Реалізація сценарію узагальненої переміжності в динамічній системі Ресслера
The realization of novel scenario involving transitions between different types of chaotic attractors is investigated for the Rössler system. Characteristic features indicative of the presence of generalized intermittency scenario in this system are identified. The properties of “chaos–chaos” transi...
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| description | The realization of novel scenario involving transitions between different types of chaotic attractors is investigated for the Rössler system. Characteristic features indicative of the presence of generalized intermittency scenario in this system are identified. The properties of “chaos–chaos” transitions following the generalized intermittency scenario are analyzed in detail based on phase-parametric characteristics, Lyapunov characteristic exponents, phase portraits, and Poincaré sections. |
| doi_str_mv | 10.20535/SRIT.2308-8893.2026.1.07 |
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O.O. Horchakov, A.Yu. Shvets, 2026
Системні дослідження та інформаційні технології, 2026, № 1 103
МАТЕМАТИЧНІ МЕТОДИ, МОДЕЛІ, ПРОБЛЕМИ
І ТЕХНОЛОГІЇ ДОСЛІДЖЕННЯ СКЛАДНИХ
СИСТЕМ
UDC 517.9:534.1
DOI: 10.20535/SRIT.2308-8893.2026.1.07
IMPLEMENTATION OF A GENERALIZED INTERMITTENCY
SCENARIO IN THE RÖSSLER DYNAMICAL SYSTEM
O.O. HORCHAKOV, A.YU. SHVETS
Abstract. The realization of novel scenario involving transitions between different
types of chaotic attractors is investigated for the Rössler system. Characteristic
features indicative of the presence of generalized intermittency scenario in this system
are identified. The properties of “chaos–chaos” transitions following the generalized
intermittency scenario are analyzed in detail based on phase-parametric
characteristics, Lyapunov characteristic exponents, phase portraits, and Poincaré
sections.
Keywords: ideal dynamical system, regular and chaotic attractors, generalized
intermittency scenario.
INTODUCTION
Scenarios of generalized intermittency describe the transition from a chaotic
attractor of one type to a chaotic attractor of another type. Such scenarios were
initially discovered in the study of non-ideal Sommerfeld–Kononenko-type
dynamical systems [1, 2]. These scenarios generalize the Manneville–Pomeau
scenarios [3, 4] and, in some cases, represent combinations of the Feigenbaum
[5, 6] and Manneville–Pomeau scenarios. The review paper [7] presents
implementations of various versions of the generalized intermittency scenario in
non-ideal pendulum, hydrodynamic, and electroelastic systems.
Moreover, transitions of the “chaos–chaos” type following the generalized
intermittency scenario have also been identified in non-isolated invariant sets, the
so-called maximal attractors. Strictly speaking, these sets do not qualify as
attractors in the classical sense. Nevertheless, even for such atypical attracting
structures, the generalized intermittency scenario can still be observed [8, 9].
OBJECTIVE AND METHODOLOGY OF THE STUDY
It was established in [10, 11] that various types of the generalized intermittency
scenario can be realized in the ideal Lorenz dynamical system. The objective of the
present study is to provide numerical evidence supporting the realization of the
generalized intermittency scenario in such classical dynamical system as the
Rössler system. The investigation employs standard techniques of chaotic
dynamics, including the Runge–Kutta method for constructing phase portraits of
TIÄC
O.O. Horchakov, A.Yu. Shvets
ISSN 1681–6048 System Research & Information Technologies, 2026, № 1 104
attractors [12], the Benettin algorithm for computing the maximal Lyapunov
exponent [13, 14], the Hénon method for constructing Poincaré sections [15], and
a computational technique based on color-shaded encoding for visualizing the
distribution of the invariant measure over the phase portrait of the attractor [16].
The detailed methodology for applying the above-mentioned numerical methods
and algorithms is described in [16–18].
RÖSSLER SYSTEM
In [20], a nonlinear system of three differential equations was considered: 𝑥 = 𝑥 − 𝑥 ; 𝑥 = 𝑥 + 𝑒𝑥 ; (1) 𝑥 = 𝑓 + 𝑥 (𝑥 − 𝑚),
Here 𝑥 , 𝑥 , 𝑥 are phase variables, and 𝑒, 𝑓, 𝑚 are system parameters. This
system later became known as the Rössler system. It should be noted that the first
two equations of system (1) are linear, while the quadratic nonlinearity appears only
in the third equation. Rössler proposed this system purely heuristically, without
relying on any physical assumptions in its derivation. His goal was to construct a
simple deterministic third-order system of differential equations exhibiting highly
complex chaotic dynamics. Over time, Rössler revisited the analysis of system (1)
in his later works [20–22]. Today, both the Rössler and Lorenz systems [23] are
widely recognized as canonical examples of chaotic dynamics in low-dimensional
deterministic systems.
