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arXiv:2406.04701v1 [nlin.AO] 7 Jun 2024
Transition to synchronization in adaptive Sakaguchi-Kuramoto model with
higher-order interactions
Sangita Dutta1,∗Prosenjit Kundu2, Pitambar Khanra3, Chittaranjan Hens4, and Pinaki Pal1†
1Department of Mathematics, National Institute of Technology, Durgapur 713209, India
2Complex Systems Group, Dhirubhai Ambani Institute of Information
and Communication Technology, Gandhinagar 382007, India
3Department of Microbiology and Immunology, Jacobs School of Medicine & Biomedical Sciences,
The State University of New York at Buffalo, Buffalo, NY 14203, USA and
4Center for Computational Natural Science and Bioinformatics,
International Institute of Informational Technology, Hyderabad, 500032, India
We investigate the phenomenon of transition to synchronization in Sakaguchi-Kuramoto model in
the presence of higher-order interactions and global order parameter adaptation. The investigation
is done by performing extensive numerical simulations and low dimensional modeling of the system.
Numerical simulations of the full system show both continuous (second order) as well as discontin-
uous transitions. The discontinuous transitions can either be associated with explosive (first order)
or with tiered synchronization states depending on the choice of parameters. To develop an in
depth understanding of the transition scenario in the parameter space we derive a reduced order
model (ROM) using the Ott-Antonsen ansatz, the results of which closely matches with that of the
numerical simulations of the full system. The simplicity and analytical accessibility of the ROM
helps to conveniently unfold the transition scenario in the system having complex dependence on
the parameters. Simultaneous analysis of the full system and the ROM clearly identifies the regions
of the parameter space exhibiting different types of transitions. It is observed that the second order
continuous transition is connected with a supercritical pitchfork bifurcation (PB) of the ROM. On
the other hand, the discontinuous teired transition is associated with multiple saddle-node (SN) bi-
furcations along with a supercritical PB and the first order explosive transition involves a subcritical
PB alongside a SN bifurcation.
I. INTRODUCTION
Network science is the keystone to study intercon-
nected systems [1, 2] surrounding us such as ecosystems,
social systems [3, 4], different kinds of biological [5, 6]
and physical systems [7–9] etc. Mathematically a net-
work is described by a graph, where the units of a system
is treated as node and the interconnection is presented
by the links/edges between these nodes. If the dynam-
ics of the isolated i-th node is governed by the ordinary
differential equation ˙xi=f(xi), xi∈RM, then in the
presence of pairwise interactions, the collective dynam-
ics of the complex network is governed by the coupled
equations
˙xi=f(xi) + K1
N
X
j1=1
Aij1G1(xi, xj1),(i= 1,2,...,N),
(1)
where K1is the strength of the coupling, A= (Aij1)N×N
is the adjacency matrix of the complex network of size
Nand G1(xi, xj1) is the coupling function between
the nodes iand j1of the network. Kuramoto intro-
duced periodic diffusive like (sinusoidal) coupling [10]
(G1(xi, xj1) = sin(xj1−xi)), where f(xi) depends on
the intrinsic frequencies of the units. This model has
∗Electronic address: sangitaduttaprl@gmail.com
†Electronic address: ppal.maths@nitdgp.ac.in
been proven to successfully describe the emergent dy-
namics of various coupled systems. One of such inter-
esting phenomena is synchronization [11, 12]. It is ob-
served in many natural as well as man made systems like
flashing of fireflies [13], clapping in a hall [14], brain neu-
ron [15, 16], cellular processes in populations of yeast [17],
metronomes, power-grids [18], etc. The route to syn-
chronization from asynchronous state may be continuous,
discontinuous or explosive depending on structural and
dynamical variability considered in the system [19–22]
Moreover, the dependence of the coupling strength on the
synchronization order parameter allows the model to re-
main updated about the dynamical state of the oscillators
at each time step. This dependence can be mathemati-
cally modeled by multiplying the coupling constant with
some function of order parameter. There are numerous
examples of such systems that adapt the dynamic states
of the units [23, 24].
Sakaguchi-Kuramoto (SK) model is dynamical system
which is often used to study the synchronization behav-
ior in coupled dynamical units of phase frustrated sys-
tems, coupled with pairwise interaction[25–28]. The cou-
pling function in this case takes the form G1(xi, xj1) =
sin(xj1−xi−α), where αis the phase frustration or phase-
lag parameter. The classic scenario of SK model reveals
non-hysteric continuous synchronization transition [29].
By contrast, an adaptive coupling (coupling constant K1
is multiplied with synchronization order parameter) may
result in hysteresic or explosive transition [30, 31]. Here
we investigate the role of higher-order interactions in SK-
2
oscillators where the temporal synchronization order pa-
rameter will control the coupling strength in each time.
A variety of systems, such as neuronal network [32–
34], social network [35, 36], ecological network [37, 38]
chemical networks involve group or community interac-
tions which can not be neglected and also have a great
impact on the dynamics [39, 40]. For instance, the out-
come of a chemical reaction between two elements can be
significantly altered when a third element is introduced
into the reaction. Also in case of disease infection, one
healthy unit can be infected in touch of multiple infected
units, takes the form of higher-order interactions [41, 42].
