Analytical treatment of the chaotic behaviour of the deterministic pseudolinear map: decay of correlations and stability of periodical systems
It was received the rate of chaotization for pseudolinear mapping. It was shown that the rate of chaotization is proportional to the dimension of the phase space and maximal Lyapunov exponent. It was shown also that the problem of the rate of chaotization is not correct and must be regularized. It w...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2001
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| Cite this: | Analytical treatment of the chaotic behaviour of the deterministic pseudolinear map: decay of correlations and stability of periodical systems / V.P. Demutsky, V.M. Rashkovan
 // Вопросы атомной науки и техники. — 2001. — № 6. — С. 238-244. — Бібліогр.: 16 назв. — англ. |
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| author | Demutsky, V.P. Rashkovan, V.M. |
| author_facet | Demutsky, V.P. Rashkovan, V.M. |
| citation_txt | Analytical treatment of the chaotic behaviour of the deterministic pseudolinear map: decay of correlations and stability of periodical systems / V.P. Demutsky, V.M. Rashkovan
 // Вопросы атомной науки и техники. — 2001. — № 6. — С. 238-244. — Бібліогр.: 16 назв. — англ. |
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| description | It was received the rate of chaotization for pseudolinear mapping. It was shown that the rate of chaotization is proportional to the dimension of the phase space and maximal Lyapunov exponent. It was shown also that the problem of the rate of chaotization is not correct and must be regularized. It was investigated also the two-dimensional dynamical system stability in the case of two and three step periodical standard maps. The stability conditions were obtained. The analytical expressions of the bounders of stability regions were written. It had been shown that the summary region of stability is expanded, when compared to the case of the one-step map, but the number of stable points decreases.
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ANALYTICAL TREATMENT OF THE CHAOTIC BEHAVIOUR
OF THE DETERMINISTIC PSEUDOLINEAR MAP: DECAY OF
CORRELATIONS AND STABILITY OF PERIODICAL SYSTEMS
V.P. Demutsky
Kharkov National University, Kharkov 61077, 4 Svoboda Sq., Ukraine
V.M. Rashkovan
National Aerospace University, Kharkov 61070, 17 Chkalova St., Ukraine
It was received the rate of chaotization for pseudolinear mapping. It was shown that the rate of chaotization is
proportional to the dimension of the phase space and maximal Lyapunov exponent. It was shown also that the
problem of the rate of chaotization is not correct and must be regularized. It was investigated also the two-
dimensional dynamical system stability in the case of two and three step periodical standard maps. The stability
conditions were obtained. The analytical expressions of the bounders of stability regions were written. It had been
shown that the summary region of stability is expanded, when compared to the case of the one-step map, but the
number of stable points decreases.
PACS: 05.45.Ac
INTRODUCTION
The problem of chaotization of deterministic
mechanical system arose after the kinetic theory was
built. This theory aims at a mechanical explanation of
thermodynamical processes [1,2]. It was found some
decades ago that a dynamical chaos arises also in
dynamical systems with a small number of degrees of
freedom [3-5]. Soon it turned out that the dynamical
chaos is a rule rather than an exception [6].
Since the equations considered are nonlinear and
nonintegrable, the analytical results were obtained only
in a few cases. The majority investigations of the
dynamical chaos rely upon numerical simulation.
However, computer calculations are badly suited for the
treatment of the most interesting final stage of the
chaotization. Besides this when analysing experimental
data, evaluating of perspectives and searching of
possible ways of optimization of various devices, the
numerical calculations considerably lose relative to the
analytical ones in clearness.
Therefore many important questions leaved
unanswered:
1. What is the law of the decay of correlation? Is it an
exponential law ( )tα−exp (only in this case we speak
about chaotization) or is not it a more slow law
( )γβ t−exp ( )1<γ ? Maybe this law is ever power δ−t
, ( )0>δ ?
2. What is the rate of chaotization α ? Is α proportional
to the maximal Lyapunov exponent or is it proportional
to the KS - entropy? Is α proportional to the dimension
of unstable subspace or is it proportional to the
dimension of all phase space?
