J.Phys.Complex. 2(2021) 035002 (10pp) https://doi.org/10.1088/2632-072X/abedc2
OPEN ACCESS
RECEIVED
16 November 2020
REVISED
19 February 2021
ACCEPTED FOR PUBLICATION
11 March 2021
PUBLISHED
15 April 2021
Original content from
this work may be used
under the terms of the
Creative Commons
Attribution 4.0 licence.
Any further distribution
of this work must
maintain attribution to
the author(s) and the
title of the work, journal
citation and DOI.
PAPER
The impact of chaotic saddles on the synchronization of
complex networks of discrete-time units
Everton S Medeiros1,∗,ReneOMedrano-T
2,3,IberêLCaldas
4and Ulrike Feudel5
1Institut für Theoretische Physik, Technische Universit¨
at Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
2Departamento de Física, Universidade Federal de S˜
ao Paulo, Campus Diadema, R. S˜
ao Nicolau, 210, 09913-030 SP, Brazil
3Departamento de Física, Universidade Estadual Paulista, Instituto de Geociências e Ciências Exatas, Campus Rio Claro, Av. 24A, 1515,
13506-900 SP, Brazil
4Institute of Physics, University of S˜
ao Paulo, Rua do Mat˜
ao, Travessa R 187, 05508-090, S˜
ao Paulo, Brazil
5Institute for Chemistry and Biology of the Marine Environment, Carl von Ossietzky University of Oldenburg, Oldenburg, Germany
∗Author to whom any correspondence should be addressed.
E-mail: medeiros@tu-berlin.de
Keywords: chaotic saddle, synchronization, networks
Abstract
A chaotic saddle is a common nonattracting chaotic set well known for generating finite-time
chaotic behavior in low and high-dimensional systems. In general, dynamical systems possessing
chaotic saddles in their state-space exhibit irregular behavior with duration lengths following an
exponential distribution. However, when these systems are coupled into networks the chaotic
saddle plays a role in the long-term dynamics by trapping network trajectories for times that are
indefinitely long. This process transforms the network’s high-dimensional state-space by creating
an alternative persistent desynchronized state coexisting with the completely synchronized one.
Such coexistence threatens the synchronized state with vulnerability to external perturbations. We
demonstrate the onset of this phenomenon in complex networks of discrete-time units in which
the synchronization manifold is perturbed either in the initial instant of time or in arbitrary states
of its asymptotic dynamics. The role of topological asymmetries of Erdös–R´
enyi and
Barabási–Albert graphs are investigated. Besides, the required coupling strength for the occurrence
of trapping in the chaotic saddle is unveiled.
1. Introduction
Chaotic saddles are ubiquitous across dynamical systems. Such non-attracting chaotic sets appear in the
system’s state-space caused by a variety of mechanisms. For instance, they emerge as a result of a boundary cri-
sis [1,2], in which a chaotic attractor loses stability [3]. They also appear via a basin boundary metamorphosis
[4], in which a smooth boundary turns into a fractal, or even, via an embedded saddle-node bifurcation at
which the basin boundary of a stable fixed point collides with the chaotic attractor converting it into a chaotic
saddle [5,7]. This mechanism can be observed when a periodic window within a chaotic parameter range is
created—a situation relevant for the study here. Moreover, there are other mechanisms for the appearance of
such nonattracting chaotic sets, e.g., outer homoclinic and heteroclinic tangencies [6,7]. Chaotic saddles play
an essential role in the dynamics of various applications, such as e.g. in chaotic scattering processes [8–10]; in
spatially extended dynamical systems [11,12]; in the hopping dynamics between different coexisting attractors
in a multistable system [13]; in controlling long chaotic transients [14]; in the control of dynamical systems
either to sustain chaos [15] or to drive the system toward desired attractors [16] or terminating chaotic tran-
sients [17]. In complex networks of oscillators chaotic saddles can be responsible for the switching between
different space–time patterns [18]. Chaotic saddles are also of great importance in hydrodynamics. There, the
signatures of chaotic saddles can be detected via the fractal boundaries in open hydrodynamical flows [19].
In the transition to turbulence in a pipe flow, the chaotic saddle acts as a threshold separating the laminar
flow from turbulent intermittency, often called the edge of chaos [20,21]. In chaotic advection, the fractal
structure of chaotic saddles is responsible for the enhancement of particle concentrations [22], and also for
© 2021 The Author(s). Published by IOP Publishing Ltd
J.Phys.Complex. 2(2021) 035002 (10pp) E S Medeiros et al
trapping aerosols in their neighborhood preventing escaping in open flows [23]. In noisy dynamics, chaotic
saddles constitute the backbone of long-living erratic trajectories in a phenomenon called noise-induced
chaos [24]. Finally, one of the most known features of chaotic saddles is the generation of finite-time chaotic
behavior in deterministic dynamical systems ranging from simple discrete-time maps to partial differential
equations [25,26].
