Paradigms of dynamic chaos
It is shown that a condition of occurrence of regimes with chaotic behavior of dynamic systems demands more steadfast studying. In particular, it is shown that if we will take into account singular solutions the chaotic behavior will be inherent also in systems with one degree of freedom. It is show...
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irk-123456789-1121752017-01-18T03:04:06Z Paradigms of dynamic chaos Buts, V.A. Нелинейные процессы в плазменных средах It is shown that a condition of occurrence of regimes with chaotic behavior of dynamic systems demands more steadfast studying. In particular, it is shown that if we will take into account singular solutions the chaotic behavior will be inherent also in systems with one degree of freedom. It is shown that linear systems can generate chaotic dynamics. The question about necessity of local instability for realization of chaotic regimes is discussed. Показано, що умови виникнення режимів з хаотичною поведінкою динамічних систем вимагають більш пильного вивчення. Зокрема, показано, що якщо прийняти в якості рішень особливі рішення , то хаотична поведінка буде притаманна й системам з одним ступенем свободи. Показано, що лінійні системи можуть породжувати хаотичну динаміку. Обговорюється питання про необхідність локальної нестійкості для реалізації хаотичних режимів. Показано, что условия возникновения режимов с хаотическим поведением динамических систем требуют более пристального изучения. В частности, показано, что если принять в качестве решений особые решения, то хаотическое поведение будет присуще и системам с одной степенью свободы. Показано, что линейные системы могут порождать хаотическую динамику. Обсуждается вопрос о необходимости локальной неустойчивости для реализации хаотических режимов. 2013 Article Paradigms of dynamic chaos / V.A. Buts // Вопросы атомной науки и техники. — 2013. — № 4. — С. 284-288. — Бібліогр.: 5 назв. — англ. 1562-6016 PACS: 05.45.Ac http://dspace.nbuv.gov.ua/handle/123456789/112175 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 |
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Нелинейные процессы в плазменных средах Нелинейные процессы в плазменных средах Buts, V.A. Paradigms of dynamic chaos Вопросы атомной науки и техники |
| description |
It is shown that a condition of occurrence of regimes with chaotic behavior of dynamic systems demands more steadfast studying. In particular, it is shown that if we will take into account singular solutions the chaotic behavior will be inherent also in systems with one degree of freedom. It is shown that linear systems can generate chaotic dynamics. The question about necessity of local instability for realization of chaotic regimes is discussed. |
| format |
Article |
| author |
Buts, V.A. |
| author_facet |
Buts, V.A. |
| author_sort |
Buts, V.A. |
| title |
Paradigms of dynamic chaos |
| title_short |
Paradigms of dynamic chaos |
| title_full |
Paradigms of dynamic chaos |
| title_fullStr |
Paradigms of dynamic chaos |
| title_full_unstemmed |
Paradigms of dynamic chaos |
| title_sort |
paradigms of dynamic chaos |
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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/123456789/112175 |
| citation_txt |
Paradigms of dynamic chaos / V.A. Buts // Вопросы атомной науки и техники. — 2013. — № 4. — С. 284-288. — Бібліогр.: 5 назв. — англ. |
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Вопросы атомной науки и техники |
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AT butsva paradigmsofdynamicchaos |
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2025-07-08T03:29:58Z |
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1837047899379204096 |
| fulltext |
ISSN 1562-6016. ВАНТ. 2013. №4(86) 284
PARADIGMS OF DYNAMIC CHAOS
V.A. Buts
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
E-mail: vbuts@ki pt.kharkov.ua
It is shown that a condition of occurrence of regimes with chaotic behavior of dynamic systems demands more
steadfast studying. In particular, it is shown that if we will take into account singular solutions the chaotic behavior
will be inherent also in systems with one degree of freedom. It is shown that linear systems can generate chaotic
dynamics. The question about necessity of local instability for realization of chaotic regimes is discussed.
PACS: 05.45.Ac
INTRODUCTION
As it is known, for realization of chaotic regimes in
considered dynamic system performance of following
conditions are necessary: 1. The system should have 1.5
or more degrees of freedom. 2. It should be nonlinear. 3.