3. Does the rate of chaotization α depend on the initial
indeterminacy or on the uncertainty of the measuring
device? Does α tend to infinity when the indeterminacy
of the initial state or the uncertainty of the measuring
device tends to zero? If it does so what is the law of this
tendency?
4. Does the rate of chaotization depend on the shape of
the initial and the final region in the phase space? If it
depends what complementary condition must be
imposed on the dynamical system in order to the rate of
chaotization was independent from the shape of the
initial and the final regions?
The aim of this review is to answer above questions
in the case of a model of the dynamical system, which
can be solved exactly.
DECAY OF CORRELATION’S AND MIXING
We consider the dynamical system, the state of
which at the time t is described by a d-dimensional
vector ( )tx
( ) ( ) ( ) ( )( )t
d
tt xxxtx ,....,, 21=
. (1)
If there is no chaos the state vector ( )tx
changes with
time deterministically. The chaos means that the point
in phase space ( )tx moves in a very complicated
manner. Therefore, deterministic description is
incorrect. If the chaos takes place the state of the system
must be characterized not by the single state vector ( )tx
, but by the distribution function (the probability density
in the phase space) ( )txf ,
. One of the manifestations of
the chaos consists in lose of the memory about the
initial state. Moreover, if the phase volume is
conserved, every distribution function tends as ∞→t
to the sole distribution function ( )xµ (equilibrium
distribution).
We see that with the chaos the notions of the
necessity and the randomness in some sense interchange
their places. Namely, without the chaos the behaviour of
the system is random in the sense that it is determined
238 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 6, p. 238-244.
by the outer relative to the system initial state ( )0x
). On
the contrary, with the chaos every quadratically
integrable initial distribution ( )0,xf
tends (in the
metric 2L ) as ∞→t to the same limiting distribution
( )x
µ .
The measure of the memory about the initial state is
the correlation
( ) ( ) ( ) ( ) ( )0,,0,, xgtxfxgtxftC
−≡ . (2)
Here ( )f x and ( )g x are any two functions and the
angle brackets mean the average over the phase space
( ) ( ) ∫
Γ
Γ
Γ
= dtx
V
tx ),(1,
ϕϕ , (3)
( ) ddxdxdxxd ...21
µ=Γ , (4)
( ) ( )∫
Γ
=Γ ddxdxdxxV ...21
µ . (5)
Thus one of the manifestations of the randomness
consists in the decay of all correlations:
( ) 0lim =
∞→
tC
t . (6)
With randomness there occurs a mixing of the phase
space. For the simplicity we limit ourselves to the case
when the equilibrium distribution is homogeneous:
( ) constx =
µ . (7)
In this case the mixing means that after elapse of
sufficiently long time t the probability ( )tP f ,Γ that
the state vector x gets in an arbitrary region fΓ of the
phase space is proportional to its volume ( )fV Γ
( ) ( )
( )Γ
Γ
=Γ
∞→ V
V
tP f
ft
,lim . (8)
The region fΓ plays a role of a measuring device,
which determines the degree of the mixing. An error of
the measuring device is characterized by the quantity
( )fV Γ . This error is the less the smaller is ( )fV Γ .
The mixing may be defined also as the decay of
correlations. Indeed the probability ( )tP f ,Γ , which is
contained in the formula (8) equals:
( ) ( ) ( ) xdxtxftP ff
Γ=Γ ∫ ,,, χ (9)
Here ( )fx Γχ ,
is the characteristic function of the
region fΓ :
( )
Γ∈
=Γ
caseoppositethein
xif
x f
f 0
,1
,
χ (10)
We note that the integral over the whole phase space of
the characteristic function equals to the volume ( )fV Γ .
Besides this the normalizing condition is fulfilled:
( )∫
Γ
= 1, dxtxf
. (11)
Taking into account for this remarks the condition of
mixing (8) may be represented as decay of correlations
( ) ( ) ( ) ( ){ } 0,,,,lim =Γ−Γ⋅
∞→ fft
xtxfxtxf χχ . (12)
If correlations damp by an exponential law
( ) ( )ttC ⋅− αexp~ ( ∞→t ) (13)
the mixing is called a chaotization.