Assume that the parameters of system (1) are 𝑒 = 0.2, 𝑓 = 0.2 and choose
the parameter 𝑚 as the bifurcation parameter. In Fig. 1, a, the phase–parameter
characteristic of the system is shown, constructed using the Hénon method, as the
parameter 𝑚 varies within the interval 5.45 ≤ 𝑚 ≤ 5.65. Here the plane 𝑥 = 0
is chosen as the secant plane. Individual lines (branches) of the phase–parameter
characteristic (the bifurcation tree) correspond to the limit cycles of system (1),
while the densely black regions of the bifurcation tree correspond to the chaotic
attractors of the system. An analysis of the phase–parameter characteristic shows
that in the range 5.56 < 𝑚 < 5.59, system (1) undergoes transitions from limit
cycles to chaotic attractors. These transitions occur via cascades of period-doubling
bifurcations of limit cycles, that is, in full accordance with the Feigenbaum scenario
[5, 6]. Once a chaotic attractor appears, it persists over a certain interval as the
parameter m increases. When m reaches a certain critical value, the chaotic attractor
disappears and a limit cycle again becomes the attractor of system (1). As m
increases further, another transition from a regular regime to a chaotic one occurs
according to the Feigenbaum scenario. It should be noted that short intervals of
limit cycle existence are referred to as periodicity windows.
It should be noted that a positive maximal Lyapunov exponent is a necessary
condition for the chaotic nature of a steady-state regime. Fig. 1, b shows the graph
of the dependence of the maximal nonzero Lyapunov exponent λ on the
bifurcation parameter m. This graph was constructed using the algorithm proposed
by Benettin et al. [13, 14]. Positive values of the Lyapunov exponent correspond to
intervals of the parameter m for which chaotic attractors exist in system (1). The
“drops” of the Lyapunov exponent graph into the region of negative values
Implementation of a generalized intermittency scenario in the Rössler dynamical system
Системні дослідження та інформаційні технології, 2026, № 1 105
correspond to the periodicity windows observed in Fig. 1, a. The most interesting
region of the phase–parameter characteristic (Fig. 1, a) is the neighborhood of the
point 𝑚 ≈ 5.585. As seen in Fig. 1, a, in the right-side neighborhood of 𝑚 ≈ 5.585, there is a significant increase in the area of the densely black region on
the phase–parameter diagram. As established in [7, 11], such an increase in the
corresponding area indicates the realization of the generalized intermittency
scenario. Various versions of this scenario are described in [7–11]. Another
indication of the realization of the generalized intermittency scenario is a noticeable
increase in the maximum Lyapunov exponent at 𝑚 > 5.585. We can see such
increasement in Fig. 1, b.
a b
c d
Fig. 1. Phase-parametric characteristic – a; maximal non-zero Lyapunov exponent – b;
distribution of the natural invariant measure at 𝑚 = 5.58 – c; at 𝑚 = 5.59 – d
Let us now examine in more detail the realization of the generalized
intermittency scenario in the Rössler system by analyzing the distributions of
natural invariant measures and Poincaré sections.
In Fig. 1, c is shown the distribution of the invariant measure over the phase
portrait of the chaotic attractor at 𝑚 = 5.58. As the parameter m increases, a hard
bifurcation occurs in system (1), as a result of which the existing chaotic attractor
disappears and a new type of chaotic attractor emerges. The distribution of the
O.O. Horchakov, A.Yu. Shvets
ISSN 1681–6048 System Research & Information Technologies, 2026, № 1 106
invariant measure over the phase portrait of this new chaotic attractor, constructed
at m = 5.59, is shown in Fig. 1, d. The distributions of the invariant measure were
constructed using the algorithm of computer encoding in shades of black [16, 17].
The trajectory motion along the new chaotic attractor exhibits phase alternation
between two phase – a coarse-grain (rough) laminar phase and a turbulent phase.
