Узагальнені сценарії переходу до хаосу в ідеальних динамічних системах
The implementation of a new scenario of transition to chaos in the classical Lorenz system has been discovered. Signs of the presence of an implementation of the generalized intermittency scenario for dynamic systems are described. Phase-parametric characteristics, Lyapunov characteristic exponents,...
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| author | Horchakov, Oleksii Shvets, Aleksandr |
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| description | The implementation of a new scenario of transition to chaos in the classical Lorenz system has been discovered. Signs of the presence of an implementation of the generalized intermittency scenario for dynamic systems are described. Phase-parametric characteristics, Lyapunov characteristic exponents, distributions of invariant measures, and Poincaré sections are constructed and analyzed in detail, which confirm the implementation of the generalized intermittency scenario in an ideal Lorenz system. |
| doi_str_mv | 10.20535/SRIT.2308-8893.2024.3.04 |
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Publisher IASA at the Igor Sikorsky Kyiv Polytechnic Institute, 2024
64 ISSN 1681–6048 System Research & Information Technologies, 2024, № 3
UDC 517.9:534.1
DOI: 10.20535/SRIT.2308-8893.2024.3.04
GENERALIZED SCENARIOS OF TRANSITION TO CHAOS
IN IDEAL DYNAMIC SYSTEMS
O.O. HORCHAKOV, A.YU. SHVETS
Abstract. The implementation of a new scenario of transition to chaos in the classi-
cal Lorenz system has been discovered. Signs of the presence of an implementation
of the generalized intermittency scenario for dynamic systems are described. Phase-
parametric characteristics, Lyapunov characteristic exponents, distributions of in-
variant measures, and Poincaré sections are constructed and analyzed in detail,
which confirm the implementation of the generalized intermittency scenario in an
ideal Lorenz system.
Keywords: ideal dynamic system, regular and chaotic attractors, generalized inter-
mittency scenario.
ITRODUCTION
The study of the ways in which deterministic chaos arises in dynamic systems
leads to the necessity to study scenarios of transition to chaos. The scenario of
transition to chaos is understood as a sequence of bifurcations, as a result of
which a regular steady state regime, for example a periodic one, is replaced by
chaotic steady state regime. If the dynamic system is dissipative, then such
steady-state regimes will be attractors of system. Despite the huge variety of dy-
namic systems, there are two types of scenarios for the transition from regular to
chaotic modes, which are implemented in almost all dynamic systems. This is the
Feigenbaum scenario, during which there is a transition from limit cycle to cha-
otic attractor through an infinite cascade of bifurcations of period doubling of
limit cycle [1; 2]. The second typical scenario is Manneville–Pomeau intermit-
tency [3–5]. When this scenario is implemented, the transition from limit cycle to
chaotic attractor occurs in one hard bifurcation. As a result of such bifurcation,
the limit cycle disappears and a chaotic attractor appears in the system. Moreover,
movement along the trajectories of a chaotic attractor consists of two unpredicta-
bly alternating phases — laminar and turbulent.
Recently, a number of new scenarios for transitions to chaos have been de-
scribed that generalize Manneville–Pomeau and Feigenbaum scenarios. An over-
view of these scenarios is given in [6]. One of these new scenarios is the so-called
generalized intermittency [7; 8]. In contrast to classical intermittency, this sce-
nario describes the transition from a chaotic (hyper-chaotic) attractor of one type
to a chaotic (hyper-chaotic) attractor of another type. The implementation of the
generalized intermittency scenario was discovered in a number of hydrodynamic,
electro-elastic and pendulum systems [7–12], both for traditional attractors and
for atypical maximal attractors.
Generalized scenarios of transition to chaos in ideal dynamic systems
Системні дослідження та інформаційні технології, 2024, № 3 65
FORMULATION OF THE PROBLEM
All implementations of the generalized intermittency scenario noted in the Intro-
duction were discovered only in dynamic systems that were not ideal according to
Sommerfeld–Kononeko or in systems with limited excitation. The fundamental
feature of systems with limited excitation is the limited power of the energy
source that excites the movement of a particular system. It is assumed that the
power of the excitation source is comparable to the power consumed by the sys-
tem itself. On the contrary, in so-called ideal systems no restrictions are imposed
on the power of the excitation source. The fundamentals of the theory of systems
with limited excitation were formulated in the monograph [13]. Various aspects
of the dynamic behavior of various systems with limited excitation were studied
in [14–18].
The main goal of this work is to demonstrate the possibility of implementing
a generalized intermittency scenario in classical ideal dynamic systems (systems
with unlimited excitation). The Lorenz system plays a unique role among ideal
systems in chaotic dynamics. In this system in 1963, a new type of steady-state
mode of a non-linear dynamic system, namely deterministic chaos, was firstly
discovered.