SETS OF ZERO MEASURE
We note that the singular (i.e. δ - type) distribution
( ) ( )0xxxf
−= δ (14)
which corresponds to the exactly determined initial state
tends to nothing. In this case correlations do not decay
and there is no mixing. But such distribution does not
correspond to any physical situation. In practice the
initial distribution ( )0,xf
differs from zero in some
initial region iΓ (initial indeterminacy).
"...in the problem of Cauchy the solution must be
unique. It must be fully determined by initial conditions
and consequently quite predictable. How an
indeterminacy can arise? It turns out that the posing of
the Cauchy problem is not legitimate while chaotic
movements are investigated. This problem never
corresponds to the conditions of an experiment (natural
or numerical) because the initial conditions in principle
cannot be absolutely exact. Therefore there is a reason
to formulate the problem in the statistical language" [7].
In the ergodic theory in order to avoid patalogical
situations in statistical considerations one neglects sets
of measure zero. It means in particular that isolated
points in the phase space are not considered. In other
words, one treats almost all sets in the phase space. In
formulations of ergodic theory theorems there are words
"almost everywhere".
"Mathematicians who do not like the speculations in
which the expressions "almost all" and "neglecting sets
of zero measure" occur, may be objected that this is the
only way to mathematically interpret what "as a rule"
takes place in the nature" [8].
"The most important principle of the theory of
measure is a neglect of sets of zero measure. In
accordance with this principle, spaces with measure and
their endomorphisms must be studied only disregarding
the sets of zero measure, or, as people say, "module 0"
(mod 0) ... often the addition "mod 0" is implied but is
not included in the wording obviously" [9].
On the same ground the final region fΓ cannot be a
single point: it must have some positive measure. In
other words the uncertainty of the measure device
cannot equal to zero.
We may say that the phase space consists not of
points but rather of infinitesimal cells. This causes the
drastical change of properties of the elements of the
phase space: a point has precisely defined place and has
no shape. On the contrary, a cell has definite shape.
With the lapse of time the shape of the cell changes.
According to the Liouville's theorem, the area of the cell
remains constant under the natural motion of the
system. It becomes more and more complicated.
Eventually it resembles a spider or a sponge. Thus the
cell has an "age". In other words, in the theory there is
the time arrow. It means that the paradoxes of
239
Loschmidt [10, p. 152] and Zermelo [10, p. 155; 17]
disappear.
THE PSEUDOLINEAR MAPPING
As the time elapses the phase point ( )tx intersects many
times the Poincare section
( ) ( )tt xTx
=+ 1 (15)
where ( )tx
is the phase point which corresponds to the
t -th intersection of the Poincare section and T is some
nonlinear operator. Instead of the investigation of ( )tx
we shall consider the sequence of the phase points
( ) ( ) ,..., 21 xx . In other words we will pass on to discrete
time t .
The dynamical chaos is intimately connected with
two effects:
1) the exponential growth of infinitesimal perturbations,
2) the non linear limitation of the perturbation growth
when perturbation increases to finite value.
The first effect is described by linearized map
( ) ( )txx jij
t
i α=+ 1 . (16)
Here ijα are given constants and it is implied the
repeated indices are summarized from 1 to d .
In order to consider the second effect we limit ourselves
for simplicity to the stochastic acceleration and the
stochastic diffusion. In the action-angle variables ( )θ,I
they are described by the standard map [4]:
( ) ( ) ( )( )ttt KII π θ2sin1 +=+ (17)
( ) ( ) ( )11 ++ += ttt Iθθ . (18)
As θ stands under the sign of sinus we can take the
fractional part of (18)
( ) ( ) ( ){ }11 ++ += ttt Iθθ . (19)
Thus our phase space is the cylinder
( ) 1,, +≡ θθθI . (20)
Now we replace the equation (17) by the more
tractable one
( ) { } ( )( ){ }ttt KII π θ2sin1 +=+ (21)
Doing so, we identify the points 0=I and 1=I . In
other words, we transform the phase space from the
cylinder (20) into the torus
( ) 1,1,, +≡+≡ III θθθ . (22)
The solution of the modified system (19), (21) is less
stochastic than the solution of the previous system.