In phase space local instability should develop. It is pos-
sible to name these three conditions as paradigms of
dynamic chaos. In this or that forms they are formulated
in all books and the reviews devoted to dynamic chaos
(see, for example, [1]).
Below we will discuss these conditions. Some words
about an essence of these conditions. They, certainly, all
are necessary for a certain class of dynamic systems.
Necessity for the phase space to have three or more di-
mension follows from the uniqueness theorem. Really in
two-dimensional phase space it is impossible to realize
mixing of integrated curves without intersection. Fur-
ther, for realize mixing phase trajectories in restricted
phase area it is necessary that they ran away from each
other, i.e. local instability is necessary. And, at last, as
the real system is described in restricted phase space,
nonlinearity is necessary for returning.
However we know that except usual solutions of the
differential equations there are singular solutions. In
points of singular solutions the uniqueness theorem is
broken. Therefore it is possible to expect that if we will
take into account these singular solutions the regimes
with chaotic behavior will be inherent also for dynamic
systems with one degree of freedom. In the second part
we will show that really such dynamics takes place.
Usually at studying of linear systems nobody is ori-
ented on studying of chaotic regimes. Really in linear
systems they are absent. However it is very frequent at
studying of linear systems that it is convenient to intro-
duce new dependent, and independent variables. At this
the linear mathematical model becomes nonlinear. The
well known example is the transition from the equations
of quantum mechanics to the equations of classical me-
chanics, and also the transition from the wave equations
to the equations of geometrical optics. In all these new
nonlinear systems the regimes with dynamic chaos are
possible. In the third partition we show that it is enough
general situation.
In the conclusion the results and their connection
with known results are discussed.
1. CHAOTIC DYNAMICS OF SYSTEMS
WITH “ONE” DEGREE OF FREEDOM
Let’s look at the first paradigm that regimes with
dynamic chaos are possible only in the dynamic systems
which number of degrees of freedom is more or equally
to 1.5. The cause of occurrence of this paradigm is that
fact that for realization of chaotic dynamics the mixing
of trajectories in phase space is necessary. The phase
space of systems with one degree of freedom represents
a plane. Owing the theorem of uniqueness on plane such
intersection can not to be. This conclusion, certainly, is
true. However it is true only to class of the differential
equations which have no singular solutions. As it is
known, in the presence of singular solutions on trajecto-
ries corresponding to these solutions, the uniqueness
theorem is broken. In this case the arguments formu-
lated above about impossibility of chaotic dynamics in
systems with one degree of freedom cease to work. Be-
low we will show that in systems with one degree of
freedom in the presence of singular solutions chaotic
regimes are possible. It is the main result of this part.
Clearly that singular solutions are characteristic and for
systems with a great number of degrees of freedom. In
these systems the regimes with dynamic chaos, which
are caused by presence of these singular solutions, also
are possible. However in these systems occurrence of
such regimes is not surprising. Therefore below we will
concentrate our attention on systems with one degree of
freedom. It is necessary to notice that, getting on a sin-
gular solution, in points where the uniqueness theorem
is broken, the system, in the general case, "does not
know" the further trajectory. The choice of the further
trajectory is defined by any external, even as much as
small perturbation. Presence of these perturbations for-
mally transforms system with one degree of freedom
into system with one and a half degree of freedom. We
will notice that the size of this perturbation can be as
much as small, up to what inevitably arise at numerical
research of studied model.
