical methods. The most informative are energetic
reflection coefficients and energetic transmission
coefficient, which have been studied.
It was shown that there is a range of parameters,
where the dependence of these characteristics on the
frequency of the incident radiation is irregular. Figs. 4-7
show two typical cases. In the first case (see Figs. 4, 5)
parameters of the inhomogeneous layers are chosen so
that the dependence of the transmission coefficient on
frequency is a regular function.
In this case the correlation function oscillates
without decreasing the amplitude of the oscillations.
However, with increasing the number of layers, this
dependence becomes irregular (see Figs. 6, 7), and the
amplitude of the correlation function decreases rapidly.
Fig .4. Energetic coefficient of the transmission
(number of layers is 10).
0 15, 2n n
Fig. 5. Autocorrelation function (number of layers is
10).
0 15, 2n n
ISSN 1562-6016. ВАНТ. 2014. №6(94) 111
Fig. 6. Energetic coefficient of the
transmission (number of layers is 80).
0 15, 2n n
Fig. 7. Correlation function (number of layers
is 80).
0 15, 2n n
CONCLUSIONS
Thus, the above presented results do additionally
confirm the following main conclusions: when
providing the study of linear systems, there is a high
probability to encounter the dependences that are
inherent in systems with dynamic chaos. This
situation arises when for analysis of the dynamics of
linear systems variables that are either themselves
nonlinear, or obey to non-linear equations are used.
In plasma physics, such a situation can be met, for
example, in the diagnosis of plasma parameters.
REFERENCES
1. V.A. Buts. Chaotic dynamic of linear systems //
Electromagnetic waves and electron systems. 2006,
v. 11, № 11, p. 65-70.
2. V.A. Buts, A.G. Nerukh, N.N. Ruzhytska, and
D.A. Nerukh. Wave Chaotic Behaviour Generated by
Linear Systems // Optical and Quantum Electronics.
2008, v. 40, p. 587-601.
3. V.A. Buts. True quantum chaos // Problems of
Atomic Science and Technology. Series “Plasma
Physics”, 2008, №6 (58), p. 120-123.
4. L.М. Brehovskih. Waves in layered media.
Moscow: “Publisher of the Academy of Sciences of
the USSR”, 1957, p. 497.
5. Sh.A. Furman, A.V. Tikhonravov. Basics of
Optics of Multilayer systems. Paris: “ADAGP”, 1992,
р. 242.
Article received 25.10.2014
РЕЖИМЫ С ДИНАМИЧЕСКИМ ХАОСОМ ПРИ АНАЛИЗЕ ДИНАМИКИ ЛИНЕЙНЫХ
СИСТЕМ
В.С. Антипов, В.А. Буц
Приведены примеры, которые показывают, что при анализе динамики линейных систем могут возникать
режимы с динамическим хаосом. Это случается, когда для анализа линейных систем используются замены
переменных, которые приводят к необходимости исследовать либо нелинейные уравнения, либо
нелинейные функции. Для физики плазмы такая ситуация может возникнуть, например, при диагностике
параметров плазмы.
РЕЖИМИ З ДИНАМІЧНИМ ХАОСОМ ПРИ АНАЛІЗІ ДИНАМІКИ ЛІНІЙНИХ СИСТЕМ
В.С. Антіпов, В.О. Буц
Наведено приклади, які показують, що при аналізі динаміки лінійних систем можуть виникати режими з
динамічним хаосом. Це трапляється, коли для аналізу лінійних систем використовуються заміни змінних,
які призводять до необхідності досліджувати, або нелінійні рівняння, або нелінійні функції. Для фізики
плазми така ситуація може виникнути, наприклад, при діагностиці параметрів плазми.
|