Indeed, if, for example, the value of I jumps erratically
from zero to one, there and back, its fractional part will
be always equal to 0.
"It may appear that it is a very special class of
dynamical systems. But it is not so: many important
dynamical systems turn out to be nonergodic. Their
phase space splits into invariant tori" [11, p. 66].
Then we replace )2sin( π θ by π θ2 for simplicity.
Thus we obtain the pseudolinear map:
( ) ( ) ( ){ }ttt KII θπ21 +=+ , (23)
( ) ( ) ( ) ( ){ }ttt KI θπθ 211 ++=+ , (24)
The pseudo linear map looks as linear. But it is
essentially non linear, as it does not admit the
transformation
θθ ccII →→ , . (25)
The pseudo linear map (23), (24) is a very crude
approximation but it preserves the simplicity of the
linearized map and, at the same time, takes into account
the non linear limitation of the perturbation growth. In
this approximation one can obtain a series of exact
results. In the d-dimensional case the pseudo linear map
has the form
( ) ( ){ }t
jij
t
i xx α=+ 1 , ( )dji ,...,2,1, = . (26)
We further assume the coefficients ijα in these
equations to be integers. If ijα in (26) are integers the
limiting distribution ( )∞,xf
will be homogeneous:
( ) ( ) 1, =≡∞ xxf
µ . (27)
When
+
=
k
k
ij 11
1
α (28)
the pseudo linear mapping was investigated in the work
[7]. If 1=k , the transformation (28) is named the
"Arnold's cat". The pseudo linear map arises also in the
one-dimensional theory of a crystal [6].
THE GENERAL SOLUTION
Let the initial distribution be
( )
Γ∈= ∏
=
caseoppositein
xwhenxf ij
d
j
i
j
0
1
1 ε
. (29)
We assume that the initial region iΓ is determined by
the relations
i
j
i
jj
i
j
i
j xxx εε
2
1
2
1 +≤≤− , ( )dj ,...,2,1= (30)
and the final region fΓ is
f
j
f
jj
f
j
f
j xxx εε
2
1
2
1 +≤≤− , ( )dj ,...,2,1= (31)
Then the correlation equals [12]
( ) ( ) ,,...,1
,...,2,1 1
1∑ ∏
∞
− ∞= =π
=
dmmm
d
j
jdd KmmFtC
(32)
( )
( )
( )
απ
α
απ
= ∑
∑
∑
=
−
=
−
=
−
d
k
f
jkj
t
kd
k
kj
t
k
d
k
f
jkj
t
k
j xm
m
xmi
K
1
1
1 sin
2exp
.
Here ( )dmmF ,...,1 are the Fourier coefficients of the
initial distribution ( )xf
:
( ) ( ) ( )∏
=
−
⋅
=
d
j
i
jji
jj
i
jj
dd xim
m
m
mmF
1
1 2exp
sin1,..., π
ε
επ
π
, (33)
240
and the stroke at the sign of summation means that the
term 021 === dmmm is omitted, ( ) ij
ta − are the
elements of the matrix tT − .
With arbitrary iΓ and fΓ the evaluation of the d-
fold sum in (32) is embarrassing. Therefore we
investigate two particular cases:
1) A crude initial state and a fine measuring device
( ) ( )
=≤≤
+≤≤−
=
caseoppositein
djx
xxxifxf
j
iiii
i
,0
,...,2,10
,
2
1
2
1
,1 111 εε
ε
, (34)
fΓ is determined by the relations (31). In this case the
expansion of the initial distribution function into a
Furier series collapses from the d -dimensional sum
into the one-dimensional:
( ) ( ) ( )∑
∞
− ∞=
=
m
imxmFxf 12exp π
. (35)
According to the well-known formula
( ) ( )xTfxTf 1−= (36)
the evolution of correlation is determined by the inverse
transformation 1−T . So
( ) ( ) ( )( )∑ ∏
∞
− ∞=
−
=
=
m
jj
t
d
j
t ximmFxfT 1
1
2exp απ
. (37)
Therefore the correlation in this case equals to [26]
( )
( )
( )[ ]
( ) ( )i
d
j
if
jj
t
dm
d
l
f
lll
t
m
md
j
j
tid
mxxm
m
K
mKtC
επαπ
εαπ
αεπ
sin2cos1
sin2
1
111
11
1
1
1
1
−=
=
∑
∏∑
∏
=
−
+
=
−
∞
=
=
−+
.