In networked dynamical systems, chaotic saddles may be present in the state-space of individual system
units. In this scenario, the attractive character of the network coupling creates a synchronization manifold,
i.e., all units behave identically in time. Normally, such an invariant manifold would oscillate chaotically in
the vicinity of the underlying chaotic saddle before converging to its asymptotic dynamics in times expo-
nentially distributed. However, under the effect of external perturbations, the synchronization manifold may
break down and, consequently, the network units go across the chaotic saddle with individual trajectories. This
mechanism gives rise to an interaction between the chaotic dynamics in the saddle and the network coupling
strength. Such interplay postpones the permanency of units in the chaotic saddle for times that are indefinitely
long, suppressing the occurrence of synchronization in the network [27,28]. Therefore, the occurrence of this
kind of vulnerability affects many systems relying on synchronization to function properly. Yet, the role of
asymmetrical network topologies such as the ones with nontrivial degree distribution is still unclear.
In this work, we examine the vulnerability of synchronized states in networks possessing chaotic saddles
in the state-space of their units. Our networks are composed of identical H´
enon maps coupled via complex
topologies such as Erdös–R´
enyi (ER) and Barabási–Albert (BA) graphs. First, we study the vulnerability of the
synchronization manifold to perturbations occurring at the initial time instant. This vulnerability is demon-
strated by attributing a different initial condition (IC) to one of the network units (perturbed unit), while the
remaining units receive identical values. By investigating the occurrence of complete synchronization in the
network for different ICs attributed to the perturbed unit, we verify that the occurrence of completely syn-
chronized states is sensitively dependent on the choice of ICs. In addition, we observe that the relative portion
of ICs attributed to perturbed units leading to synchronized states is significantly lower for perturbing units
with a high degree. These results are demonstrated by computing the single-node basin stability as a function
of the unit’s degree in a Barabási–Albert graph.
Next, we investigate the state-dependent vulnerability, i.e., perturbations are applied directly to asymp-
totic dynamical states visited by the synchronization manifold. By performing a comprehensive analysis of
the perturbation amplitudes, we find that the perturbation sets leading the network to desynchronization are
intermingled with the ones at which synchronization is restored. In addition, we verify that the vulnerability
of the synchronization manifold also depends on the dynamical state at which the perturbation is applied.
This dependence is demonstrated by assessing the effects of a given perturbation in consecutive states of the
synchronization manifold trajectory. Finally, we investigate the intervals of the network coupling strength for
which the interaction with the chaotic saddle generates vulnerabilities.
2. System dynamics
To illustrate the impact of chaotic saddles on the long-term dynamics of networked discrete-time systems, we
consider an ER graph with H´
enon maps oscillating in its nodes. The high-dimensional equations describing
this system are given by:
xt+1
i=f(xt
i,yt
i)+σ
Di
j∈Di
[f(xt
j,yt
j)−f(xt
i,yt
i)],
yt+1
i=g(xt
i,yt
i)+σ
Di
j∈Di
[g(xt
j,yt
j)−g(xt
i,yt
i)], (1)
where the functions f(x,y)=1−ax2+yand g(x,y)=bx providethedynamicsoftheH
´
enon maps in the
network. The parameter set (a,b) specifies the dynamical behavior of each individual map. In this study, we
fix these parameters at a=1.433 883 and b=0.178 063, whereas after a chaotic transient, the isolated maps
oscillate in a period-5 attractor Adenoted by the blue dots in figure 1.Thevectorvt
i=(xt
i,yt
i)definesthestate
space of each H´
enon unit iin the network with i=1, ...,N. The constant σcontrols the coupling strength
among the units. Following the ER graph shown in figure 2(a), Dispecifies the set of network units adjacent
to a given unit i, while the parameter Diis the number of units to which the node iis connected.
First, by considering σ=0inequation(1), we investigate the individual dynamics of the H´
enon maps
composing the network. Besides the aforementioned period-5 attractor, A,anunstablechaoticsetλ(chaotic
saddle) occurs in the system’s state-space. In figure 1, we show in red an approximation of the chaotic saddle
λ, obtained via the sprinkler method [26]. The gray dots are an approximation of the stable manifold of this
2
J.Phys.Complex. 2(2021) 035002 (10pp) E S Medeiros et al
Figure 1. State-space of an individual H´
enon map. The blue dots denote the period-5 attractor, A. The red dots are an
approximation of the chaotic saddle λcoexisting with A. The gray dots indicate the long times for trajectories starting at (x0,y0)
to reach a -neighborhood of A. The control parameters are fixed at a=1.433 883 and b=0.178 063.