The mathematical model of the Lorenz system can be written as a system of
three differential equations [5; 19]:
)( xyx ;
xzyrxy ; (1)
xybzz ,
here zyx ,, are phase variables, and br ,, are some parameters. E.N. Lorenz
derived a system of equations (1) at studying convection in a liquid layer, which
is heated from below. In his work [19], the variable x characterizes the speed of
rotation of the convection shafts, and the variables zy, are responsible for the
horizontal and vertical temperature distribution, respectively. The parameter b is
determined by the geometry of the convection cell, is the Prandtl number,
which is the ratio of kinematic viscosity to the thermal diffusivity coefficient, and
r is the Rayleigh number, dimensionless quantity that determines the behavior of
the liquid under the influence of temperature.
Subsequently, it turned out that the Lorenz system of equations is applicable
not only to the problem of convection in a layer of liquid, but also to describe the
dynamics of many other physical systems. These include the convection problem
in a closed loop, dissipative oscillator with inertial excitation, etc. [5; 20].
METHODS OF NUMERICAL CALCULATIONS AND MAIN NUMERICAL
RESULTS
The system of differential equations (1) is nonlinear, therefore the main methods
for studying the dynamic behavior of such a system are various numerical meth-
ods. Such methods are: Runge–Kutta method (both with constant and variable
steps of numerical integration) for constructing phase portraits, Hénon method for
constructing Poincaré sections and phase-parametric characteristics, the Benettin–
O.O. Horchakov, A.Yu. Shvets
ISSN 1681–6048 System Research & Information Technologies, 2024, № 3 66
Galgani method for calculating Lyapunov characteristic exponents, computer al-
gorithms for encoding images with shades of different brightness for constructing
distributions of the natural invariant measure [5; 21; 22]. The general methodol-
ogy for conducting such studies is described in [23].
A characteristic feature of the Lorenz system is the existence of pairs of
symmetric attractors, which could be either regular or chaotic. When the bifurca-
tion parameter changes, a pair of separately existing symmetric attractors can
“merge” into one attractor. Then, with a further change of the bifurcation parame-
ter, a pair of attractors may arise again. Let us illustrate this process for some spe-
cific values of parameters of Lorenz system. Let the parameters of system (1) be
respectively equal
3
8
,1.203 br .
We take parameter b to be the same as in [19], and choose the Rayleigh
number sufficiently large. We choose the Prandtl number as the bifurcation
parameter.
Numerical calculations have shown that for values of the bifraction parame-
ter 109 , two symmetric attractors simultaneously exist in system (1). When
10 these attractors “merge” and only one attractor remains in the system. In
Fig. 1, a are shown the phase-parametric characteristics (bifurcation trees) of sys-
tem (1). The values of the bifurcation parameter are plotted along the abscissa
axis, and the values of the phase variable x are plotted along the ordinate axis.
These characteristics were constructed using the Hénon method, when the plane
170z was chosen as secant plane.
In Fig. 1, a, at 109 , the phase-parametric characteristic of one of the
attractors is plotted in gray, and the other is plotted in black. The individual
branches of bifurcation trees correspond to limit cycles of system (1). The points
at which new individual branches of bifurcation trees appear are clearly visible.
These points correspond to period doubling bifurcations of limit cycles. We note
that such of period doubling bifurcations occur for both symmetric limit cycles
for the same value of the bifurcation parameter . And, at last, the densely gray
and the densely black areas of the bifurcation trees correspond to the chaotic at-
tractors of the Lorenz system.
In Fig. 2, a are constracted the phase portraits of two chaotic attractors of the
Lorenz system at 10 . One of the attractors is plotted in gray, the other in
x
σ a
x
σ b
Fig. 1. Phase-parametric characteristics
Generalized scenarios of transition to chaos in ideal dynamic systems
Системні дослідження та інформаційні технології, 2024, № 3 67
black. Both chaotic attractors exist simultaneously, and each of them has its own
basin of attraction. The transition from limit cycles to chaos for both attractors
occurs according to Feigenbaum’s scenario [1; 2]. These chaotic attractors are
symmetric in the phase variables x and y . Both attractors have the same spec-
trum of Lyapunov characteristic exponents, and the maximum characteristic ex-
ponent must be positive. This is a necessary condition for the existence of chaos.