The coarse-grain laminar phase corresponds to chaotic wanderings of the trajectory
in the region of localization of the disappeared chaotic attractor (dense black region
in Fig. 1, c). At an unpredictable moment in time, the trajectory leaves the
localization region of the vanished chaotic attractor and “escapes” to more distant
areas of the phase space (gray points in Fig. 1, d). Such motions correspond to the
turbulent phase of intermittency. Alternations between the coarse-grained laminar
phase and the turbulent phase are observed an infinite number times. The transition
time from one phase to another is also unpredictable. On average, the duration of
the coarse-grained laminar phase exceeds that of the turbulent phase. This process
fully corresponds to the scenario of generalized intermittency [7, 10, 11]. The
scenario of generalized intermittency can also be identified by analyzing the
Poincaré sections. In Fig. 2, the Poincaré sections of chaotic attractors at 𝑚 = 5.58
and 𝑚 = 5.59 are constructed using the Hénon method. Both sections exhibit a
quasi-ribbon structure and represent chaotic sets of discrete points. It is worth
noting that such a quasi-ribbon structure is characteristic of chaotic attractors in the
Rössler system. As shown in Fig. 2, b, the structure of the Poincaré section at 𝑚 =5.59 contains all the fragments present in the Poincaré section of the chaotic
attractor at 𝑚 = 5.58 (Fig. 2, a).
a b
Fig. 2. Poincaré sections at 𝑚 = 5.58 – a; at 𝑚 = 5.59 – b
These fragments form the coarse-grained laminar phase of the attractor at 𝑚 = 5.59. Accordingly, new points appear in the Poincaré section at 𝑚 = 5.59,
corresponding to the turbulent phase.
Let us now consider the bifurcations in the Rössler system as the parameter m
varies within the interval (16, 18.5). The values of the parameters e and f remain
unchanged. As before, using the methods of Hénon, Benettin, and computer-based
color coding, we construct a series of dynamic characteristics of the Rössler system.
Thus, in Fig. 3, a, the phase–parameter characteristic of the Rössler system is
Implementation of a generalized intermittency scenario in the Rössler dynamical system
Системні дослідження та інформаційні технології, 2026, № 1 107
presented. The constructed bifurcation tree provides a clear representation of the
types of attractors in system (1). The individual branches of the bifurcation tree
correspond to limit cycles, while the densely black regions of the tree represent
chaotic attractors. Moreover, this figure makes it possible to identify transition
scenarios, including both “limit cycle-to-chaos” and “chaos-to-chaos” transitions.
The constructed bifurcation tree demonstrates a symmetry in the transitions to
chaos, both with increasing and decreasing values of the parameter 𝑚. As 𝑚
increases, starting from 𝑚 = 16.7, in the system begins an infinite cascade of
period-doubling bifurcations of limit cycles, followed by the emergence of a
chaotic attractor with a relatively small localization region in the phase space. This
represents a transition to chaos following the Feigenbaum scenario. A similar
scenario is observed as m decreases, beginning from 𝑚 = 18.05. Particular
attention should also be paid to two bifurcation points: 𝑚 ≈ 17.35 and 𝑚 ≈ 17.795.
a b
c d
Fig. 3. Phase-parametric characteristic – a; Maximal non-zero Lyapunov exponent – b;
and projections of distribution of the natural invariant measure at 𝑚 = 17.8 – c; at 𝑚 = 17.79 – d
O.O. Horchakov, A.Yu. Shvets
ISSN 1681–6048 System Research & Information Technologies, 2026, № 1 108
In the right-hand neighborhood of 𝑚 ≈ 17.35 (and the left-hand
neighborhood of 𝑚 ≈ 17.795), a significant increase in the area of the densely
black chaotic region in Fig. 3, a is observed, indicating the realization of a
generalized intermittency scenario of the transition from one type of chaotic
attractor to another.
In addition, two bifurcation points are clearly visible at 𝑚 ≈ 16.6 and 𝑚 ≈18.1. As the system passes through these points, a “limit cycle–chaos” transition
occurs following the Pomeau–Manneville scenario. In Fig. 3, b, the graph of the
dependence of the maximal nonzero Lyapunov exponent on the bifurcation
parameter m is presented. As seen from the graph, for 𝑚 > 17.35 and 𝑚 < 17.795,
the value of the maximal Lyapunov exponent nearly doubles. This increase is
further evidence of the realization of a generalized intermittency scenario in the
Rössler system.
Finally, let us consider the realization of the generalized intermittency
scenario through the phase portraits of chaotic attractors of different types.