One can also take the example of a collaboration net-
work, where the dynamics of multi-author collaboration
can not be suffices to describe with the combination of
pairwise collaborations [43]. In such systems, the multi-
author interactions can be represented as higher-order
interactions which can not always be expressed as a sum
of pairwise interactions. As a result, the concept of hy-
pergraph [44], simplicial complex [45–49] are introduced
to encode these higher-order interactions framework. To
examine the rich dynamics of a network, the higher-order
interactions should be considered along with the pairwise
interactions. The dynamical system with higher-order in-
teractions can be expressed in the form,
˙xi=f(xi) + K1
N
X
j1=1
Aij1G1(xi, xj1)
+K2
N
X
j1=1
N
X
j2=1
Bij1j2G2(xi, xj1, xj2) + ...
+Km
N
X
j1=1
···
N
X
jm=1
Cij1...jmGm(xi, xj1,...,xjm),
i= 1,2,...,N
where Bij1j2and Cij1...jmaccount respectively the con-
nections between three and (m+ 1) units. G1(K1)
presents the coupling function (strength) for pairwise in-
teraction, G2(K2) is for triadic interaction and Gm(Km)
is for (m+ 1) units interaction and so on [50]. The
structural properties of these higher-order interactions
and the impact of those structures in the emergent dy-
namics of networked systems draw the attention of many
researchers [46, 51]. One of the most interesting effect of
such structures is to produce explosive synchronization
irrespective of the choice of natural frequencies or with-
out any correlation between the structural and dynamical
properties in different kinds of networks [52–56].
Recently, distinct transition pathways known as tiered
synchronization have been reported in the presence of
both pairwise and higher-order interactions [57, 58], in
addition to the continuous and explosive transition to
synchronization. In tiered synchronization the system
transits from the incoherent state to a weak synchro-
nization state via continuous path and then it abruptly
jump from the weak synchronization state to strong syn-
chronization state. Where as in the backward transition,
depending on the hysteresis width, it jumps from the
synchronization state to the weak synchronization or in-
coherent state. Several modification in the Kuramoto
system can induce tiered paths. For example introduc-
tion of time delay in the coupling function in a system
with higher-order interactions originate tiered paths [57].
Also adapting the global order parameter in the higher-
order terms can give rise to tiered route to synchroniza-
tion [58]. An interplay between adaptation in higher-
order term in absence of any adaptation in pairwise term
leads the system to transit from the explosive to continu-
ous paths of synchronization via tiered paths. Whereas,
in absence of higher-order interactions the adaptation pa-
rameter in the pairwise term promotes only explosiveness
of synchronization transitions in a phase frustrated sys-
tem [30]. Till now the effect of the adaptation of order
parameter in a phase frustrated system with higher-order
interactions has not been explored. In this paper, we fo-
cus on the reciprocity between the global order param-
eter adaptation, higher-order coupling strength and the
phase-lag term. Since we are aware of the role of these
parameters separately in the synchronization transitions
from the previous literature, we are eager to elaborate the
combined effect of these parameters in a phase oscillator
system.
Here, we accomplish it through numerical simulations
of all-to-all connected networks. Tiered synchronization
paths are observed for certain selections of parameter val-
ues. To investigate the mechanism behind the numerical
outcomes, we utilize the Ott-Antonsen reduction tech-
nique, which reduces the dimension of the entire network.
Next, we examine the reduced order model and identify
the bifurcations at various parameter regimes that gen-
erally guarantee three distinct kinds of synchronization
transitions. This analytical procedure can be used to
fine-tune the parameters and acquire the necessary tran-
sition routes to synchronization. We note that in some
parameter regimes, the phase-lag parameter and higher-
order coupling strength favor first-order and tiered syn-
chronization techniques.
II. PHASE OSCILLATOR MODEL
In this paper, we consider an adaptive system of N
coupled Sakaguchi-Kuramoto phase oscillators interact-
ing through pairwise and triadic connection. The dy-
namics of the system is governed by the equations
˙
θi=ωi+K1ra
1
N
N
X
j=1
sin(θj−θi−β)
+K2rb
1
N2
N
X
j=1
N
X
k=1
sin(2θj−θk−θi−β),(2)
i= 1,2,...,N,
where θiis the phase and ωiis the natural frequency of
the ith oscillator. K1and K2are the coupling strengths
3
corresponding to the pairwise and triadic interactions re-
spectively. aand bare two real constants, and β∈[0,π
2),
denotes the phase frustration or phase-lag in the system.
The natural frequencies ωiare drawn from a distribution
g(ω). The level of synchronization are measured by two
global order parameters r1and r2corresponding to pair-
wise and higher-order interactions respectively and are
defined by
z1=r1eiψ1=1
N
N
X
j=1
eiθj,(3)
and z2=r2eiψ2=1
N
N
X
j=1
e2iθj.(4)
The complex valued order parameters z1and z2describe
the macroscopic dynamics of the whole oscillator popula-
tion. ψ1and ψ2are the average phases of the oscillators.