When 10 , the existing two chaotic attractors merge into one. As can be
seen from the consideration of the bifurcation tree, for all values of )15,10( ,
there is a single attractor in the system, which can be either a limit cycle or a cha-
otic attractor. The phase portrait of such a “united” chaotic attractor, constructed
at 01.10 , is shown in Fig. 2, b. Note that the spectrum of Lyapunov character-
istic exponents of this attractor is practically no different from the corresponding
spectrum of attractors are shown in Fig. 2, a.
Since the classical scenarios for the transition from regular attractors to cha-
otic attractors for the Lorenz system are quite well studied [4; 5; 21; 22], we will
focus on the implementation of a new scenario of generalized intermittency.
For this, consider an enlarged fragment of the phase-parametric characteris-
tic shown in Fig. 1, b. Let us study this phase-parametric characteristic in neigh-
borhood of the bifurcation point 764.12 . Both in left and right semi-
neighborhood of this bifurcation point in system (1) there are exist chaotic attrac-
tors (densely black areas of the bifurcation tree). However, as can be seen from
Fig. 1, b, area of the densely black region of localization of the points of the bi-
furcation tree to the right of the point 764.12 noticeably exceeds the corre-
sponding area to the left of the bifurcation point. This feature of the phase-
parametric characteristic is one of the signs of the implementation of the general-
ized intermittency scenario [6–12].
In Fig. 3, a is constracted the distribution of natural invariant measure over
the phase portrait of the attractor at value 76.12 . The maximum Lyapunov
characteristic exponent of this attractor, calculated using the method of Benettin
et al., is positive and equal 289.11 . The positivity of the maximum Lyapunov
exponent is another evidence of chaotic nature of this attractor.
z z
x x
y y
a b
Fig. 2. Phase portrait of chaotic attractors at )( 10 a and ) (01.10 b
O.O. Horchakov, A.Yu. Shvets
ISSN 1681–6048 System Research & Information Technologies, 2024, № 3 68
At the value of bifurcation parameter σ increases, one hard bifurcation hap-
pens at point 764.12 . After which the chaotic attractor that existed at
764.12 disappears and a chaotic attractor of a different type appears in the
system (1). The transition from a chaotic attractor of one type to a chaotic attrac-
tor of another type occurs according to the scenario of generalized intermittency.
In Fig. 3, b is constructed distribution of the natural invariant measure over phase
portrait of the new chaotic attractor at 775.12 . The attractor that arises after
passing the bifurcation point has a significantly larger positive maximum Lya-
punov characteristic exponent. This characteristic exponent is equal 71.11 .
Such a noticeable increase in the value of the maximum characteristic exponent
after passing the bifurcation point is another sign of the implementation of the
scenario of generalized intermittency. The movement of the trajectory along the
phase portrait of the arising chaotic attractor consists of two phases, so called,
rough laminar and turbulent. Rough laminar phase corresponds to chaotic wander-
ings of the trajectory in the region of localization of the disappeared chaotic at-
tractor (densely black area in Fig. 3, b). For turbulent phase corresponds to depar-
tures of the trajectory to more distant regions of the phase space (gray points on
the distribution of the invariant measure in Fig. 3, b). Such transitions from the
rough laminar phase of motion to the turbulent phase occur an infinite number of
times. Note that the moment of transition from the rough laminar phase to the tur-
bulent phase and the moment of return of the trajectory from the turbulent phase
to the rough laminar phase are unpredictable. Also note, the time during which the
trajectory is in a rough laminar phase significantly exceeds duration of time, in
which trajectory in turbulent phase.
The scenario of generalized intermittency at transitions “a chaotic attractor
of one type a chaotic attractor of another type” can also be identified by studying
Poincaré sections. In Fig. 4, a, b, using the Hénon method, Poincaré sections of
chaotic attractors at 76.12 and at 775.12 are constructed. Here 200z
plane is selected as the secant plane. Both Poincaré sections have a quasi-ribbon
structure and represent chaotic sets of individual points. Note that the quasi-
ribbon structure is typical for chaotic attractors of the Lorenz system. As can be
seen from Fig. 4, b, the structure of the Poincaré section of this attractor includes
all fragments of the Poincaré section of the chaotic attractor shown in Fig. 4, a.
z z
x x
y y
a b
Fig. 3. Distribution of the natural invariant measure at )( 76.12 a and )( 775.12 b
Generalized scenarios of transition to chaos in ideal dynamic systems
Системні дослідження та інформаційні технології, 2024, № 3 69
These fragments of the Poincaré section of the attractor at 76.12 form a rough
laminar phase of the attractor at 775.12 . Accordingly, in the structure of the
Poincaré section of the attractor at 12.775 , new points additionally appear,
which form the turbulent phase.