As a characteristic example we will consider dynam-
ics of system which is described by following equations:
0 1x x=& ;
2
1
1 0
0
0.5
2
xx x
x
⎛ ⎞
= − ⋅⎜ ⎟
⎝ ⎠
& . ` (1)
The phase portrait of system (1) is presented on pic-
ture 1. Integral curves in this case are circles:
( )2 2 2
0 1 0x R x Rϕ = − + − = (2)
and the circle centers settle down on an axis 1 0x = . Ra-
diuses of these circles are equal to distance of these cen-
tres to zero point ( 0 10; 0x x= = ). This point is the com-
mon for all circles. Besides, this point is a singular solu-
tion of system (1) (see below). The system (1) was ana-
lyzed numerically. Results are presented in drawings 1-
6. In the second and third drawings characteristic de-
pendences of a variable 0x on time are presented. Com-
ISSN 1562-6016. ВАНТ. 2013. №4(86) 285
paring Fig. 1 to Fig. 2 and 3, it is possible to make the
conclusion that a representative point, moving on one of
circles after crossing the zero point gets on other circle.
Fig. 1. Phase portrait of system (1)
Fig. 2. Time dependence of the variable 0x . Transitions
of representative points from one circle on another are
visible
Fig. 3. Time dependence of the variable 0x .
( 50 85t≤ < )
Fig. 4. Phase portrait
Fig. 5. Spectrum of the variable 0x
Рис. 6. Correlation function of the variable 0x
Transferring from one circle on another is well visi-
ble on a phase plane (Fig. 4). And, transferring from one
circle on another circle occur under the casual law.
Really, the spectral analysis of dynamics of system (1)
shows that spectrums of this dynamics are wide (see
Fig. 5), and correlation function enough quickly falls
down (Fig. 6).
Let's notice that casual the transitions from one cir-
cle to another at crossing zero point depend on accuracy
of the computation. Changes, for example, a step of
calculations change concrete character of these transi-
tions. However, as a whole statistically, dynamics re-
mains the same.
Let's show now that the zero point is a singular solu-
tion of system (1). At that, we will understand as singu-
lar solution those solutions on which points the unique-
ness theorem is broken. Really, this point belongs to
family of circles (2). The Same circles are integral
curves of system (1). These integral curves are conven-
ient rewrite in a such kind: 2
1 0 0/x x x R+ = . From a kind
of these integral curves follows that in vicinity of zero
point the Lipchitz conditions for system (1) are violated.
Really, the Lipchitz conditions for system (1) can be
written down in a kind:
( )
2 2
1 1
0 0 1 1
0 0
x x L x x x x
x x
− ≤ − + −
%
% %
%
, (3)
where L − positive constant.
In vicinity of zero point the left part of inequality (3)
can be estimated by size ( )R R−% , where R% and R ra-
diuses of two arbitrary circles. Generally, differences of
these radiuses can be arbitrary size. Thus, in zero point
the Lipchitz condition is not carried out, i.e. conditions
of the theorem of uniqueness for system (1) are violated.
Besides, taking partial derivative of function (2) on
R and equating it to zero, we find that really point
( 0 10; 0x x= = ) is a singular solution of system (1), and
also it is envelope line around integral curves.
The system (1) is not unique. It is possible to show
that for example dynamics of system which is described
by set of equations
0
0 1 1 1
dx x x x F
dt
γ= ⋅ + ⋅ ≡ , 2 41
1 0 0 2
dx x x x F
dt
γ= − − ⋅ ≡ (4)
also appears chaotic dynamics. Moreover, sets of such
systems can be constructed. We will show how such set
can be constructed. Let we have an integral curve which
is specified by the equation: ( )0 1, 0x xϕ = . Then the set
of equations for which this integrated curve will be as
integral, can be presented in a following kind:
ISSN 1562-6016. ВАНТ. 2013. №4(86) 286
( ) ( )
1
0
1 0 1 0
1
, , ,dx F x x M x x
dt x
ϕϕ ∂
= −
∂
;
( ) ( )
1
1
2 0 1 0
0
, , ,dx F x x M x x
dt x
ϕϕ ∂
= +
∂
, (5)
where 0 1( , , )sF x xϕ − arbitrary functions, which has
properties: 0 1(0, , ) 0sF x x = ; ( )
10 ,M x x − arbitrary func-
tion.
Using system (5), it is possible to construct the big
diversity of the dynamic systems possessing the neces-
sary properties. As an example we will consider a case
when integrated curves is the family of circles with ra-
dius R .