Here r1= 0 indicates the incoherence state and r1= 1
indicates the perfect synchronization state, whereas, r2
measures the 2-cluster synchronization states of the os-
cillators.
III. NUMERICAL OBSERVATIONS
First we numerically simulate the system (2) to under-
stand the effect of the phase-lag βon the transition to
synchronization in the presence of global order parameter
(r1) adaptation . The numerical simulation of the system
(2) for a network of size N= 1000 is performed by using
the 4-th order Runge-Kutta method. Intrinsic frequen-
cies of the phase oscillators are drawn from a Lorentzian
distribution with mean 0 and half-width 1.
For primary investigation, we adapt the order param-
eter r1with the higher-order coupling only, i.e. we set
a= 0 and b6= 0 in the equation (2). The transitions
are then studied by looking at the variation of the order
parameter r1either (i) with the pairwise coupling K1for
fixed K2,band βor (ii) with the triadic coupling K2for
fixed K1,band β. In both the cases, forward and back-
ward continuations of the solutions are done by setting
a= 0 and b= 2.
In case (i), we take K2= 10 and compute the variation
of r1with K1for different values of β. The forward
continuation for each βstarts with K1= 0 by drawing
the initial phases of the oscillators uniformly from the
range −πto π. The value of r1is then computed after
removing the transients. Subsequently, the value of K1is
increased in small steps till K1= 6 and in each step the
system (2) is integrated using the last point of previous
solution trajectory as the initial condition. While, for
backward continuation, for each β, simulation starts with
K1= 6 and the value K1is then reduced in small steps,
and in each case, the last point on the solution trajectory
of the previous solution is used as initial condition for
the present simulation. Thus, the variations of r1with
123456
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20 25
0
0.2
0.4
0.6
0.8
1
(a)
(b)
FIG. 1: Synchronization profile as obtained from the forward
(filled circles connected by solid lines) and backward (filled
circles connected by dashed dot lines) numerical continuation
of the solutions of the system (2) for a= 0 and b= 2. (a)
Variation of r1with K1for K2= 10 and different β. (b)
Variation of r1with K2for K1= 2.5 and different β. The
purple, sky blue, red, green and light green colors respectively
represent the curves for β= 0, 0.5, 0.6, 0.8 and 1.
K1are obtained for different βboth for the forward and
backward continuations.
FIG. 1(a) shows the variation of r1with K1in the
range 1 to 6 for four different β, namely, β= 0,0.5,0.8
and 1. The forward continuation data points are denoted
by filled circles of different colors connected by solid lines,
while, the backward continuation data points are shown
with filled circles connected by dashed dot lines. From
the figure it is seen that for all values of β, the system re-
mains in desynchronized state (r1≈0) near K1= 1. The
order parameter follows different paths during the for-
ward and backward continuation of the solutions which
4
results in a hysteresis loop for lower values of β. However,
the width of the hysteresis loop is gradually decreased
with the increment of βand the transition becomes con-
tinuous from discontinuous. It is also observed that for
the discontinuous transitions, the forward or backward
or both transitions from incoherence to coherence or vice
versa occurs via an intermediate weak synchronization
state. This type of discontinuous transition involving a
continuous path of weekly synchronized states is termed
as tiered transition in the literature [57–60]. As the value
of βis increased the length of the continuous path ex-
pands and persists over a broader range of the coupling
strength K1and as βis raised beyond a critical value,
continuous transition to synchronization is observed.
On the other hand, in the case (ii), as K2is varied by
fixing K1= 2.5, we observe a different scenario. Note
that the forward and backward continuation of the solu-
tions are done similarly as those have been done for the
case (i). The variation of the global order parameter r1
with K2for different βare shown in the FIG. 1(b). For
β= 0, the forward and backward continuation follow the
same path and the transition to synchronization is con-
tinuous. As βis increased, the transition becomes dis-
continuous involving partially synchronized states. For
relatively lower values of β, the discontinuous transition
occurs involving tiered synchronization states. However,
with the increase of β, the transition increasingly be-
comes explosive in nature.
Thus, from the simulation results discussed above, the
observation of the occurrence of tiered synchronization
due to the adaptation of order parameter value in the
triadic coupling in presence of phase-lag term appears to
be quite interesting. To understand the origin of such dy-
namical behaviours of the considered system, we proceed
with an analytical treatment in the next section.
IV. LOW DIMENSIONAL MODELLING
We now use the Ott-Antonsen ansatz [61] to reduce
the dimension of the globally coupled system of equations
(2). To proceed further, we simplify the system (2) and
rewrite it in terms of the order parameters defined in (3)
as
˙
θi=ωi+1
2ihe−i(θi+β)H−ei(θi+β)¯
Hi,(5)
where H=K1ra
1z1+K2rb
1z2¯z1and ¯
Hdenotes the com-
plex conjugate of H. Considering the thermodynamic
limit as N→ ∞, let the density of the oscillators at time
twith phase θand frequency ωbe given by the continu-
ous function f(θ, ω, t), and we normalize it as
Z2π
0
f(θ, ω, t)dθ =g(ω).