The implementation of the generalized intermittency scenario can be found
when studying bifurcations in other parameters, in particular for r (the Rayleigh
number in the classical work of Lorenz [19]). Let the parameters of system (1) be
respectively equal
.
3
8
, 10 b
And now we choose the Rayleigh number r as the bifurcation parameter.
The technique for performing numerical calculations is similar to that used when
studying bifurcations with respect to the parameter .
In Fig. 5, a are constructed parts of two phase-parametric characteristics of
system (1). These phase-parametric characteristics, “gray” and “black,” corre-
spond to two attractors existing in the system. These two attractors exist in the
system simultaneously. Moreover, the only possible situation is when either two
Fig. 4. Poincaré section at )(76.12 a and )( 775.12 b
y y
x xa b
x x
r r
a b
Fig. 5. Phase-parametric characteristics
O.O. Horchakov, A.Yu. Shvets
ISSN 1681–6048 System Research & Information Technologies, 2024, № 3 70
limit cycles or two chaotic attractors simultaneously exist in the system. A careful
study of Fig. 5, a allows us to identify a number of transitions to chaos according
to Feigenbaum’s scenario, which occur as the parameter r decreases. In turn, as
the parameter r increases, transitions to chaos occur according to the Manne-
ville–Pomeau scenario. We especially emphasize that all bifurcations of these two
attractors occur at the same values of the parameter r .
However, we are primarily interested in the implementation of the general-
ized intermittency scenario. To do this, we will construct a significantly enlarged
fragment of the “black” phase-parametric characteristic. Such a fragment is
shown in Fig. 5, b. At decreasing the values of parameter r in Fig. 5, b is clear
visible the process of branching of individual branches of the bifurcation tree. As
is known, such a process describes the transition to chaos through a cascade of
bifurcations of period doubling of the limit cycles. After the appearance of a cha-
otic attractor, with a further decrease in the values of the parameter r , a suffi-
ciently small window of periodicity arises in which the cascade of bifurcations of
period doubling again repeats and a chaotic attractor arises again.
After the appearance of a chaotic attractor, with a further decrease in the
values of the parameter r , a sufficiently small window of periodicity appears in
which the period-doubling bifurcation sequence again repeats and a chaotic at-
tractor appears again. Finally, when 885.213 r , one of the signs of the imple-
mentation of the generalized intermittency scenario can be seen on the phase-
parametric characteristic. Namely, a significant increase in the region of chaos in
the phase-parametric characteristic. We studied bifurcations only for one of exist-
ing chaotic attractors of the system. However, the same processes can be seen
when studying an enlarged fragment of the “gray” phase-parametric characteris-
tic, that correspond to symmetric chaotic attractor.
In Fig. 6, a the distributions of the natural invariant measure are plotted over
the phase portraits of a pair, symmetric with respect to x and y, attractors at
89.213r . These symmetrical attractors are shown in black and gray. In Fig. 6,
b shows the distributions of the natural invariant measure over the phase portraits
of another pair of symmetric attractors at 88.213r . In this figure, each of the
symmetrical attractors is also shown in black and gray. Both pairs of constructed
attractors are chaotic attractors, since they have positive maximum Lyapunov
characteristic exponents. As shown by numerical calculations carried out using
the method of Benettin et al., each of the chaotic attractors shown in Fig. 6, a
have the same maximum exponent 18.01 . Accordingly, the second pair of
chaotic attractors has maximum exponent 28.01 . So, when passing the bifur-
cation point 885.213r , the second sign of the implementation of the general-
ized intermittency scenario is observed, namely a significant increase of the max-
imum Lyapunov characteristic exponent.
For a more visual description of the phases of generalized intermittency in
Fig. 6, c, d, respectively, show significantly enlarged fragments of the distribu-
tions of invariant measures for one of the attractors before and immediately after
the bifurcation point. Note that for the second of the symmetric attractors we will
have a similar description of the phases of generalized intermittency. The rough
laminar phase in Fig. 6, d form thicker black areas, the shape of which is similar
to the areas of distribution of invariant measures in Fig. 6, c. The turbulent phase
is formed by individual points between the “turns” of the rough laminar phase. As
Generalized scenarios of transition to chaos in ideal dynamic systems
Системні дослідження та інформаційні технології, 2024, № 3 71
in the case of studying bifurcations with respect to parameter, a picture typical
for generalized intermittency can also be found when studying the similarities of
Poincaré sections before and after the bifurcation point 885.213r .
CONCLUSIONS
Thus, for the classical ideal dynamic Lorenz system, the implementation of transi-
tions from “a chaotic attractor of one type to a chaotic attractor of another type”
was established according to the scenario of generalized intermittency. It is shown
that such a scenario can be realized both in the case of existence of a single attrac-
tor in the Lorenz system, and in the case of existence of a pair of symmetric at-
tractors.