( )2 2 2
0 1 0x R x Rϕ = − + − = . (6)
Set of integral curves (6) are presented in Fig. 1. By
choosing of functions sF and M it is possible to
achieve an elimination of parameter R from set of equa-
tions (5). Really, let us choose these functions in a kind:
1 0F = ; ( )2 0 1,F f x xϕ= ⋅ ; ( )0 0 1M x f x x= − ⋅
here ( )0 1f x x − an arbitrary function. Substituting these
expressions in system (5), we will get set of equations in
which the parameter R is already excluded:
( )0
0 1 0 12dx x x f x x
dt
= ⋅ ⋅ ; ( ) ( )2 20
1 0 0 1 .
dx
x x f x x
dt
= − ⋅ (7)
Choosing function ( )0 1f x x in a kind
( )0 1 01 / 2f x x x= , we will get set of equations (1). Dy-
namics of this system is chaotic. It is presented in
Fig. 1-6.
Let's point out in one another possibility of construc-
tion of systems of the differential equations with one
degree of freedom which dynamics can display chaotic
character. Let we have two families of integral curves
( )1 0 1, 0x xϕ = and ( )2 0 1, 0x xϕ = . Using set of equations
(5), it is easy to find system of the differential equations
which solutions are these integral curves:
0 2 1
1 1 0 1 2 2 0 1
1 1
( , , , ) ( , , , )dx F x x t F x x t
dt x x
ϕ ϕϕ ϕ∂ ∂
= −
∂ ∂
,
1 2 1
1 1 0 1 2 2 0 1
0 0
( , , , ) ( , , , )dx F x x t F x x t
dt x x
ϕ ϕϕ ϕ∂ ∂
= − +
∂ ∂
, (8)
here 0 1( , , , )s sF x x tϕ − the arbitrary functions possessing
property 0 1(0, , , ) 0sF x x t = . Imposing on integral curves
and on function sF necessary conditions it is possible to
construct extensive enough set of the dynamic systems
possessing chaotic dynamics.
2. THE DYNAMIC CHAOS GENERATED
BY LINEAR SYSTEMS
In works [2-5] examples when linear dynamic sys-
tems generate chaotic dynamics of studied systems are
in detail enough considered. Below we will shortly de-
scribe the key moments of such consideration. Let for us
is available three linear connected oscillators. Two of
them are identical. Frequency of the third slightly dif-
fers from frequency of two others. Set of equations
which describe this dynamic system, it is possible to
present in a kind:
( )
0 0 1 1 2 2
1 1 1 0
2 2 2 01
q q q q
q q q
q q q
μ μ
μ
δ μ
+ = − −
+ = −
+ + = −
&&
&&
&&
, (9)
where ,dqq
dτ
≡& 1δ << , 1iμ << connection coeffi-
cients.
As coefficients of coupling are small, (9) it is con-
venient to search the solution of these equations in a
form:
( )( ) expi i iq A i tτ ω= . (10)
In the solution (10) dependence of complex ampli-
tudes ( )iA τ on time is caused by connection presence
between oscillators. In that case when this connection is
small, it is possible to consider that these amplitudes are
slowly changing functions. For a finding of these ampli-
tudes it is possible to use averaging method. As a result
we will get the following system of the linear truncated
equations for these amplitudes:
0 1 1 2 22 exp( )iA A A iμ μ δτ= − −&
1 1 02iA Aμ= −& 2 2 02 exp( )iA A iμ δτ= − −& . (11)
For the further analysis of dynamics of complex am-
plitudes ( )iA τ we will present them in a kind:
( ) ( ) exp( ( ))i i iA a iτ τ ϕ τ= , (12)
here ia , iϕ − real amplitudes and real phases.
The transformation (12) is key transformation for us.
