Phase chaos and multistability in the discrete Kuramoto model
The paper describes the appearance of a novel, a high-dimensional chaotic regime, called phase chaos, in the discrete Kuramoto model of globally coupled phase oscillators. This type of chaos is observed at small and intermediate values of the coupling strength. It is caused by the nonlinear intera...
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irk-123456789-1785762021-02-28T01:26:09Z Phase chaos and multistability in the discrete Kuramoto model Maistrenko, V.L. Vasylenko, A.A. Maistrenko, Y.L. Mosekilde, E. The paper describes the appearance of a novel, a high-dimensional chaotic regime, called phase chaos, in the discrete Kuramoto model of globally coupled phase oscillators. This type of chaos is observed at small and intermediate values of the coupling strength. It is caused by the nonlinear interaction of the oscillators, while the individual oscillators behave periodically when left uncoupled. For the four-dimensional discrete Kuramoto model we outlined the region of the phase chaos in the parameter plane and distinguished the region where the phase chaos coexists with other periodic attractors, and demonstrate, in addition, that the transition to the phase chaos takes place through the torus destruction scenario. Описано виникнення нового багатовимiрного хаотичного режиму, який називають фазовим хаосом, у дискретнiй моделi Курамото глобально зв’язаних фазових осциляторiв. Цей тип хаосу спостерiгають при малих та середнiх значеннях параметра зв’язку. Вiн спричиняється нелiнiйною взаємодiєю осциляторiв, тодi як поведiнка незв’язаних iндивiдуальних осциляторiв є перiодичною. Для чотиривимiрної дискретної моделi Курамото окреслено область фазового хаосу в площинi параметрiв, видiлено область, де фазовий хаос спiвiснує з iншими перiодичними атракторами, та продемонстровано, що перехiд до фазового хаосу вiдбувається через сценарiй руйнування тора. 2008 Article Phase chaos and multistability in the discrete Kuramoto model / V.L. Maistrenko, A.A. Vasylenko, Y.L. Maistrenko, E. Mosekilde // Нелінійні коливання. — 2008. — Т. 11, № 2. — С. 217-229. — Бібліогр.: 22 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/178576 517.9 en Нелінійні коливання Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| language |
English |
| description |
The paper describes the appearance of a novel, a high-dimensional chaotic regime, called phase chaos, in
the discrete Kuramoto model of globally coupled phase oscillators. This type of chaos is observed at small
and intermediate values of the coupling strength. It is caused by the nonlinear interaction of the oscillators,
while the individual oscillators behave periodically when left uncoupled. For the four-dimensional discrete
Kuramoto model we outlined the region of the phase chaos in the parameter plane and distinguished the
region where the phase chaos coexists with other periodic attractors, and demonstrate, in addition, that the
transition to the phase chaos takes place through the torus destruction scenario. |
| format |
Article |
| author |
Maistrenko, V.L. Vasylenko, A.A. Maistrenko, Y.L. Mosekilde, E. |
| spellingShingle |
Maistrenko, V.L. Vasylenko, A.A. Maistrenko, Y.L. Mosekilde, E. Phase chaos and multistability in the discrete Kuramoto model Нелінійні коливання |
| author_facet |
Maistrenko, V.L. Vasylenko, A.A. Maistrenko, Y.L. Mosekilde, E. |
| author_sort |
Maistrenko, V.L. |
| title |
Phase chaos and multistability in the discrete Kuramoto model |
| title_short |
Phase chaos and multistability in the discrete Kuramoto model |
| title_full |
Phase chaos and multistability in the discrete Kuramoto model |
| title_fullStr |
Phase chaos and multistability in the discrete Kuramoto model |
| title_full_unstemmed |