In Fig. 3, c, the projection of the invariant measure distribution for the chaotic
attractor at m = 17.8 is shown, while Fig. 3, d presents the projection of the invariant
measure distribution for the chaotic attractor at m = 17.79. As the value of the
parameter m decreases, the chaotic attractor that existed in the right-hand
neighborhood of the bifurcation point m = 17.795 disappears, and for
m < 17.795, a new type of chaotic attractor emerges. The motion of trajectories on
this new attractor includes two phases, clearly identifiable in Fig. 3, d: a coarse-
grain laminar phase and a turbulent phase. In the coarse-grain laminar phase (the
densely black fragment in Fig. 3, d), the trajectory performs chaotic wandering in
a neighborhood of the phase-space localization region of the attractor that existed
for m > 17.795. The turbulent phase (the gray fragments in Fig. 3, d) corresponds
to the trajectory’s excursions into more distant regions of the phase space.
Similarly, the generalized intermittency scenario can be illustrated through
Poincaré sections, as was done in Fig. 2. It should be noted that, in contrast to the
previously analyzed case, the transition to chaos through the generalized
intermittency scenario can occur both with increasing and decreasing values of the
parameter m.
The implementation of the generalized intermittency scenario can also be
observed in other regions of the parameter space of the Rössler system. Let us assume
that e = 0.2 and m = 17.4, while the bifurcation parameter is chosen to be f.
We will investigate the dynamical behavior of system (1) within the range 0.2308 < 𝑓 < 0.2311. For these parameter values, the Rössler system has two
coexisting attractors, each possessing its own basin of attraction. Fig. 4, a, b
presents the phase–parameter characteristics of two different attractors constructed
using the Hénon method.
As before, individual branches of the bifurcation trees correspond to limit
cycles, while the densely black regions represent chaotic attractors. Despite a
certain similarity between these phase–parameter characteristics, it is clearly
seen—by examining the intervals of variation of the coordinate 𝑥 —that the
corresponding attractors are localized in different regions of the phase space.
Let us now focus exclusively on the realization of the generalized
intermittency scenario. As noted earlier, an indicator of this scenario is a significant
increase in the area of the densely black (chaotic) regions on the phase–parameter
Implementation of a generalized intermittency scenario in the Rössler dynamical system
Системні дослідження та інформаційні технології, 2026, № 1 109
characteristic. Such increases in the areas of the densely black regions can be
observed on both phase–parameter characteristics. This indicates the possibility of
a transition “chaotic attractor of one kind → chaotic attractor of another kind”
according to the generalized intermittency scenario.
a b
c d
Fig. 4. Phase-parametric characteristics – a, b, c; fragment of distribution of
invariant measure at 𝑓 = 0.23082 – d
Let us examine this scenario in more detail using one of the coexisting
attractors as an example. Fig. 4, c shows a fragment of the phase–parameter
characteristic from Fig. 4, a. The enlarged scale in Fig. 4, c makes it possible to
identify the bifurcation point f = 0.23085, at which a “chaos → chaos” transition
occurs according to the generalized intermittency scenario. In Fig. 4, d is shown an
enlarged fragment of the distribution of the invariant measure over the phase
portrait of the attractor at f = 0.23082. This attractor appears as the parameter
f decreases immediately after the bifurcation point f ≈ 0.23085. The use of the
enlarged scale makes it possible to clearly visualize the features of this distribution.
One can distinguish a coarse-grain laminar phase of the trajectory (the densely
black region in the figure) and a turbulent phase (the gray-shaded areas).
Let us emphasize once again that the coarse-grain laminar phase almost
coincides with the region of localization in phase space of the chaotic attractor that
O.O. Horchakov, A.Yu. Shvets
ISSN 1681–6048 System Research & Information Technologies, 2026, № 1 110
exists for f > 0.23085 and disappears after the bifurcation point is passed. Another
confirmation of the generalized intermittency scenario is a noticeable increase in
the value of the maximal Lyapunov exponent. Specifically, for the chaotic attractor
at f = 0.23086, the maximal Lyapunov exponent is 𝜆 = 0.005, while for the
chaotic attractor at f = 0.23082, it increases to 𝜆 = 0.010.
CONCLUSIONS
Thus, the generalized intermittency scenario, previously identified in non-ideal
dynamical systems, is also realized in ideal dynamical systems such as the classical
ideal Rössler system. Future research will focus on identifying the realization of
other types of the generalized intermittency scenario in various ideal dynamical
systems.