(38)
When ∞→t
( ) ( )dLtconsttC min1 exp −⋅≤ (39)
where minL is the minimal Lyapunov's exponent of the
linearized system (16).
2) A fine initial state and a crude measuring device
iΓ is defined by the formula (30) and fΓ is defined by
the relations:
ffff xxx εε
2
1
2
1
111 +≤≤− , 10 ≤≤ jx , ( )dj ,...,2,1= . (40)
The calculations analogous to the preceding case lead to
the following expression for the correlation [26]:
( )
( )
( )[ ]
( ) ( )f
d
k
f
d
i
kdk
t
dm
d
j
i
jdj
t
d
m
md
l
dl
ti
l
d
mxxm
m
K
xmKtC
επαπ
απ
αεπ
sin2cos1
sin2
1
1
11
1
1
2
−=
=
∑
∏∑
∏
=
+
==
=
−+
. (41)
When ∞→t
( ) ( )dtLconsttC max2 exp −⋅≤ , (42)
here maxL is the maximal Lyapunov's exponent.
We see that the rate of chaotization in both cases is
proportional to the dimension of the phase space and
has nothing to do with the dimension of the unstable
subspace. If the initial state is crude and the measuring
device is fine, the rate of chaotization is proportional to
the minL , whereas on the opposite case the rate of
chaotization is proportional to maxL . We note that the
rate of chaotization is nothing to do with the KC-
entropy. The later equals to the sum of all positive
Lyapunov's exponents
∑
>
=
0iL
iLK . (43)
We note further that the rate of chaotization depends on
the choice of ( )0,xf
and ( )fx Γ,χ . This dependence
disappears if
minmax LL = . (44)
The last condition is fulfilled for a Hamilton system.
TWO-DIMENSIONAL CASE
In the case 2=d , ∞→t the correlation 1C takes the
form [12]
( ) ( )
{ } { } { }( ) { } { } { }( )[ ]22
max
1222
2
1
231231
2exp
3
ggguuui
iLi
aaaaaaA
AtL
e
LshC
+−−+−=
−
−
= ∑
±
− ααε (45)
Here ga are various sums of the form
( ) ( ) ftftiP 212122 2
1
2
1
2
1 εαεαε ±±± (46)
with even number of minuses, and ua - the same sums
with odd number of minuses, and
( ) ( ) iftft xxxP 1212122 −α−α= . (47)
There is analogous expression for ( )tC2 .
Note that in the two-dimensional case
minmax LL −= (48)
Formula (45) shows that the decay of correlation is
not exponential, but rather erratic. In contrast to the
thermodynamics the correlation (45) always approaches
zero nonmonotonically ever after elapse of an arbitrary
long interval of time.
However, the majoranta of correlation is exponential
( ) ( ) ( )tL
e
LshtCSup
Li max
1222
max
2
1 2exp
39
4
max
−
−
=
− ααε . (49)
We note that the multiplier before the exponent in
(45) remains finite when 0→ε , ∞→t .
CONTINUOUS FUNCTIONS
Up to now we assumed the functions ( )0,xf
and
( )xχ to be piece-wise constant and discontinuous. Let
241
us now consider more complicated function ( )xχ .
Namely
( ) .
0
3
1
6
1
3
1
10
6
10
11
211
≤≤
−
≤≤≤≤
=χ
caseoppositein
xifx
xxifpx
x
(50)
As to the function ( )0,xf
it is defined by the expression
( )
≤≤≤≤=
caseoppositein
xxifxf
0
10,
2
10,10, 21
(51)
Then for the Arnold's cat the asymptotic value of the
correlation when ∞→t is
( ) ( ) ,1 32 LtLt
t
BeepAtCSup −− +−= (52)
here
2
53ln +=L .