It transforms linear set of equations (11) into the nonlin-
ear. We will notice that such transformation is widely
used in the physics, especially in the radiophysics. Sub-
stituting (12) in (11) for a finding of the real amplitudes
and phases, we will get the following set of equations:
( ) ( ) ( ) ( )0 1 1 2 2 1/ 2 sin / 2 sina a aμ μ= − ⋅ Φ − ⋅ ⋅ Φ& ,
( ) ( )1 1 0/ 2 sina aμ= ⋅ Φ& ( ) ( )2 2 0 1/ 2 sina aμ= ⋅ Φ& ,
( ) ( ) ( ) ( )0 1 2
1 2 1
1 0 0
/ 2 cos / 2 cosa a a
a a a
μ μ
⎛ ⎞ ⎛ ⎞
Φ = − Φ − Φ⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
& ,(13)
( ) ( ) ( ) ( )0 2 1
1 2 1 1
2 0 0
/ 2 cos / 2 cosa a a
a a a
μ μ δ
⎛ ⎞ ⎛ ⎞
Φ = − Φ − Φ +⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
& ,
where 1 0 1 2 0,ϕ ϕ ϕ ϕ δτΦ ≡ − Φ ≡ − + .
The set of equations (13) is the simplified system in
comparison with initial system (9). However this system
is nonlinear. Generally, dynamics of such system can be
chaotic. It is possible to show that its dynamics is simi-
lar to dynamics of two nonlinear connected oscillators.
The condition of a overlapping of these nonlinear reso-
nances is condition for regimes with chaotic dynamics
occurrence. This condition is simple and looks like:
( )1 2μ μ δ+ > . Here δ − is distance between nonlinear
resonances. More detailed results of investigation of
system (13), the analytical estimations of conditions of
dynamic chaos occurrence, and also results of numerical
researches are represented in [2-4].
2.1. QUANTUM SYSTEMS
The appearing the regimes with chaotic motion in
quantum systems are special interest. Below we will see
that such regimes are quite inherent to quantum sys-
tems. At this the key model is three-level systems, but
ISSN 1562-6016. ВАНТ. 2013. №4(86) 287
not two-level systems. Really set of equations (11) is
equivalent to system which is used for description of
quantum three-level system under influence on it the
perturbation. We will show it. We will consider quan-
tum system which is described by such Hamiltonian:
0 1
ˆ ˆ ˆ ( )H H H t= + . (14)
The second summand in the right part describes per-
turbation. Wave function of system (14) is subject to
Schrödinger equation. The solution of the Schrödinger
we will search in the form of a series on own functions
of unperturbed system:
( ) ( ) exp( )n n n
n
t A t i tψ ϕ ω= ⋅ ⋅∑ , (15)
where /n nEω = h ; 0
ˆn
n n ni H E
t
ϕ
ϕ ϕ
∂
= = ⋅
∂
h .
Let’s substitute (15) in Schrödinger equation and by
usual way we can get system of equations for complex
amplitudes nA :
( )n n m m
m
i A U t A⋅ = ⋅∑&h , (16)
where 1
ˆ ( ) exp[ ( ) / ]n m m n n mU H t i t E E dqϕ ϕ∗= ⋅ ⋅ ⋅ ⋅ ⋅ − ⋅∫ h .
Let’s consider the more simple case - the case of
harmonic perturbation: 1
ˆ ˆ( ) exp( )H t U i t= ⋅ Ω . Then the
matrix elements of interaction will get following ex-
pression:
exp{ [( ) / ]}n m n m n mU V i t E E= ⋅ ⋅ − +Ωh ,
ˆ
n m n mV U dqϕ ϕ∗= ⋅ ⋅∫ . (17)
Let's consider dynamics of three-level systems
( 0 , 1 , 2 ).
We will consider that the frequency of harmonic
perturbation and own energy of these levels satisfy to
such conditions:
1, 0m n= = , 1 0E EΩ = −h ; 2, 0m n= =
2 0( ) E EδΩ+ = −h δ << Ω . (18)
These relations specify in that fact that frequency of
external perturbation is resonant for transitions between
zero and the first levels, and energy of the third level is
slightly differs from energy of the second level. Using
these relations in system (16) it is possible to leave only
three equations:
0 01 1 02 2 exp( )i A V A V A i tδ⋅ ⋅ = + ⋅ ⋅ ⋅&h ;
1 10 0i A V A⋅ ⋅ =&h ; 2 20 0 exp( )i A V A i tδ⋅ ⋅ = ⋅ − ⋅ ⋅&h . (19)
Let matrix elements of interaction of the direct and
the backward transitions are equal ( 0 0 , ( 1;2)i iV V i= = ).