ACKNOWLEDGMENTS
This work was supported by a grant from the Simons Foundation International
(SFI-PD-Ukraine-00014586, O.O. Horchakov).
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Received 19.06.2025
INFORMATION ON THE ARTICLE
Aleksandr Yu. Shvets, ORCID: 0000-0003-0330-5136, National Technical University of
Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Ukraine, email: aleksandrshvetskpi@gmail.com
Oleksii O. Horchakov, ORCID: 0009-0006-3664-8812, Institute of Mathematics NAS of
Ukraine, Ukraine, e-mail: o.horchakov@imath.kiev.ua
РЕАЛІЗАЦІЯ СЦЕНАРІЮ УЗАГАЛЬНЕНОЇ ПЕРЕМІЖНОСТІ В ДИНАМІЧНІЙ
СИСТЕМІ РЕССЛЕРА / О.О. Горчаков, О.Ю. Швець
Анотація. Досліджено реалізацію нового сценарію переходу між різними
типами хаотичних атракторів для системи Ресслера. Виявлено характерні
ознаки, що вказують на наявність сценарію узагальненої переміжності в цій
системі. Властивості переходів типу «хаос–хаос» за сценарієм узагальненої
переміжності детально проаналізовано на основі фазо-параметричних
характеристик, ляпуновських характеристичних показників, фазових портретів
і перерізів Пуанкаре.
Ключові слова: ідеальна динамічна система, регулярний і хаотичний
атрактори, сценарій узагальненої переміжності.
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| spelling | journaliasakpiua-article-3580772026-04-19T21:53:19Z Implementation of a generalized intermittency scenario in the Rössler dynamical system Реалізація сценарію узагальненої переміжності в динамічній системі Ресслера Horchakov, Oleksii Shvets, Aleksandr ideal dynamical system regular and chaotic attractors generalized intermittency scenario ідеальна динамічна система регулярний і хаотичний атрактори сценарій узагальненої переміжності The realization of novel scenario involving transitions between different types of chaotic attractors is investigated for the Rössler system. Characteristic features indicative of the presence of generalized intermittency scenario in this system are identified. The properties of “chaos–chaos” transitions following the generalized intermittency scenario are analyzed in detail based on phase-parametric characteristics, Lyapunov characteristic exponents, phase portraits, and Poincaré sections. Досліджено реалізацію нового сценарію переходу між різними типами хаотичних атракторів для системи Ресслера. Виявлено характерні ознаки, що вказують на наявність сценарію узагальненої переміжності в цій системі. Властивості переходів типу «хаос–хаос» за сценарієм узагальненої переміжності детально проаналізовано на основі фазо-параметричних характеристик, ляпуновських характеристичних показників, фазових портретів і перерізів Пуанкаре. The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" 2026-03-31 Article Article application/pdf https://journal.iasa.kpi.ua/article/view/358077 10.20535/SRIT.2308-8893.2026.1.07 System research and information technologies; No. 1 (2026); 103-111 Системные исследования и информационные технологии; № 1 (2026); 103-111 Системні дослідження та інформаційні технології; № 1 (2026); 103-111 2308-8893 1681-6048 en https://journal.iasa.kpi.ua/article/view/358077/344002 |
| spellingShingle | ідеальна динамічна система регулярний і хаотичний атрактори сценарій узагальненої переміжності Horchakov, Oleksii Shvets, Aleksandr Реалізація сценарію узагальненої переміжності в динамічній системі Ресслера |
| title | Реалізація сценарію узагальненої переміжності в динамічній системі Ресслера |
| title_alt | Implementation of a generalized intermittency scenario in the Rössler dynamical system |
| title_full | Реалізація сценарію узагальненої переміжності в динамічній системі Ресслера |
| title_fullStr | Реалізація сценарію узагальненої переміжності в динамічній системі Ресслера |
| title_full_unstemmed | Реалізація сценарію узагальненої переміжності в динамічній системі Ресслера |
| title_short | Реалізація сценарію узагальненої переміжності в динамічній системі Ресслера |
| title_sort | реалізація сценарію узагальненої переміжності в динамічній системі ресслера |
| topic | ідеальна динамічна система регулярний і хаотичний атрактори сценарій узагальненої переміжності |
| topic_facet | ideal dynamical system regular and chaotic attractors generalized intermittency scenario ідеальна динамічна система регулярний і хаотичний атрактори сценарій узагальненої переміжності |
| url | https://journal.iasa.kpi.ua/article/view/358077 |
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