Phase chaos and multistability in the discrete Kuramoto model |
| title_sort |
phase chaos and multistability in the discrete kuramoto model |
| publisher |
Інститут математики НАН України |
| publishDate |
2008 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/178576 |
| citation_txt |
Phase chaos and multistability in the discrete Kuramoto model / V.L. Maistrenko, A.A. Vasylenko, Y.L. Maistrenko, E. Mosekilde // Нелінійні коливання. — 2008. — Т. 11, № 2. — С. 217-229. — Бібліогр.: 22 назв. — англ. |
| series |
Нелінійні коливання |
| work_keys_str_mv |
AT maistrenkovl phasechaosandmultistabilityinthediscretekuramotomodel AT vasylenkoaa phasechaosandmultistabilityinthediscretekuramotomodel AT maistrenkoyl phasechaosandmultistabilityinthediscretekuramotomodel AT mosekildee phasechaosandmultistabilityinthediscretekuramotomodel |
| first_indexed |
2025-07-15T16:43:48Z |
| last_indexed |
2025-07-15T16:43:48Z |
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1837732021439299584 |
| fulltext |
UDC 517 . 9
PHASE CHAOS AND MULTISTABILITY
IN THE DISCRETE KURAMOTO MODEL
ФАЗОВИЙ ХАОС ТА МУЛЬТИСТIЙКIСТЬ
У ДИСКРЕТНIЙ МОДЕЛI КУРАМОТО
V. Maistrenko, A. Vasylenko
Inst. Math. Nat. Acad. Sci. Ukraine
Tereshchenkivska St. 3, Kyiv, 01601, Ukraine
Yu. Maistrenko
Inst. Math. Nat. Acad. Sci. Ukraine
Tereshchenkivska St. 3, Kyiv, 01601, Ukraine
and
Inst. Medicine
Research Centre Jülich, 52425, Jülich, Germany
E. Mosekilde
Techn. Univ. Denmark
DK-2800, Kongens Lyngby, Denmark
The paper describes the appearance of a novel, a high-dimensional chaotic regime, called phase chaos, in
the discrete Kuramoto model of globally coupled phase oscillators. This type of chaos is observed at small
and intermediate values of the coupling strength. It is caused by the nonlinear interaction of the oscillators,
while the individual oscillators behave periodically when left uncoupled. For the four-dimensional discrete
Kuramoto model we outlined the region of the phase chaos in the parameter plane and distinguished the
region where the phase chaos coexists with other periodic attractors, and demonstrate, in addition, that the
transition to the phase chaos takes place through the torus destruction scenario.
Описано виникнення нового багатовимiрного хаотичного режиму, який називають фазовим ха-
осом, у дискретнiй моделi Курамото глобально зв’язаних фазових осциляторiв. Цей тип хаосу
спостерiгають при малих та середнiх значеннях параметра зв’язку. Вiн спричиняється нелi-
нiйною взаємодiєю осциляторiв, тодi як поведiнка незв’язаних iндивiдуальних осциляторiв є
перiодичною. Для чотиривимiрної дискретної моделi Курамото окреслено область фазового
хаосу в площинi параметрiв, видiлено область, де фазовий хаос спiвiснує з iншими перiодичними
атракторами, та продемонстровано, що перехiд до фазового хаосу вiдбувається через сценарiй
руйнування тора.
1. Introduction. Unraveling the collective behavior of a group of interacting oscillators repre-
sents a challenging problem that arises in many areas of science and technology [1 – 5]. Examples
from biology and the lifesciences are the generation of cyclic AMP pulsors in slime mold
cultures of Dictyostelium discoideum [6], the propagation of waves of cytoplasmic Ca2+ across
islets of bursting pancreotic cells [7], the newly discovered synchronization of metabolic proces-
ses in suspensions of yeast cells [8], and the entrainment of the pressure and flow regulation
between neighboring nephrons in the kidny [9]. Examples from physics, chemistry and engi-
c© V. Maistrenko, A. Vasylenko, Yu. Maistrenko, E. Mosekilde, 2008
ISSN 1562-3076. Нелiнiйнi коливання, 2008, т . 11 , N◦ 2 217
218 V. MAISTRENKO, A. VASYLENKO, YU. MAISTRENKO, E. MOSEKILDE
neering have been published, for instance, by Hadley et al. [10], Kiss et al. [11] and Rulkov [12].