Then from (19) we can find the following connection
between squares of complex amplitudes nA :
( )2 2 2
0 1 2 2 0 22 sind A A A A A
d
μ δτ
τ
⎡ ⎤− − = ⋅ ⋅⎣ ⎦ . (20)
From this relation follows that if the third level coin-
cides with the second (two-level system, 0δ = ) the
system (19) has only one degree of freedom. Develop-
ment of dynamic chaos in such system is impossible.
Above we saw that the size δ defines distance between
nonlinear resonances. For the further analysis of dynam-
ics of the complex amplitudes ( )iA τ we will present
them in a kind:
( ) ( ) exp( ( ))i iA a iτ τ ϕ τ= . (21)
It’s clearly that dynamics of such quantum system
will be similar to dynamics of system (13), i.e. in it the
regimes with dynamic chaos are possible. It is necessary
to notice that these regimes can correspond to essen-
tially quantum values of the parameters (not quasi-
classical). Results of more detailed studying of dynam-
ics of such quantum system are contained in [5].
2.2. MORE COMPLICATED SYSTEMS
The examples considered above are simple enough.
We knew (in second part) an analytical kind of integral
curve these systems. This fact has allowed us to define
solutions and areas of phase space in which the unique-
ness theorem is not carried out. We have only three lin-
ear oscillators in third part. In more complex cases such
possibility arises seldom enough, therefore it would be
desirable to find more simple and general criteria which
will allow to define areas of phase space in which ex-
hibiting of elements of unpredictability is possible.
One of possibilities consists in measure use. Really,
let we will introduce of the measure of «an interval»
xΔr : ( )ip x xμΔ = ⋅Δ
r r . Here ( )ip xr − is density of prob-
ability that this representative point is inside this inter-
val. Let, as a result of time dynamics of considered sys-
tem, the point ixr passes in a point zr . Thus, we have
( )iz f x=
rr
− a image of a point ixr ; and the point ixr is
preimage of zr . The number of prototypes (preimages)
can be many. We will consider now certain "piece" zΔr :
[ ]/ 2; / 2z z z z− Δ + Δ
r r r r . The measure of this piece will
be defined now by the formula:
( ) ( )z i i
i
g z z p x xμΔ = ⋅Δ = ⋅Δ∑ r rr r . (22)
Here ( )g zr − density of probability to find of a repre-
sentative point in phase volume zΔr . From this formula
we find expression for density ( )g zr :
( )( ) ( ) i i
i
i i i
x p xg z p x
z J
Δ
= =
Δ∑ ∑
r r
rr
r . (23)
Here iJ − Jacobean transformations of new variables
through old variables.
The formula (23) practically is the Perron-Frobenius
formula. From this formula it is visible that in those
areas where Jacobean transformations will have any
singularity (for example, to go to infinity or to zero), is
possible to expect that relations between initial density
of probabilities and transformed − become uncertain.
These areas can be sources of chaotic motion.
CONCLUSIONS
Thus, abandoning from the theorem of uniqueness
essentially increase quantity of the dynamic systems
having regimes with chaotic behavior. However it is
necessary to remember that the chaotic behavior in this
case by their nature differs from dynamic chaos. This
chaos in dynamic systems is generated by not consid-
ered casual forces. These forces can be as much as
small, but they define a trajectory of integral curves
when they pass through ambiguity area. For this cause
we in the title of the second part used inverted commas
when we spoke about one degree of freedom. Actually
ISSN 1562-6016. ВАНТ. 2013. №4(86) 288
the behavior of such dynamic system with one degree of
freedom is defined by as much as small numerical fluc-
tuations. The real systems constructed on such model,
also will chaotically behave. It’s absolutely clearly that
the considered mechanism of occurrence of chaotic dy-
namics will be inherent also for systems with a great
number of degrees of freedom. In particular, dynamics
of system (13) can chaotically behave as a result of such
mechanism. Really, for values of the amplitudes which
are going to zero ( 0ia → ), there is an infringement of
the theorem of uniqueness. Therefore, generally, the
chaotic behavior of system (13) can be caused as occur-
rence gomoclinic structures (when nonlinear resonances
overlapped), and as a result of infringement of the theo-
rem of uniqueness.