Next we take g(ω) as Lorentzian distribution given by
g(ω) = ∆
π[∆2+(ω−ω2
0)] and expanding f(θ, ω, t) in a Fourier
series in θ, we obtain
f=g(ω)
2π(1 +
∞
X
n=1 hfn(ω, t)einθ +ˆ
fn(ω, t)e−inθ i),
(6)
where fn(ω, t) is the co-efficient of the n-th term of the
series and ˆ
fnstands for the complex conjugate of fn(ω, t).
Now following [61], we consider an additional ansatz fn=
αn(ω, t), where αis an analytic function satisfying the
condition |α| ≤ 1 for convergence.
Since we are dealing with a static network, to maintain
the conservation of the oscillators in the network, the
density function fsatisfies the continuity equation
∂f
∂t +∂
∂θ (f v) = 0,(7)
where, v=dθ
dt is the velocity field on the circle that drives
the dynamics of fand is given by the equation (5). Sub-
stituting the expression of ffrom the equation (6) into
the continuity equation (7), we obtain the one dimen-
sional differential equation for αas
˙α+iαω +1
2Hα2e−iβ −¯
Heiβ = 0.(8)
In the continuum limit, the complex order parameters
can be written as
z1=Z∞
−∞ Z2π
0
f(θ, ω, t)eiθ dθdω,
=Z∞
−∞
g(ω)¯αdω. (9)
To reduce the dimension of our model, we now intro-
duce one more restriction on our assumed form of f;
we consider here that α(ω, t) has no singularities in the
lower half of the ω-plane, and that |α(ω , t)| → 0 as
Im(ω)→ −∞. Therefore, the above integration can now
be done by taking a close contour in the lower half the
ω-plane and for the Lorentzian distribution, we obtain
z1= ¯α(ω0−i∆, t).
In a similar manner, we can find the order parameter
z2as
z2=Z∞
−∞ Z2π
0
f(θ, ω, t)e2iθ dθdω,
=Z∞
−∞
g(ω)¯α2dω,
= ¯α2(ω0−i∆, t),
=z2
1.(10)
This provides the relation between two order parameters
z1and z2. Now setting ω=ω0−i∆ in the equation (8)
we obtain the evolution equation for z1as
˙z1=iz1ω0−z1∆ + 1
2[(K1ra
1z1+K2rb
1z2
1¯z1)e−iβ
−z2
1(K1ra
1¯z1+K2rb
1¯z2
1z1)eiβ ].(11)
5
Substituting the relation z1=r1eiψ1in the equation (11)
and comparing the real as well as imaginary parts we
obtain the 2-dimensional system of first order ordinary
nonlinear differential equations
˙r1=−∆r1+cos β
2(K1ra+1
1+K2rb+3
1)[1 −r2
1],(12)
˙
ψ1=ω0−sin β
2(K1ra
1+K2rb+2
1)[1 + r2
1],(13)
to investigate the dynamics of the system (2). Subse-
quently we use the reduced order model (ROM) given by
the equations (12)-(13) for a detailed analysis.
V. ROM VS NUMERICAL SIMULATION
To understand the origin of different kinds of transi-
tions to synchronization including the ones observed in
the numerical simulation, described in the section (III)
we perform detailed bifurcation analysis of the reduced
order model (12) using the MATCONT software [62].
The results are then compared with that of the numeri-
cal simulations of the full system (2) at different points
of the parameter space for validation.
Before proceeding further, we focus on the synchro-
nization curves obtained from the numerical simulation
of the system (2) and presented in the FIG. 1(a) for
β= 0.5 (sky blue), and figure 1(b) for 0.6 (green) re-
spectively where transitions are related with tiered syn-
chronization. To understand the origin of such interest-
ing transition in the presence of triadic interactions, we
construct two bifurcation diagrams using the ROM (12)
for the same set of parameter values used in the numer-
ical simulation. The bifurcation diagrams are shown in
the figures 2(a) and (c) respectively along with the cor-
responding numerical simulation data.
In the FIG. 2(a), the solid and dashed black curves
respectively represent the stable and unstable solutions
as obtained from the ROM (12). The r1= 0 solution
is stable for lower values of K1and it becomes unsta-
ble through supercritical pitchfork bifurcation (PB) at
K1= 2.279. A stable branch (solid black curve) with
nonzero r1is generated from there and the zero solution
continue to exist as unstable one for K1>2.279. The
nonzero branch move forward a bit to undergo a saddle-
node (SN1) bifurcation at K1= 2.457 (pink square).
The resulting unstable blanch (dashed black curve) move
backward and undergoes another saddle-node (SN2) bi-
furcation (pink dot) at K1= 2.038. The stable branch
originated out of the SN2 bifurcation then continue to
exist for higher values of K1. Now the occurrences of
a pair of saddle-node bifurcations leads to discontinuous
transition to synchronization in the system with the vari-
ation of K1in the presence of adaptation in the triadic
interaction.