Problems of this type have been investigated from many different points of view. Postnov et al.
[13], for instance, have suggested a mechanism for the development of high-order chaos and,
more recently, De Monte et al. [14] have established an order parameter expansion by which the
mean field dynamics of a population of oscillators with parameter diversity and global coupling
can be described in terms of a few macroscopic degrees of freedom.
Pulse- and phase-coupled oscillators constitute a paradigmatic class of interacting dynami-
cal units. Over the years, particular emphasis has been paid to the so-called Kuramoto system
[1, 4], i.e., to models of coupled phase oscillators. Such models represent the simplest approach
to the coupled oscillator problem, namely an approach in which the individual oscillator is
described by a single variable, the value of its instantaneous phase. Nonetheless, models of this
type have been surprisingly effective in designing new ways to desynchronize clusters of brain
cells in patients with Parkinsonian and other forms of tremor [15, 16]. Only recently, investigati-
ons have begun based on more physiologically realistic nerve and glia cell models. However,
the dynamics of the finite-dimensional Kuramoto system is still far from being understood in
detail, and one of the recent results of the above neurological applications of the model was
the observation of a new high-dimensional chaotic phenomenon called phase chaos [17]. Moti-
vated by these results, the present paper considers a time-discrete version of the Kuramoto
model which has some features in common with the continuous-time system, but also manifests
some important differences [18, 19].
Typical questions that arise in the investigations of complex dynamical systems are related to
the collective dynamics of an ensemble of oscillators (such as synchronization, clustering or the
appearance of chaos and higher-dimensional dynamics), how the dynamics can be changed with
varying the system parameters, and the bifurcations that lead to the appearance of different
dynamical states. In the present paper we concentrate on the studying the characteristics of
phase chaos and the mechanisms by which this form of chaos appears.
The paper is organized as follows. In Section 2 we introduce the N -dimensional and 4-
dimensional discrete Kuramoto models of globally coupled phase oscillators and describe the
change of variables which allows us to reduce the dimension of the system by one. The following
sections are devoted to the study of the four-dimensional Kuramoto model and the correspon-
ding three-dimensional map in the case when the frequencies of the individual oscillators are
distributed equidistantly over some interval. By this condition we reduce the number of system
parameters. In Section 3 we present the regions of stability of periodic orbits in the plane of
system parameters, discuss phenomenon of multistability and show some characteristic phase
portraits of the system. In Section 4 we describe the structure of the phase-locking regions
for the Poincaré map of our three-dimensional system and, using the method of Lyapunov
exponents to detect the chaotic behavior, we outline the region of phase chaos in the parameter
plane. In Section 5 the mechanism of appearance of the phase chaos is described in more detail,
and we show how this form of chaos arises through a torus destruction scenario.
2. The model. Let us consider the discrete Kuramoto model of N globally coupled phase
oscillators
ψ
(n+1)
i = ψ
(n)
i + ωi +
K
N
N∑
j=1
sin
(
ψ
(n)
j − ψ(n)
i
)
, i = 1, N, (1)
where N ∈ Z and n = 0, 1, . . . represents the discrete time, ψi are the phase variables, ωi —
ISSN 1562-3076. Нелiнiйнi коливання, 2008, т . 11, N◦ 2
PHASE CHAOS AND MULTISTABILITY IN THE DISCRETE KURAMOTO MODEL 219
the natural frequencies of individual oscillators and K > 0 is the coupling parameter.
Such a discrete-time Kuramoto model has some features in common with the continuous-
time system [16 – 18]. However, in general, the discrete-time models exhibit more complicated
and rich dynamics than the continuous-time ones.