Figure 2. (a) ER graph with N=25 nodes and average k=6.0. (b) BA graph with N=100 nodes. The node sizes represent
their respective degree.
saddle obtained via the long time for trajectories to reach a -neighborhood of the attractor (=10−3). The 5
blue dots indicate the period-5 attractor.
At the level of the units, all trajectories starting in the (x0,y0)-interval shown in figure 1will ultimately
converge to the period-5 attractor. However, the ICs signaled in gray in this figure will first approach the chaotic
saddle, oscillate chaotically in its neighborhood, and finally will escape toward the attractor. This state-space
characteristics of each individual H´
enon map will play an essential role in the dynamics of the coupled system.
3. Vulnerability with respect to the initial state
Recently, for a symmetrically coupled network of Duffing oscillators, the chaotic saddle has been found to play
a decisive role in the final state sensitivity of the network [27]. Specifically, the convergence to a completely
synchronized state or a desynchronized one depends sensitively on the system’s ICs. This is due to a nontrivial
interaction between the chaotic saddle and the network coupling, which in reference [27] is symmetric. Here,
we investigate the effects of the saddle–coupling interaction in the presence of network asymmetries such as
theonesinERandBAgraphsshowninfigures2(a) and (b), respectively. An ER graph, also called random
graph, consists of Nnodes connected by Lrandomly distributed links. Due to the random character of the link
assignments, the number of links varies among the nodes. Specifically, the number of links of a given node,
i.e., the node degree k, follows a Poisson distribution with average value k. Conversely, in the construction
of a BA graph, also called scale-free network, the concept of preferential attachment is incorporated resulting
in a growth algorithm where new nodes are more likely to be attached to more connected nodes. This proce-
dure results in large variabilities in the number of connections among the nodes, with the presence of highly
connected nodes, so-called hubs, absent in random graphs. Based on this construction, BA graphs possess a
power-law distribution of degrees as a stationary scale-free state, i.e., the power-law correlation is independent
of the graph size and growth time. For the purpose of our investigations, both ER and BA graphs provide the
required topological asymmetries. Nonetheless, in figure 5, we assess their vulnerability to the effects of the
chaotic saddle. For the network described by equation (1), the synchronized state of the high-dimensional sys-
tem lies in a synchronization manifold S,definedasvt
1=vt
2=···=vt
N. We denote the states visited by the
synchronization manifold Sin the period-5 orbit as vt
S=(xt
S,yt
S).
In order to quantify the influence of the chaotic saddle λon the global dynamics of the network, we per-
turb the synchronization manifold Sin the initial instant of time (t=0), i.e., in the initial state v0
S=(x0
S,y0
S).
3
J.Phys.Complex. 2(2021) 035002 (10pp) E S Medeiros et al
Figure 3. Space–time plots and time snapshot of the synchronization error Et
i. Top: the IC of perturbed unit i=6(D6=6) is
fixed at (0.46–0.338). The network converges to a synchronized configuration. Bottom: the IC of the perturbed unit is fixed at
(0.46–0.332). The network completely desynchronizes. The network has N=25 units distributed in an ER graph. The coupling
strength is fixed at σ=0.0005. The remaining network units receive (−0.02, 0.01) as ICs.
This strategy is realized by defining the ICs of the units at a common value v0
i=(−0.02, 0.01) except for one
unit chosen to be the perturbation to S. For instance, in the space–time plot shown in figure 3(a), the unit
i=6withdegreeD6=6actsasaperturbationtoSby receiving the IC v0
6=(0.460, −0.338). The color code
indicates the local synchronization error defined as Et
i=1/Dij∈Divt
i−vt
jwith i=1, ...,N.Withthis,
we observe that after an initial phase of desynchronized behavior in a large portion of the network, all units
approach Sfollowing the period-5 attractor. In figure 3(b), we show a spatial snapshot of the synchronization
error, E5000
i.Infigure3(c), we attribute a slightly different IC to the same unit i=6, this time, v0
6=
(0.460, −0.332). Differently from the previous scenario, the network goes toward a state of complete desyn-
chronization. The spatial snapshot at t=5000 time steps shown in figure 3(d) confirms the desynchronized
behavior. Hence, such sensitivity to the ICs observed in figure 3indicates that the chaotic saddle presents in
every unit in the uncoupled case influences the global dynamics of the network.