For the validation of the ROM results we now super-
pose the data obtained from the numerical simulation
2 4 6
0
0.5
1
0 10 20
0.2
0.6
1
024
0
0.5
1
0 10 20 30
0
0.5
1
1.5
(b)
(d)
(a)
weakly/
strongly sync/
incoherent
weakly sync/
incoherent
(c)
continuous transition
weakly
sync/
incoherent
weakly/
strongly sync/
incoherent
continuous
transition strongly
sync
strongly sync
FIG. 2: Bifurcation diagrams constructed from the ROM (12)
for a= 0 and b= 2: (a) r1vs K1for K2= 10 and β= 0.5,
and (c) r1vs K2for K1= 2.5 and β= 0.6. The stable and un-
stable branches are shown with solid and dashed black curves
respectively. The corresponding data points obtained from
the numerical simulation of (2) are shown with filled circles
connected by solid (forward) and dashed dot (backward) lines.
The saddle-node and pitchfork bifurcation points on the bifur-
cation curve are marked with SN1, SN2 and PB respectively.
Two parameter diagrams showing different synchronization
regimes on the: (b) K1−βand (d) K2−βplanes. The solid
(dashed) blue curve indicates the backward (forward) saddle
node points for different β. The pink squares and dots respec-
tively indicate the forward and backward saddle node points
shown in the bifurcation diagrams (a) and (c).
both for forward and backward continuation of the solu-
tions on top of the bifurcation diagram. The forward and
backward continuation data points are presented with
cyan dots connected by solid and dashed dot lines re-
spectively. The numerical simulation data points show a
very close match with the ROM results. We note that
between the SN2 and SN1 points, three stable states
of the system, namely, incoherent, weakly synchronized
and strongly synchronized coexist. As a result, with the
variation of K1, transition occurs in the system through
tiered synchronization states. Thus, we claim that the
occurrence of a pair of saddle-node bifurcations is re-
sponsible for the appearance of discontinuous transition
in the system involving tiered synchronization.
For detailed understanding, we now prepare a two pa-
rameter diagram using the ROM showing different syn-
chronization regimes on the K1−βplane (see FIG. 2(b)).
The dashed and solid blue curves in the diagram respec-
tively show the locations of the successive saddle-node
bifurcation points as a function of β. The pink square
6
0 2 4 6
0
0.2
0.4
0.6
0.8
1
1 2 3 4 5 6
(a) (b)
FIG. 3: Bifurcation diagram constructed from the ROM: r1
vs K1for (a) b= 0.1, β= 0.1 and (b) b= 2, β= 1 with
fixed K2= 10, a= 0. The solid and dashed black curves
respectively show the stable and unstable fixed points of the
ROM (12). The data points obtained from the numerical
simulation of the system (2) are shown with filled colored
circles.
and dot on these lines correspond to the SN1 and SN2
points of the FIG. 2(a) respectively. As long as two dis-
tinct saddle-node bifurcation points exist, the transition
becomes discontinuous involving tiered synchronization.
The saddle-node curves meet at β= 0.8298, and the tran-
sition becomes continuous thereafter. The different syn-
chronization regimes are clearly demarcated and shown
in the FIG. 2(b). Thus, from the two parameter dia-
gram we observe that the phase-lag parameter βinhibits
discontinuous transition for fixed K2. This gradual sup-
pression of the discontinuous transition by βis consistent
with the numerical observations presented in the section
(III).
Next, the variation of the order parameter r1for dif-
ferent stable and unstable solutions with K2for fixed
β= 0.6 and K1= 2.5 are shown in the bifurcation di-
agram presented in the FIG. 2(c). The stable and un-
stable solutions are shown with solid and dashed black
curves. The r1= 0 solution is unstable in the entire
range of K2.The non-zero solutions (r16= 0) show a
pair of saddle-node bifurcations marked by pink square
and dot. In between these two saddle-node bifurcation
points, the system is bistable which introduces a large
hysteresis loop leading to discontinuous transition. Here
also the transition involves tiered synchronization. The
data points obtained from the numerical simulations of
the full system (2) are then superposed on the bifurcation
diagram. The forward and backward continuation data
are shown with red dots connected by solid and dashed
dot red lines respectively. In this case also the numeri-
cal simulation results show a very close match with the
ROM results.
The detailed bifurcation scenario can be seen from the
two parameter diagram shown in the FIG. 2(d) present-
ing different synchronization regimes on the K2−βplane
for fixed K1= 2.5, a= 0 and b= 2. The solid and dashed
blue curved respectively show the variation of the back-
ward and forward saddle-node bifurcation points with
β. The pink square and dot correspond to the saddle-
node bifurcation points shown in the FIG. 2(c). We note
here that unlike the two parameter diagram shown in the
FIG . 2(b), the region enclosed by the blue curves is not
closed. Both the forward and backward saddle-node bi-
furcation curves diverge towards very high value of K2
with the enhancement of β, where the rate of divergence
of the forward SN point is much faster compared to the
backward one. This is also consistent with the numerical
observation shown in the FIG. 1(b).