For the time-continuous Kuramoto model it was found that systems of at least N ≥ 4
display a novel type of chaotic behaviour called phase chaos [17]. The present paper considers
the four-dimensional discrete-time Kuramoto system, as an extention of our recent work [19],
where the same system with N = 2 and 3 was investigated.
In this work we consider the four-dimensional discrete Kuramoto model and the case of the
main resonance, ωi are distributed equidistantly in the interval [−l; l], l > 0 :
ψ
(n+1)
i = ψ
(n)
i + ωi +
K
4
4∑
j=1
sin
(
ψ
(n)
j − ψ(n)
i
)
, i = 1, 4, (2)
where ω1 = −l, ω2 = −1
3
l, ω3 =
1
3
l, ω4 = l :
u u u u
0−l
−1
3
l
1
3
l
l
ω1 ω2 ω3 ω4
When studying the collective dynamics and interaction of the phase oscillators, it is useful to
consider differences of the individual phase variables. Indeed, introducing new phase difference
variables:
ϕi = ψi+1 − ψi, i = 1, N − 1, (3)
we can reduce the dimension of the system (1) by one. The effective dynamics then take place
on the torus TN−1.
Three-dimensional system. After change of the variables (3) in the system (1) the dynamics
are governed by the following three-dimensional system:
ϕ
(n+1)
1 = ϕ
(n)
1 + ∆1 −
K
4
(
2 sinϕ(n)
1 + sin(ϕ(n)
1 + ϕ
(n)
2 ) + sin(ϕ(n)
1 + ϕ
(n)
2 + ϕ
(n)
3 )−
− sinϕ(n)
2 − sin(ϕ(n)
2 + ϕ
(n)
3 )
)
,
ϕ
(n+1)
2 = ϕ
(n)
2 + ∆2 −
K
4
(
2 sinϕ(n)
2 + sin(ϕ(n)
1 + ϕ
(n)
2 ) + sin(ϕ(n)
2 + ϕ
(n)
3 )−
− sinϕ(n)
1 − sinϕ(n)
3
)
, (4)
ϕ
(n+1)
3 = ϕ
(n)
3 + ∆3 −
K
4
(
2 sinϕ(n)
3 + sin(ϕ(n)
2 + ϕ
(n)
3 ) + sin(ϕ(n)
1 + ϕ
(n)
2 + ϕ
(n)
3 ) −
− sinϕ(n)
2 − sin(ϕ(n)
1 + ϕ
(n)
2 )
)
,
where the variables ϕ1, ϕ2, ϕ3 are given by (3), and frequencies differencies ∆i = ωi+1 − ωi,
i = 1, 2, 3. From the form of the equations one can conclude that in the case ∆1 = ∆3 the
system (4) has a symmetric invariant manifoldM = {ϕ1 = ϕ3}.
ISSN 1562-3076. Нелiнiйнi коливання, 2008, т . 11, N◦ 2
220 V. MAISTRENKO, A. VASYLENKO, YU. MAISTRENKO, E. MOSEKILDE
Fig. 1. Sketch of the dynamics of the circle map (5).
We will consider the case of main resonance ∆i = ∆, i = 1, 2, 3. When K = 0 the dynamics
are determined by the one-dimensional circle map
ϕ(n+1) = ϕ(n) + ∆ (mod 2π), n ≥ 0. (5)
Here ∆ =
2
3
l according to our designations. This map describes a simple rotation, its dyna-
mics are characterized by the rotation number ρ = lim
n→∞
ϕ(n) − ϕ(0)
2πn
=
∆
2π
, see Fig. 1. If ρ is
rational, i.e., ρ =
p
q
, then all trajectories are periodic with period T = q. If ρ is irrational, all
trajectories are quasiperiodic.
3. Phase-locking regions. When adding perturbation KF to the map (5):
ϕ(n+1) = ϕ(n) + ∆ +KF (ϕ(n)) (mod 2π), (6)
regions of periodicity corresponding to rational rotation numbers
p
q
emerge from the points(
p
q
; 0
)
in the parameter plane (∆;K). Such regions are also called phase-locking regions, regi-
ons of synchronization or Arnol’d tongues.