We see that when p differs from unity, i.e. when the
function ( )xχ is discontinuous, the rate of the decay of
correlation is L2 as before. On the other hand, if 1=p ,
i.e. if the function ( )xχ is continuous, the decay rate of
the correlation equals L3 . This means that infinitesimal
change of the coefficient p provides an essential change
of the correlation. We see that the problem of the
calculation of the correlation is incorrect.
REGULARIZATION OF THE PROBLEM
A mathematical problem is called to be correct in the
sense of Hadamard if following conditions are satisfied:
1. The solution exists.
2. The solution is unique.
3. The solution continuously depends on initial data.
As we have seen in the preceding section the
problem of determining of the rate of decay of
correlation is incorrect. We conjecture that in this case
the correctness must be understood in the sense of
Tikhonov [13] rather than in the sense of Hadamard. In
other words the problem, which is incorrect in the sense
of Hadamard must be regularized. The regularization
consists in the reduction of the class of admissible
functions. For example, the problem of Cauchy for
the Laplace equation becomes correct, if the solution
is searched in the class of bounded functions [14].
In the case of the problem of the decay of
correlations the regularization is based on the fact
that almost all chosen by chance functions are
discontinuous (except of the set of zero measure).
This reasoning is analogous to the conclusion about
incommensurability of frequencies in a conditionally
- periodical motion [15].
According to this we divide the phase space
into cells and specify the number of particles in each cell
[2].
In this case the rate of chaotization for two-
dimensional phase space is the doubled maximal
Lyapunov exponent L . More detailed specifying the
functions ( )0,xf
and ( )xχ influences only on a
multiplier before the exponent.
STABILITY OF A PERIODICAL SYSTEM
We will investigate a periodical structure in a period
of τ time:
NTTTT ⋅⋅⋅=τ 21 (53)
Here N is a number of stages (steps) in that period.
As it is well known (see, for example, [16]), the
question of periodical systems stability is determined by
means of investigating of eigen value signification ρ of
the matrix of monodromy M̂ that satisfies the equation
01ˆ2 =+⋅⋅ρ−ρ MSp , (54)
(with significant regard of the phase volume (3)). Here
M̂ – is a matrix of monodromy that displaces the
solution of equation (54) in a period of time
τ=⋅ jjij xxM 0ˆ . (55)
It is supposed [16] that
φ=ρ ie (56)
and for the phase φ we have the equation
MSp ˆ
2
1cos ⋅=φ . (57)
Real meaning of φ corresponds to the conditions of
system stability, e.g. the condition of stability is:
2ˆ2 ≤⋅≤− MSp . (58)
For determination of the value MSp ˆ⋅ we will find two
solutions for the system (16). The first solution ( )It
jx is
for such initial conditions is
( ) 10
1 =
I
x , ( ) 00
2 =
I
x . (59)
Then, from (55) follows that
( ) 111 M̂x
I
=τ (60)
The second decisión for the system (16) ( ) IIt
jx made up
for the following initial conditions
( ) 00
1 =
II
x , ( ) 10
2 =
II
x . (61)
Then, from (55) follows that
( ) 222 M̂x
II
=τ . (62)
Consequently
( ) ( ) III
xxMSp ττ +=⋅ 21
ˆ , (63)
and the periodical structure conditions of stability can
be written as
( ) ( ) 22 21 ≤+≤− ττ III
xx (64)
It is recalled that in the case of the standard map [4,6],
the conditions of stability have the following aspect
04 ≤≤− k . (65)
242
TWO STEP PERIODICITY
We will now investigate the periodic map T with a
period of time τ , which consists of two successive
standard maps
21 TTT ⋅=τ , (66)
here
( )2,1
11
1
=
+
= q
k
k
T
q
q
q . (67)
The conditions of stability in this case are
1
2
111 2121 ≤+++≤− kkkk . (68)
We have to note that the inequality (68) is symmetrical
in relation to the substitution of the members 21 kk ⇔ ,
e.g. in relation to the order of 1T and 2T in (66).
Fig. 1. The stability regions
In the plane ( )21kk in Fig. 1, the stability regions of
the periodical structure (66), determined by the
inequality (68), are pointed out in shaded lines. [The
square formed by the points (0,0), (-4,0); (0,-4), (-4,4)
represents the stability region of nonperioducal
structure].