In the FIG.2 (a) and (c), we have either varied the pa-
rameters K1or K2keeping the other parameters fixed
and observed tiered synchronization. From the bifur-
cation point of view, apart from tiered synchronization,
we have also have identified the points in the parameter
space where we observe second order and explosive transi-
tions to synchronization both in the ROM and numerical
simulations. FIG. 3(a) and (b) show the bifurcation dia-
grams obtained from the ROM (12) where explosive and
second order continuous transitions are observed with the
variation of K1. The numerically obtained data points
for the same set of parameter values are also plotted in
these figures which show a very close match. Inspired by
this observation, we now wish to develop a deeper un-
derstanding of the parameter space using the ROM. In
this context, a closer look at the bifurcation curves pre-
sented in the FIGs. 2(a), 3(a) and (b), reveal that for
tiered, explosive and continuous transitions to synchro-
nization, the synchronization curves possess three, two
and one extrema points respectively of the graph of K1
as a function of r1given by
K1=2
ra
1(1 −r2
1) cos β−K2r(b+2−a)
1.(14)
Similarly, as K2is varied by keeping the other parameters
fixed, the number of extrema of the function
K2=2
r(b+2)
1(1 −r2
1) cos β−K1ra
1
r(b+2)
1
(15)
will determine the nature of transitions in the system.
Thus, using the equations (14) and (15) we prepare the
diagrams shown in the figures (4), (6) and (8. Different
transition regions are nicely depicted in these figures.
FIG. 4 shows the transition regimes on the K2−bplane
for β= 0.5 and 1 with the variation of K1.
The figure clearly shows that βpromotes second or-
der transition in the absence of adaptation in the pair-
wise coupling when K1is varied for fixed K2and b. On
the other hand, βsuppresses the tiered and first order
transitions for similar variations of the parameters. To
explore the transitions in more detail, we take the point
(K2, b) = (2,2) inside the green region of the FIG. 4
where second order transition occurs with the variation of
K1and then we move horizontally to enter the tiered re-
gion. The bifurcation diagrams obtained from the ROM
(12) alongside the numerical simulation data are shown
7
FIG. 4: Different synchronization transition regimes on the
K2−bplane obtained with the variation of K1in the ROM
for (a) β= 0.5 and (b) β= 1 in the absence of adaptation in
the pairwise coupling (a= 0). First order (red), tiered (grey)
and second order (green) regimes are separated by dashed
black curves, indicates the critical parameter sets.
012345
0
0.2
0.4
0.6
0.8
1
0 2 4
0
10
20
30
strongly sync
(a)
incoherent/
weakly sync
incoherent/
weakly sync/
strongly sync
(b)
continuous transition
FIG. 5: Bifurcation diagram constructed from the ROM: (a)
Synchronization profile for different K2with fixed a= 0, b= 2
and β= 0.5. Numerically calculated points (colored filled cir-
cles) are plotted along with the solid black curves obtained
from the ROM. Dashed black curve indicates the unstable
states. (b) Two parameter diagram showing different synchro-
nization regimes on the K1−K2plane. The existence of both
forward (dashed blue curve) and backward (solid blue curve)
saddle node points implies tiered synchronization states in the
system.
in the FIG. 5 (a) for three values of K2for fixed b= 2. A
very good match of the bifurcation structure with the
numerical simulation data is observed. As mentioned
earlier, the tiered synchronization are characterized by a
pair of saddle-node bifurcations. The variation of those
saddle-node bifurcation points on the K1−K2planes
are shown with the solid and dashed blue lines in the
FIG. 5(b). These lines separate different synchronization
regimes on that plane.
FIG. 6 determined from the equation (14) shows the
synchronization transition regimes on the β−bplane
for two values of K2. In this case, K2suppresses the
second order transition, and promotes first order and
tiered transitions. To compare the ROM results with
FIG. 6: Different synchronization transition regimes on the
β−bplane obtained with the variation of K1in the ROM for
(a) K2= 10 and (b) K2= 20 in the absence of adaptation in
the pairwise coupling (a= 0). First order (red), tiered (grey)
and second order (green) regimes are separated by dashed
black curves, indicates the critical parameter sets.
-1 0 1 2 3 4 5
0
0.2
0.4
0.6
0.8
1
0 2 4
0
2
4
6
incoherent/
weakly sync
(a) (b)
strongly sync
incoherent/
weakly sync/
strongly sync
FIG. 7: Bifurcation diagram constructed from the ROM: (a)
r1vs K1for different bwith K2= 10, a= 0 and β= 0.5.