Three-dimensional discrete Kuramoto model (4) can be considered as a perturbation of the
map (5), where K ≥ 0 is a parameter of the perturbation. We find that for K > 0 the system
(4) can exhibit periodic, quasiperiodic and chaotic dynamics. To overview the system dynamics
for different values of the parameters, we have determined main parameter regions of existence
and stability of the orbits in the plane (l,K). Fig. 2 depicts the widest Arnol’d tongues as well
as the region of phase chaos, which will be discussed in more detail in the following sections.
For the parameter of discreteness l, the following relation takes place:
∆ =
2
3
l = 2πρ.
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PHASE CHAOS AND MULTISTABILITY IN THE DISCRETE KURAMOTO MODEL 221
Fig. 2. Regions of stability for the main periodic orbits of the three-dimensional system (4).
Numbers denote periods of the stable orbits inside the Arnol’d tongues; these cycles
lose the transverse stability in the bifurcation Ksb — symmetry-breaking; the curve
Kcr shows where boundary crisis for the chaotic attractor takes place; K is the coupling
parameter, and l is a measure of the spread in natural frequencies.
As shown in Fig. 2 the Arnol’d tongues of the rotation numbers ρ =
p
q
emanate from the points
lq =
3
2
2πρ = 3π
p
q
. In the figure, we denoted the tongues of the q-periodic orbits with the
corresponding numbers, and for some of the tongues we have shown points lq of their origin on
the axis l. For example, the region of stability of the periodic orbit with period 5 corresponds to
the rotation number ρ =
1
5
and emanates from the point l5 = 3π
1
5
≈ 1.885. Analogously l6 =
= 3π
1
6
≈ 1.57; l8 = 3π
1
8
≈1.178; l10 = 3π
1
10
≈ 0.94. As we have found, periodic orbits, exis-
ting in these phase-locking regions, belong to the diagonal M = {ϕ1 = ϕ3}. Two phase port-
raits in Fig. 3 (a, b) show typical attractors inside the diagonal. With decrease in K, the attractor
in the diagonal M loses its transverse stability at the bifurcation curve Ksb — symmetry-
breaking. The Arnol’d tongues go down to the axis l, but the periodic orbits which are stable
inside M, are no longer stable transversally to M. At the same time, for all K < Kcr other
attractors exist outside the manifold M, an example is shown in Fig. 3 (c). In the parameter
region between Ksb and Kcr, we have multistability: two attractors, one inside M and another
outside M coexist. This can be illustrated with graphs of the Lyapunov exponents for these
attractors, see Fig. 4, in this case the attractor outside M is chaotic, and we will discuss such
dynamics in more detail later.
The structure of the Arnol’d tongues is standard, and they are arranged in accordance with
the rule of Fibonacci numbers. The widest tongues belong to the first level and their rotation
numbers are ordered according to the period-adding rule. In between every pair of the tongues
of the rotation numbers
1
q
and
1
q + 1
there is a smaller tongue of
ISSN 1562-3076. Нелiнiйнi коливання, 2008, т . 11, N◦ 2
222 V. MAISTRENKO, A. VASYLENKO, YU. MAISTRENKO, E. MOSEKILDE
a b c
Fig. 3. Phase portraits of the three-dimensional system (4): (a) fixed point of the system (4) inside M, l = 0.126,
K = 0.2; (b) quasiperiodic trajectory of (4) inside M, it intersects the plane ϕ2 = 0 only once this
intersection is the fixed point of the Poincaré map, l = 0.14, K = 0.2; (c) quasiperiodic torus for the
system (4) outsideM, stable periodic orbit of the period 4 for the Poincaré map, l = 0.08623, K = 0.1.