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O.O. Horchakov, A.Yu. Shvets
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Received 02.04.2024
INFORMATION ON THE ARTICLE
Aleksandr Yu. Shvets, ORCID: 0000-0003-0330-5136, National Technical University of
Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Ukraine, e-mail: aleksandrshvet-
skpi@gmail.com
Oleksii O. Horchakov, ORCID: 0009-0006-3664-8812, Institute of Mathematics NAS of
Ukraine, Ukraine, e-mail: jarkafres@gmail.com
УЗАГАЛЬНЕНІ СЦЕНАРІЇ ПЕРЕХОДУ ДО ХАОСУ В ІДЕАЛЬНИХ
ДИНАМІЧНИХ СИСТЕМАХ / О.О. Горчаков, О.Ю. Швець
Анотація. Виявлено реалізацію нового сценарію переходу до хаосу в класич-
ній системі Лоренца. Описано ознаки наявності реалізації сценарію узагальне-
ної переміжності для динамічних систем. Побудовано та детально проаналізо-
вано фазово-параметричні характеристики, показники Ляпунова, розподіли
інваріантних мір та перерізи Пуанкаре, які підтверджують реалізацію сценарію
узагальненої переміжності в ідеальній системі Лоренца.
Ключові слова: ідеальна динамічна система, регулярний і хаотичний атракто-
ри, сценарій узагальненої переміжності.
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| last_indexed | 2025-07-17T10:28:28Z |
| publishDate | 2024 |
| publisher | The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" |
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| spelling | journaliasakpiua-article-3026652024-11-16T18:06:34Z Generalized scenarios of transition to chaos in ideal dynamic systems Узагальнені сценарії переходу до хаосу в ідеальних динамічних системах Horchakov, Oleksii Shvets, Aleksandr ідеальна динамічна система регулярний і хаотичний атрактори сценарій узагальненої переміжності ideal dynamic system regular and chaotic attractors generalized intermittency scenario The implementation of a new scenario of transition to chaos in the classical Lorenz system has been discovered. Signs of the presence of an implementation of the generalized intermittency scenario for dynamic systems are described. Phase-parametric characteristics, Lyapunov characteristic exponents, distributions of invariant measures, and Poincaré sections are constructed and analyzed in detail, which confirm the implementation of the generalized intermittency scenario in an ideal Lorenz system. Виявлено реалізацію нового сценарію переходу до хаосу в класичній системі Лоренца. Описано ознаки наявності реалізації сценарію узагальненої переміжності для динамічних систем. Побудовано та детально проаналізовано фазово-параметричні характеристики, показники Ляпунова, розподіли інваріантних мір та перерізи Пуанкаре, які підтверджують реалізацію сценарію узагальненої переміжності в ідеальній системі Лоренца. The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" 2024-09-28 Article Article application/pdf https://journal.iasa.kpi.ua/article/view/302665 10.20535/SRIT.2308-8893.2024.3.04 System research and information technologies; No. 3 (2024); 64-73 Системные исследования и информационные технологии; № 3 (2024); 64-73 Системні дослідження та інформаційні технології; № 3 (2024); 64-73 2308-8893 1681-6048 en https://journal.iasa.kpi.ua/article/view/302665/306060 |
| spellingShingle | ідеальна динамічна система регулярний і хаотичний атрактори сценарій узагальненої переміжності Horchakov, Oleksii Shvets, Aleksandr Узагальнені сценарії переходу до хаосу в ідеальних динамічних системах |
| title | Узагальнені сценарії переходу до хаосу в ідеальних динамічних системах |
| title_alt | Generalized scenarios of transition to chaos in ideal dynamic systems |
| title_full | Узагальнені сценарії переходу до хаосу в ідеальних динамічних системах |
| title_fullStr | Узагальнені сценарії переходу до хаосу в ідеальних динамічних системах |
| title_full_unstemmed | Узагальнені сценарії переходу до хаосу в ідеальних динамічних системах |
| title_short | Узагальнені сценарії переходу до хаосу в ідеальних динамічних системах |
| title_sort | узагальнені сценарії переходу до хаосу в ідеальних динамічних системах |
| topic | ідеальна динамічна система регулярний і хаотичний атрактори сценарій узагальненої переміжності |
| topic_facet | ідеальна динамічна система регулярний і хаотичний атрактори сценарій узагальненої переміжності ideal dynamic system regular and chaotic attractors generalized intermittency scenario |
| url | https://journal.iasa.kpi.ua/article/view/302665 |
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