The solid and dashed black curves respectively show the sta-
ble and unstable fixed points of the ROM (12). The data
points obtained from the numerical simulation of the system
(2) are shown with filled circles of different colors. (b) Two
parameter diagram showing different synchronization regimes
on the K1−bplane. Solid and dashed blue line respectively
show the backward and forward saddle node points.
that of the numerical simulations, we take three points
(0.5,0), (0.5,2) and (0.5,4) from the β−bplane shown in
the FIG. 6(a) where, respectively, first order, tiered and
second order transitions to synchronization occur in the
ROM. We then construct bifurcation diagrams from the
ROM using the parameter sets corresponding to these
points and show in the FIG. 7(a) alongside the numeri-
cal simulation data points. A very close match between
the ROM and full system results are observed. We then
go ahead to identify different synchronization regimes on
the K1−bplane using the ROM (12) and present in
the FIG. 7(b). The solid and dashed blue curves en-
closing the cyan region in the figure shows the backward
and forward saddle-node points associated with the tiered
transition respectively and the blue solid line denotes the
8
location of pitchfork bifurcation points. Other synchro-
nization regimes are also clearly depicted in this figure.
FIG. 8: Different synchronization transition regimes on the
K1−bplane obtained with the variation of K2in the ROM
for (a) β= 0.5 and (b) β= 1 setting a= 0. The solid
and dashed black curves separate the regimes of first order
(light green), tiered (magenta) and second order (light red)
synchronization transitions.
0 10 20 30 40 50
0.2
0.4
0.6
0.8
1
0 10 20 30 40 50
0
1
2
3
4
(a)
incoherent/
weakly sync/
strongly sync
strongly sync
incoherent/
weakly sync
(b) continuous transition
FIG. 9: (a) Bifurcation diagrams constructed using the ROM
for a= 0, b= 3.5 and β= 0.5 and different values of K1.
The solid and dashed black curves respectively represent the
stable and unstable fixed points of the ROM. The data points
obtained by the numerical simulation of the system (2) are
also shown on top of the bifurcation curves. (b) Two pa-
rameter diagram showing different synchronization regimes
as obtained from the ROM on the K2−K1plane. Solid and
dashed blue curves respectively show the backward and for-
ward saddle node bifurcation points of the ROM.
Next we use the equation (15) to investigate the effect
of phase frustration on the transitions to synchronization.
FIG. 8 shows the two parameter diagram prepared using
the equation (15) depicting various transition regimes on
the K1−bplane for β= 0.5 and 1 with the variation
of K2. From the diagram it is seen that in this case,
phase-lag promotes first order as well as tiered transi-
tion to synchronization, while, it suppresses second or-
der transition. In this case also, we take three points,
namely, (1.5,3.5), (2.5,3.5) and (4,3.5) from the K1−b
plane shown in the FIG. 8(a) where first order, tiered
and second order transitions respectively are observed in
the ROM. We construct bifurcation diagrams from the
ROM using the parameter set corresponding to those
points and present in the FIG. 9(a) along with the as-
sociated numerical simulation data points obtained from
the full system (2). The comparison of the ROM results
with that of the numerical simulation of the full system
not only show similar transition type but also show a
very close match in the order parameter values. As done
in the previous two cases, here also we identify different
synchronization regimes on the K2−K1plane using the
ROM and present in the FIG. 9(b).
12345
0
0.2
0.4
0.6
0.8
1
0 5 10
0
2
4
6
8
2 2.5
0
0.05
0.1
2 2.5 3
0
0.5
1
(a) (b)
incoherent/
strongly sync
incoherent
FIG. 10: (a) Bifurcation diagram constructed from the ROM
for K2= 10, b= 2 and β= 0.5 and different values of a.
The stable and unstable fixed points of the ROM are shown
with solid and dashed black curves. The corresponding r1
values obtained from numerical simulation of the system (2)
are shown with filled circles of different colors. The inset
presents the bifurcation diagram for a= 0.02 showing three
saddle node bifurcation points with red dots. (b) Two pa-
rameter diagram depicting different synchronization regimes
on the K1−aplane. The inset shows an enlarged view of the
indicated region. Three red dots correspond to the saddle-
node points shown in the inset of (a).
So far we have investigated different types of transi-
tions to various synchronization states in the absence of
the adaptation in the pairwise coupling term (a= 0). We
have extensively used the ROM (12) for the investigation
and validated the results with the ones obtained from the
numerical simulation of the full system (2). The analysis
revealed a clear picture of the transition scenario involv-
ing continuous as well as discontinuous transitions in a
wide region of the parameter space. The discontinuous
transitions include both explosive and tiered synchroniza-
tions.
Encouraged by the satisfactory match of the results,
we now consider the case where the adaptation of the
order parameter is present in both pairwise and higher-
order term. The FIG. 10 demonstrates the role of the
adaptation exponent aon the transition to synchroniza-
tion. In the FIG. 10(a), the bifurcation diagrams pre-
pared from the ROM are shown for different values of
a. The values of the other parameters K2, band βare
kept fixed. For this parameter choice, the system shows
tiered transition for a= 0 and explosive transition for
9
other values of a. The numerically computed order pa-
rameters from the full system (2) are seen to fit nicely
with the ROM curve. Beside this synchronization di-
agram, we also plot the stability diagram in Fig.10(b)
to look into the bifurcation points. In the figure, the
solid blue curve represents the saddle-node bifurcation
points as a function of a. For very small values of a
(0 < a ≤0.07238), more than one saddle-node bifurca-
tion points are obtained and for a > 0.07238, there is
only one SN point. Thus, for 0 < a ≤0.07238, tiered
synchronization is observed, while, for a > 0.07238, we
obtain explosive synchronization only. The blue curve
in the figure clearly divides the region into different syn-
chronization regimes.