Fig. 4. Forward and backward calculations of the Lyapunov exponents
for the attractors inside and outside the manifoldM, l = 0.2.
the second level of the rotation number
2
2q + 1
; between the tongues corresponding to
1
q
and
2
2q + 1
there is a tongue of the third level with ρ =
3
3q + 1
and so on. Boundaries of the tongues
are curves of saddle-node bifurcations for the periodic orbits existing inside the tongues. The
dynamics of the system (4) are one-to-one (invertible) for all 0 < K ≤ 1. For the values of K
grater than 1, the dynamics become non-invertible and transition to chaos takes place through
the period-doubling cascades.
ISSN 1562-3076. Нелiнiйнi коливання, 2008, т . 11, N◦ 2
PHASE CHAOS AND MULTISTABILITY IN THE DISCRETE KURAMOTO MODEL 223
Construction of the Poincaré map. To further simplify the analysis we will use the classical
idea of Poincaré, who proposed a method to reduce the studies of continuous-time systems by
considering the associated Poincaré maps. For anN -dimensional flow the core of this method li-
es in considering respective discrete dynamics on some (N − 1)-dimensional surface transverse
to the vector field. This method allows one to reduce the dimension of the system and it also
provides a useful display of the global dynamics of the system. A similar approach can be appli-
ed for studying discrete-time models that exhibit quasiperiodic behavior. For the parameter
values that correspond to quasiperiodic or chaotic trajectories we can define the Poincaré map
by considering only the intersections of the trajectories with some particular plane.
For the three-dimensional system (4) we choose to analize the Poincaré map on the torus
section ϕ2 = 0. As shown in Fig. 2 there exist a wide region of stability of the fixed point
and many regions corresponding to stability of some periodic orbits. But for smaller values of
coupling K the Arnol’d tongues are very narrow, and quasiperiodic trajectories prevail. The
region of quasiperiodicity is coloured in white on the figure, and for the corresponding values
of parameters we will apply the method of Poincaré. Our main interest is in considering the
parameter values in the lower left corner of Fig. 2, since it is in this region that we find chaotic
behavior unexpectedly.
As shown in Fig. 2 we have found that the system exhibits chaotic behavior for unusually
small values of the coupling K, far away from the region where the Arnol’d tongues overlap
due to non-invertibility of the map (4), that appears for K > 1, which causes the well-known
chaoticity observed, for instance, in the sine circle map [19, 20]. Following [17], we will call
this type of chaos, which appears for small K and l, — phase chaos. The key to understanding
the appearance of phase chaos in this region can be found by considering the dynamics of the
Poincaré map.
To get a better insight into the system dynamics several phase portraits of the three-dimen-
sional system (4) are depicted in Fig. 3. Fig. 3 (a) corresponds to existence of a stable fixed point
for the system (4). Fig. 3 (b, c) show the cases when the attractor of the system is an invariant
curve, and the intersection of this curve with the plane ϕ2 = 0 implies the existence of the fixed
or periodic points for the Poincaré map considered.
4. Arnol’d tongues for the Poincaré map. To consider the transition to phase chaos in more
detail let us study the occurrence of periodic orbits for the Poincaré map. The general picture
of the main tongues is presented in Fig. 5 and, enlarged, in Fig. 6. In these figures, the stability
regions of different periodic orbits are shown. The existence of an Arnol’d tongue for some
m-periodic orbit of the Poincaré map signifies the existence of an invariant curve for the three-
dimensional system (4), and this curve intersects the plane ϕ2 = 0 in m points, as illustrated
by phase portraits in Fig. 3. Phase chaos occurs in the region where the Arnol’d tongues for the
Poincaré map intersect and overlap each other. The chaotic attractor is destroyed in a boundary
crisis which is marked with the bidurcation curve Kcr in Fig. 5.