To distinguish these two mechanisms of occurrence
of chaotic dynamics, in general cases, difficultly. How-
ever it can be made thus (by such way). At the chaotic
dynamics caused by dynamic processes, the combina-
tion of some functions for example, such as
( )( ) cos ( )i ia τ ϕ τ or ( )( )sin ( )i ia τ ϕ τ , will behave regu-
larly, despite fact that each of multipliers of this func-
tion behaves chaotically. Really, each of these combina-
tions, according to the formula (12) represents simply
real and imaginary part of the function which dynamics
is regular. This fact is similar to known result that the
combination of the functions representing integral, is
conserved, despite chaotic behaviors of everyone the
components, which are entering into this integral. If
chaotic dynamics is caused by infringement of the theo-
rem of uniqueness any such combination of functions
will remain chaotic because their dynamics it is defined
by fluctuations (though as much as small).
It is necessary to tell some words about local insta-
bility which are necessary for appearing of dynamic
chaos. If the chaotic behavior in system is caused by
infringement of the theorem of uniqueness, in general
cases, presence of such instability is not necessarily.
Really, let’s look at system (1). Its trajectories after
crossing the zero point are going away from each other.
However after it they again direct to the zero point and
the distance between them contracts.
Lyapunov's index calculated, for example, under the
Benetin schema (see, for example, [1]) will be equal to
zero.
REFERENCES
1. A.J. Liechtenberg, M.A. Lieberman. Regular and
Stochastic Motion / by Springer-Verlag. 1983, New
York inc., 499 p.
2. V.A. Buts. Chaotic dynamics linear systems // Elec-
tromagnetic Waves and Electron Systems. 2006,
v. 11, № 11, p. 65-70
3. V.A. Buts and. A.G. Nerukh. Elements chaotic dy-
namics in linear systems // The sixth International
Kharkov Symposium on Physics and Engineering of
Microwaves, Millimeter and Submillimeter waves
and workshop on Terahertz Technologies. Kharkov,
Ukraine, June 25-30, 2007, v. 1, p. 363-365.
4. V.A. Buts, A.G. Nerukh, N.N. Ruzhytska, D.A. Nerukh.
Wave Chaotic Behaviour Generated by Linear Sys-
tems // Opt Quant Electron. Springer. 2008, v. 40,
p. 587-601.
5. V.A. Buts. True quantum chaos // Problems of
Atomic Science and Technology. Sеries “Plasma
Physics” (14). 2008, № 5, p. 120-122.
Article received 13.05.2013.
ПАРАДИГМЫ ДИНАМИЧЕСКОГО ХАОСА
В.А. Буц
Показано, что условия возникновения режимов с хаотическим поведением динамических систем требуют
более пристального изучения. В частности, показано, что если принять в качестве решений особые решения,
то хаотическое поведение будет присуще и системам с одной степенью свободы. Показано, что линейные
системы могут порождать хаотическую динамику. Обсуждается вопрос о необходимости локальной неус-
тойчивости для реализации хаотических режимов.
ПАРАДИГМИ ДИНАМІЧНОГО ХАОСУ
В.О. Буц
Показано, що умови виникнення режимів з хаотичною поведінкою динамічних систем вимагають більш
пильного вивчення. Зокрема, показано, що якщо прийняти в якості рішень особливі рішення , то хаотична
поведінка буде притаманна й системам з одним ступенем свободи. Показано, що лінійні системи можуть
породжувати хаотичну динаміку. Обговорюється питання про необхідність локальної нестійкості для реалі-
зації хаотичних режимів.
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