In our opinion, interesting results were obtained:
some stable points standing in the boundary of an
nonperiodical structure [for example, the points (-3,
-1); (-1, -3) become unstable, but at the same time, some
unstable points [for instance, the points (1, -1); (-5, -3);
(3-, -5); (-1,1) become stable. In other words, in the case
of a periodical map, there’s a change in the stability
conditions in the standard map.
THREE STEP PERIODICITY MAP
This time, we’ll investigate the periodical map τT (
τ – is any period), which consists of three steps. In
order to get obvious results of stability regions on the
plane, we’ll consider only one variant, when two steps
coincide, i.e.
211 TTTT ⋅⋅=τ (69)
Following the mentioned scheme of the calculations
MSp ˆ⋅ we can obtain the stability conditions for the
variant (69):
( )( ) 11313
2
! 1
2
111
2 ≤+++++≤− kkkk
k
(70)
The stability regions on the plane ( )12 , kk are indicated
in Fig. 2 as the shaded regions
The boundary curves of the stability regions
conformable to the sign of the equality (70) are given by
the following equations:
1
22
1
2 +
+−=
k
k (the curves A), (71)
3
22
1
2 +
+−=
k
k (the curves B). (72)
Fig. 2. The regions of stability
Therefore, the horizontal rect-line 22 −=k and two
vertical rect-lines 11 −=k and 31 −=k represent the
boundary lines. Notice that the obtained results do not
depend on the step order in (69).
In Fig. 2, we can see that two points, just (-3,-4); and
(-1, 0) standing on the boundary of stability square (65)
of two inperiodical maps, become stable in the case of
periodical maps with the same steps T1 and T2.
Therefore, the other two points, just (-3, -5) and (-1, 1)
standing outside of the stability square of the
inperiodical map stability, fall on the bounds of the
stability regions of the three step periodical map.
In our opinion, it’s interesting the decay of the one
stability region (65) in the three stability regions in the
case of the three-step periodicity. (We could regard that
in the case of two-step periodicity, such regions were
only two). The points (-3,3) and (-1,1) of the bounders
of the stability regions crossing are interesting too. The
investigation of the region around these two points in
the case of small disturbances qk in the ( )2,11 =lT
shows that if the significance of disturbances falls into
in the stability regions, it means that the additions in the
right part of (57) decreases the module of its
signification, e.g. to convert the equalities (58) into
inequalities.
243
1
1
2
2
3
3
4 5-1
-1
-2
-2
-3
-3
-4
-4
-5
-5
-6
-6
-7
K2
CONCLUSIONS
As this paper is divided into two sections two groups
of conclusions are made. The first group of conclusions
concerns the decay of correlations. They are:
1. The decay of correlations is going in a
complicated non-exponential way.
2. The majoranta of the correlation function is an
exponent.
3. The problem of the rate of chaotization is
incorrect in the sense of Hadamard: the rate of the decay
of correlation depends on the smoothness of the initial
and the final functions.
4. We conjecture that the algorithm of regularization
of this problem consists in dividing phase space into
cells and then specifying a number of particles in each
cell.
5. The rate of chaotization is proportional to the
dimension of the phase space.
6. 1n general the rate of chaotization essentially
depends on the initial and the final functions. This
dependence disappears if the system is invariant under
the time inversion. In that case the rate of chaotization is
proportional to the maximal Lyapunov exponent.
The second group includes the conclusions about
stability of periodical mapping. The two-dimensional
dynamical system stability was investigated in the case
of two- and three-step periodical maps. The conditions
of stability were obtained. The influence of periodicity
of standard maps on the stability of chaotic dynamical
systems was investigated. In the case of a two step
periodical mapping there’s a change in the stability
conditions in the standard map: some stable points
standing in the boundary of an inperiodical structure
become unstable, but at the same time, some unstable
points become stable. In the case of the three-step
periodicity it’s interesting the decay of the one stability
region (65) in to the three stability regions.