The structure and the disposition of the tongues are shown in more detail in Fig. 6. These
regions are ordered in a standard way: the widest tongues are located in accordance with a
period-adding rule. In between every pair of tongues with periods p and p+ 1 there is a smaller
tongue for the period 2p+ 1, between the regions for periods p and 2p+ 1 there is a tongue for
the period 3p+ 1 and so on. The boundaries of the tongues are curves of saddle-node bifurcati-
ons and in the top of the tongues, where period-doubling cascades occur, one can observe the
transition to chaotic behavior.
ISSN 1562-3076. Нелiнiйнi коливання, 2008, т . 11, N◦ 2
224 V. MAISTRENKO, A. VASYLENKO, YU. MAISTRENKO, E. MOSEKILDE
Fig. 5. Some of the Arnol’d tongues for the Poincaré map in the region of appearance of
phase chaos; the numbers show the periods of the periodic orbits; the bifurcation
curve Kcr denotes the boundary crisis of the chaotic attractor; the points a-c correspond
to the parameter values for the phase portraits in Fig. 3.
Fig. 6. Structure of the phase-locking regions for the Poincaré map of the system (4);
the numbers denote the periods of the periodic orbits for the corresponding
tongues.
ISSN 1562-3076. Нелiнiйнi коливання, 2008, т . 11, N◦ 2
PHASE CHAOS AND MULTISTABILITY IN THE DISCRETE KURAMOTO MODEL 225
Fig. 7. Region of phase chaos, outlined with the method of Lyapunov exponents by the
condition λmax ≥ 0.001; for comparizon, the zero-valued Lyapunov exponent
takes values of the order 0.0001.
Fig. 8. Maximum with K of the maximal Lyapunov exponents in the region of appearance
of the phase chaos.
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226 V. MAISTRENKO, A. VASYLENKO, YU. MAISTRENKO, E. MOSEKILDE
Fig. 9. Lyapunov exponents of the system (4), l = 0.1
(a), 0.23 (b), 0.35 (c); the numbers denote the periods
for the tongues crossed while calculating the exponents.
Fig. 10. Parameter region where the transition to phase
chaos occurs; different regions of stability of peri-
odic orbits for the Poincaré map, the arrow shows the
parameter values for which the Lyapunov exponents in
the last graph of Fig. 9 where calculated.
Lyapunov exponents and phase chaos. To detect the chaotic, quasiperiodic or periodic regi-
mes of the system dymanics we calculate the Lyapunov exponents. The system has three Lyapu-
nov exponents with one of them being always equal to zero. The maximal exponent λ1, when it
becomes positive, signals the appearance of phase chaos. The region of parameter values where
phase chaos occurs is shown in Fig. 7. To outline this region we used the criteria λmax ≥ 0.001,
considering all the smaller values to be zero. It is also interesting to observe how the value of
the maximal Lyapunov exponent varies in the parameter region where the phase chaos occurs.
For the range of l-values we considered such values of K where λ1 is positive and compared λ1
for different K in order to find the maximum. The resulting graph is shown in Fig. 8. One can
see the correspondance between the region of phase chaos and the dynamics of the Lyapunov
exponents.
Fig. 9 shows several graphs of the Lyapunov exponents for the parameter values that corres-
pond to appearance of chaotic attractors. As shown, the maximal exponent reaches the values
of order 0.01, while the zero Lyapunov exponent takes values of order 0.0001. Fig. 10 shows the
main Arnol’d tongues in the region where the Lyapunov exponents were calculated. The arrow
corresponds to the third graph in Fig. 9. Fig. 10 shows that if we increase the parameter K then
the transition to chaos occurs after leaving the tongue of periodicity.
The region of phase chaos in the (l,K)-parameter plane is found to possess a complicated
topological structure due to multiple intersections with tongues of periodicity, causing multi-
stability and coexistence of the chaotic behavior with the phase locked states.