The exhibition of different types of transitions to syn-
chronization in the system depends on the complex in-
teraction among the pairwise and triadic coupling terms,
and the phase frustration parameter β. The phase frus-
tration βand the pairwise coupling strength K1are
known to promote second order transition by inhibit-
ing discontinuous transition [53], while, the higher-order
coupling K2promotes discontinuous transition in the sys-
tem [56]. Since the global order parameter r1ranges from
0 to 1, for any value of a, b ≥0, 0 ≤ra
1, rb
1≤1. Multipli-
cation of K1by ra
1and K2by rb
1reduces the respective
effective coupling strengths for a, b > 1. Thus, relative
dominance of the pairwise and triadic coupling terms in
the presence of phase frustration depends on the parame-
ter ranges and the type of the transitions are determined
accordingly in the system.
Finally, we compute the mean field frequency Ω us-
ing the equations (12) and (13), and the full system (2).
FIG. (11) represents the variation of Ω with both pair-
wise coupling K1and higher-order coupling K2. The
figure shows that the mean frequency of the system is
also closely captured by the reduced order model derived
from the full system.
0246
-6
-4
-2
0
0 5 10 15 20 25
-12
-8
-4
0
(a) (b)
FIG. 11: (a) Mean frequency Ω as a function of pairwise
coupling strength K1for phase-lag β= 0.5 and (b) Ω as
a function of higher-order coupling strength K2for β= 0.6.
Filled green circles indicate numerically calculated points and
the solid (dashed) red curves indicate the stable (unstable)
points calculated from the ROM.
VI. CONCLUSIONS
In this paper, we have investigated the phenomenon
of transition to synchronization in globally coupled sys-
tem of Sakaguchi-Kuramoto oscillators in the presence
of higher-order interactions (up to triangular) and order
parameter adaptation. The system involves five control
parameters, namely, K1,K2,a,band βrepresenting the
coupling constants and adaptation exponents in the pair-
wise and triadic coupling terms, and phase frustration
respectively.
Primary numerical simulations of the full system at
some specific points in the parameter space show inter-
esting continuous as well as discontinuous transitions of
different types. However, investigating the role of dif-
ferent parameters in determining the type transitions to
synchronization in the system by performing numerical
simulations only is a challenging task and computation-
ally expensive. Thus, along with the performance of nu-
merical simulations of the full system, we also derive a
simple reduced order model of the system using the Ott-
Antonsen ansatz to understand the transition scenario in
detail in a large region of the parameter space.
The comprehensive bifurcation analysis of the ROM
clearly demarcates different regions of the parameters
space exhibiting different types of transitions to synchro-
nization. The analysis reveals a complex dependence of
the transitions on the parameters. It is observed that the
second order continuous transition is connected with a su-
percritical pitchfork bifurcation of the ROM. While, the
discontinuous teired transition is associated with multi-
ple SN bifurcations along with a supercritical PB and the
first order explosive transition involves a single subcrti-
cal PB alongside a SN bifurcation. The results obtained
from the ROM are compared with that of the numerical
simulations at several points of the parameter space and
are found to match closely.
The investigation of the system both numerically as
well as analytically shows that the joint role of different
parameters of the system in the transition to synchro-
nization can not be understood simply by superposing
their individual roles. For example, the phase frustration
is known to promote second order transition in the the
absence of adaptation and higher order coupling [25, 26].
However, as seen in the FIG. 8, under certain conditions,
the phase frustration may promote discontinuous transi-
tions. Nonetheless, in some situations as depicted in the
FIG. 4, the phase frustration can suppress the discontin-
uous transitions and exhibits its usual role in promoting
second order transition in the system. Therefore, a gen-
eral conclusion about the role of the individual parame-
ters on the transitions is found to be difficult. Best way
to look at the pro jections of the parameter space on two
dimensional planes as has been done in this paper for a
better understanding of the transitions in the parameter
space.
The results of the investigation presented in the paper
show that in the sole presence of adaptation in the tri-
10
adic coupling, as K1is varied for fixed b,K2and β, the
system exhibits all three types of transitions depending
on the values of the fixed parameters as seen in the FIG.
4(a). It is also seen that with the enhancement of β, the
discontinuous transition regions are suppressed (see FIG.
4(b)). On the contrary, FIG. 8 suggests the promotion
of discontinuous transition by βon the K1−bplane.
Similarly, the promotion of discontinuous transitions by
the higher order coupling parameter K2is evident from
the FIG.5. Thus, the present analysis provide an elegant
methodology to unfold the transition scenario in the sys-
tem.
ACKNOWLEDGEMENTS
S.D. acknowledges the support from DST, India under
the INSPIRE program (Code No. IF190605).
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