Given the topological asymmetries of the ER graph specifying the network structure in equation (1), we ask
the question of how synchronization would be affected by perturbing units with different degrees, Di.Before
addressing this issue, we define a measure to assess the levels of synchronization in the entire network. With
this, by analyzing the local synchronization error Et
i, we define the global synchronization error Zas:
Z=1
N
N
i=1
Ki,Ki=⎧
⎨
⎩
1, if Etf
i>δ
0, if Etf
i<δ. (2)
The parameter δestablishes the synchronization quality (δ=0.01) and tfis the overall number of iterations of
the network (tf=5×104steps). Hence, the completely synchronized state returns a global synchronization
error Z=0, while the completely desynchronized state implies a global synchronization error Z=1.
Now, in figure 4, we compute Zfor two-dimensional intervals of ICs attributed to perturbed units. In this
figure, the dots marked in blue corresponds to ICs leading to synchronization (Z=0), while the ones marked
in gray indicate ICs leading the entire network to a desynchronized configuration (Z=1). These diagrams
constitute two-dimensional sections of the system’s 2N-dimensional basin of attraction. First, in figure 4(a),
we show the interval of ICs attributed to the unit i=17, specifically, x0
17 ∈[−2.0, 2.0] and y0
17 ∈[−2.4, 2.4].
This unit has a relatively low degree, D17 =5. For this case, we observe that the ICs leading the network to syn-
chronized behavior, the sync basin (blue dots), are predominant. Nevertheless, the sparse ICs corresponding
to desynchronized behavior in the network appears intermingled with the sync basin. Next, in figure 4(b), we
consider the same interval of ICs but perturb the unit i=6withD6=6. In this case, the sync basin is already
smaller than in the previous scenario giving rise to more ICs leading to desynchronized behavior. In the interest
of further increase of the degree of the perturbed units, in figure 4(c), we show the effects of perturbing the unit
i=20 with D20 =7. The fraction of ICs leading the network to desynchronization is now larger than the ones
leading to synchronized behavior. The intermingled aspect of the basins is still visible even though areas with
a higher density of desynchronizing ICs appear. Finally, in figure 4(d), we show the basin of attraction in the
cross-section (xi,yi)oftheuniti=1withD1=10. In this case, the ICs leading to desynchronized behavior
dominate the picture leaving only a small number of ICs for the sync basin.
Theresultsshowninfigure4illustrate that the network is highly sensitive to the choice of the ICs attributed
to the perturbed unit regardless of the degree of connections of such a unit. However, the relative volume of
the basin of attraction leading to synchronization, computed as the number of ICs in the sync basin divided
by the total number of ICs, is lower for perturbed units possessing a higher degree, see figures 4(a) and (d).
4
J.Phys.Complex. 2(2021) 035002 (10pp) E S Medeiros et al
Figure 4. Set of ICs attributed to the different units outside the synchronization manifold of the ER network. White color
corresponds to ICs leading to an attractor at infinity. Blue corresponds to ICs leading to the synchronized state (Z=0), gray
corresponds to desynchronization (Z=1). Perturbation applied to: (a) unit 17 (D17 =5). (b) Unit 6 (D6=6). (c) Unit 20
(D20 =7). (d) Unit 1 (D1=10). The network parameters are N=25 and σ=0.0005. The units at the synchronization
manifold receive (−0.02, 0.01) as ICs.
Figure 5. The single-node basin stability Siis shown as a function of the units degree Difor 10 network realizations with different
topologies. The gray dots correspond to realizations of a network with N=100 units distributed in a BA graph. The orange dots
stand for the realizations of an ER network with N=25 units and average degree k=6.0 (same properties as in figures 3and 4).
The connected dots stand for the mean value of Siover the realizations. The coupling strength is fixed at σ=0.0005 for all
realizations of both networks.
To further elucidate this dependence, we look at ensembles of network realizations with different topolo-
gies. First, we generate 10 different ER graphs of the same size and average degree as addressed in figure 4,i.e.
N=25 and k=6.0. In addition, we consider 10 realizations of networks with N=100 units arranged in
BA graphs possessing a power-law degree distribution [29], see figure 2(b). Next, we investigate the effect of
perturbing units with different degrees for the ensembles of the two network categories. For this, we estimate
the relative volume of the basin of attraction of the synchronized state for different units using the concept
of basin stability [30]. Here, similarly to reference [31], basin stability is computed for single-nodes, i.e. in a
two-dimensional cross-section of the 2N-dimensional state-space. For every unit i, we consider a finite por-
tion of the cross-section (x0
i,y0
i), namely, x0
i∈[−2.0, 2.0] and y0
i∈[−2.4, 2.4]. Hence, in figure 5,weshow
the single-node basin stability Siobtained for every unit ias a function of its degree Diin both proposed graphs
categories for every network realization as well as the averaged Siover the ensembles.