5. By torus destruction to phase chaos. Let us finally consider the types of phase dynamics
that are typical for our system and precede the transition to phase chaos. Fig. 11 defines several
points (a — f) in the parameter plane, which were chosen for our examination, and Fig. 12
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PHASE CHAOS AND MULTISTABILITY IN THE DISCRETE KURAMOTO MODEL 227
Fig. 11. Some of the mode-locking regions near the transition to phase chaos. Points a – f correspond to the parameter
values for the phase portraits in Fig. 12. Point a is situated in a region of quasiperiodic dynamics, point b —
in a region with period-14 dynamics, and c — in a parameter region where the torus is wrinkled.
The point f falls in a region of fully developed phase chaos. The black boundary shows the bifurcation
line that corresponds to the boundary crisis of the chaotic attractor.
demonstrates the phase portraits for the corresponding parameter values. Between the phase-
locking regions there exists a quasiperiodic attractor for the Poincaré map. This is illustrated
in Fig. 12 (a). With the change of the parameters the quasiperiodic torus becomes wrinkled,
and the transition to chaos takes place, as shown in Fig. 12 (c). This confirms that phase chaos
appears through the torus destruction scenario. With further variation of the parameters, wri-
nkles and nonsmoothness spread along the torus, and it breaks up into a fractal or strange
structure.
The torus destruction scenario applies to smooth systems and to systems that can be repre-
sented by two-dimensional invertible maps. It is a common scenario for the loss of smoothness
in systems that display quasiperiodicity and phase locking [20].
For the parameter values inside the Arnol’d tongues the behavior on the plane ϕ2 = 0 is
periodic. Examples of the periodic orbits are depicted in Figs. 12 (b, e), and Figs. 12 (d, f) show
the phase portraits in the case of the chaotic behavior. As discussed earlier, the phase chaos can
coexist with some other attractors, and in Fig. 12 (f) one can see the fixed point coexisting with
the chaotic attractor.
6. Conclusions. Phase chaos is a novel phenomenon in nonlinear systems of coupled osci-
llators [17]. In the present paper we investigate the appearance of this type of chaos in the
discrete Kuramoto model of globally coupled phase oscillators. The peculiarity of the phase
chaos is that this type of chaotic behavior takes place even for very small values of the coupling
parameter. When changing the coupling from zero one would intuitively expect emergence of a
coherence among the oscillators, but we find chaotic behaviour there. Nonlinear interactions of
the oscillators cause chaoticity of the collective phase dynamics even for those parameter values
which correspond to periodic behaviour of the uncoupled individual oscillators. The same type
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228 V. MAISTRENKO, A. VASYLENKO, YU. MAISTRENKO, E. MOSEKILDE
a b
c d
e f
Fig. 12. Phase portraits obtained for different parameter values corresponding to the points a – f in the Fig. 11:
(a) shows a quasiperiodic torus for l = 0.28 and K = 0.26, (b) a stable periodic orbit of period 14,
l = 0.28, K = 0.272, (c) phase chaos, λ1 ≈ 0.0026, l = 0.277, K = 0.28, (d) phase chaos, λ1 ≈
≈ 0.007, l = 0.28, K = 0.3, (e) two stable periodic orbits of period 2, coexisting with a fixed
point, l = 0.28, K = 0.303, (f) phase chaos, λ1 ≈ 0.01, and fixed point, l = 0.28, K = 0.31.
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PHASE CHAOS AND MULTISTABILITY IN THE DISCRETE KURAMOTO MODEL 229
of chaos has been found in other systems both regular and chaotic [21, 22], and this allows us
to conclude that phase chaos is, probably a common phenomenon of networks of very different
nature. However, the detailed analysis of this phenomena has not yet been taken, and as a si-
gnificant achievement, in the present paper we present the parameter region of the phase chaos
existence, region of multistability and the regions of existence of the periodic orbits which help
to understand the appearance of the chaotic behavior.
In this work we find typicalness of phase chaos for a discrete analog of the Kuramoto
model, and analyse how it depends on a new parameter l induced by discreteness. We describe
the structure of the Arnol’d tongues and show how this form of chaos arises through a torus
destruction scenario.
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Received 01.06.07
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