However, the most interesting, to our opinion, is:
internal points from the squire of stability (65) of one-
step mapping, there are the point (-2,-2) in Fig. 1 and
the point (-1,-1) and (-3,-3) in Fig. 2 do not correspond
to absolute stability. But these point correspond to one
step mapping with k=1,-2 and -3 there are, for the stable
value of k. Therefore the stability of these points
depends from the meaning of perturbations (which
appear in any moment). So we can make such
conclusion: mapping (28) with integer k has no points
of absolute stability.
REFERENCES
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5. G.M. Zaslavsky. Chaos in dynamical
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6. G.M. Zaslavsky, R.Z. Sagdeev. Vvedenie v
nelineynuyu fiziku. 1988, Moscow: “Fizmatlit”,
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244
Kharkov National University, Kharkov 61077, 4 Svoboda Sq., Ukraine
Decay of CORRELATION’S and mixing
SETS OF zero measure
THE PSEUDOLINEAR MAPPING
The general solution
Two-dimensional case
Continuous functions
Regularization of the problem
Stability of a periodical system
Two Step Periodicity
Three Step Periodicity Map
Conclusions
References
|
| id | nasplib_isofts_kiev_ua-123456789-79897 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T18:19:43Z |
| publishDate | 2001 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Demutsky, V.P. Rashkovan, V.M. 2015-04-06T16:38:43Z 2015-04-06T16:38:43Z 2001 Analytical treatment of the chaotic behaviour of the deterministic pseudolinear map: decay of correlations and stability of periodical systems / V.P. Demutsky, V.M. Rashkovan
 // Вопросы атомной науки и техники. — 2001. — № 6. — С. 238-244. — Бібліогр.: 16 назв. — англ. 1562-6016 PACS: 05.45.Ac https://nasplib.isofts.kiev.ua/handle/123456789/79897 It was received the rate of chaotization for pseudolinear mapping. It was shown that the rate of chaotization is proportional to the dimension of the phase space and maximal Lyapunov exponent. It was shown also that the problem of the rate of chaotization is not correct and must be regularized. It was investigated also the two-dimensional dynamical system stability in the case of two and three step periodical standard maps. The stability conditions were obtained. The analytical expressions of the bounders of stability regions were written. It had been shown that the summary region of stability is expanded, when compared to the case of the one-step map, but the number of stable points decreases. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Anomalous diffusion, fractals, and chaos Analytical treatment of the chaotic behaviour of the deterministic pseudolinear map: decay of correlations and stability of periodical systems Analytical treatment of the chaotic behaviour of the deterministic pseudolinear map: decay of correlations and stability of periodical systems Article published earlier |
| spellingShingle | Analytical treatment of the chaotic behaviour of the deterministic pseudolinear map: decay of correlations and stability of periodical systems Demutsky, V.P. Rashkovan, V.M. Anomalous diffusion, fractals, and chaos |
| title | Analytical treatment of the chaotic behaviour of the deterministic pseudolinear map: decay of correlations and stability of periodical systems |
| title_alt | Analytical treatment of the chaotic behaviour of the deterministic pseudolinear map: decay of correlations and stability of periodical systems |
| title_full | Analytical treatment of the chaotic behaviour of the deterministic pseudolinear map: decay of correlations and stability of periodical systems |
| title_fullStr | Analytical treatment of the chaotic behaviour of the deterministic pseudolinear map: decay of correlations and stability of periodical systems |
| title_full_unstemmed | Analytical treatment of the chaotic behaviour of the deterministic pseudolinear map: decay of correlations and stability of periodical systems |
| title_short | Analytical treatment of the chaotic behaviour of the deterministic pseudolinear map: decay of correlations and stability of periodical systems |
| title_sort | analytical treatment of the chaotic behaviour of the deterministic pseudolinear map: decay of correlations and stability of periodical systems |
| topic | Anomalous diffusion, fractals, and chaos |
| topic_facet | Anomalous diffusion, fractals, and chaos |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/79897 |
| work_keys_str_mv | AT demutskyvp analyticaltreatmentofthechaoticbehaviourofthedeterministicpseudolinearmapdecayofcorrelationsandstabilityofperiodicalsystems AT rashkovanvm analyticaltreatmentofthechaoticbehaviourofthedeterministicpseudolinearmapdecayofcorrelationsandstabilityofperiodicalsystems |