From figure 5, one can conclude that the relative volume of the unit’s state-space leading the network to
synchronization becomes significantly lower with the increasing degree for both, the ER and the BA networks.
Therefore, the trapping of trajectories in the chaotic saddle as a consequence of the interaction between net-
work units due to their coupling is much more prominent for perturbing units with a higher degree. These
results confirm the evidence raised in figures 4(a)–(d) for only one realization of an ER network. Besides, in
figure 5, for most of the investigated degree interval, we observe that the averaged single-node basin stability
5
J.Phys.Complex. 2(2021) 035002 (10pp) E S Medeiros et al
Figure 6. (a) Projection of the network state-space (xi×yi) of each network unit at t=5000 time steps for ICs leading to
desynchronization. The red dots indicate the chaotic saddle λ. The coupling strength is fixed at σ=0.0005 and the network size
is N=25 (ER). (b) For 200 network realizations, the probability of reaching the period-5 orbit PSas a function of the coupling
strength σ. The colors correspond to ICs attributed to different perturbed units. Network size is N=25 (ER).
is lower for ER networks compared to units on BA networks with the same degree. This suggests that the ER
networks are more susceptible to the effects of the chaotic saddle. However, the general characteristics of the
investigated network topologies such as their size and their average degree play an essential role in the global
stability of the synchronized state. Since the impact of these aspects is beyond the scope of this study, any con-
clusion on which network category is most influenced by the chaotic saddle is left open for future investigations
aiming at a comprehensive view on the consequences of the topological characteristics.
The similarities between the basin of the desynchronized state in figure 4and the approximation of stable
manifold of the chaotic saddle shown in figure 1suggest a close relationship between them. To emphasize this
relationship, we show snapshots of the network trajectories at the time t=5000 times steps (figure 6(a)). The
large dots represent the network units (N=25), the colors correspond to a snapshot of trajectories originating
from different perturbed units. Indeed, we verify that the interaction between the chaotic dynamics in the sad-
dle and the network coupling traps the network trajectory in the vicinity of the saddle (red dots in figure 6(a)).
This phenomenon occurs for a limited range of the network coupling strength σasshowninfigure6(b). In this
figure, we consider n=200 realizations of the ER network with different ICs attributed to a given perturbed
unit (i=1, i=6, i=17, or i=20). From this ensemble, we compute the probability PS=nS/n,wherenS
corresponds to realizations in which all units are close to the period-5 attractor A, i.e. escape from the chaotic
saddle. Hence, in figure 6(b) we show PSas function of σand for different perturbed units exhibiting different
degrees (color code).
The results shown in figure 6(b) illustrate that the occurrence of the network trapping in the chaotic saddle
depends strongly on the choice of σrather than the degree of the perturbed unit. We also point out that this
phenomenon is very robust since it occurs for many orders of magnitude of the coupling strength σ.Only
for very strong coupling strength, the synchronized state is globally stable. Please note, that for intermediate
coupling strength not a single out of 200 realizations leads to synchronization.
The mechanism at which the synchronized period-5 solution exhibits the aforementioned intricate depen-
dence on the perturbation applied to an arbitrarily chosen unit lies in the interplay between the chaotic saddle
and the coupling between the units. When one unit is perturbed by a specific vector, its state variables will be
placed in another point of the state space. In the case that this point is close to the stable manifold of the chaotic
saddle, the trajectory of the perturbed unit will approach the chaotic saddle along its stable manifold. Since the
perturbed unit is coupled to other units, it will pull additional units also in a direction to leave the synchro-
nization manifold toward the chaotic saddle. At this point, the interplay between the chaotic saddle and the
coupling can then lead the network to a complete desynchronized state. By contrast, when the perturbation
vector is slightly different, the state variables of the perturbed unit will end up at a different point in state space
which might not be close enough to the saddle’s stable manifold. In that case, the network coupling can restore
the synchronized state. Since the stable manifolds of the chaotic saddle have a very intricate structure in the
system state-space, points close to it and points far from it are complexly interwoven. Consequently, the syn-
chronized solution is highly sensitive to perturbations at the initial instant of time. In summary, the motion of
the perturbed unit close to the chaotic saddle initially desynchronizes a subset of the network units by bringing
it also close to the chaotic saddle. The longer the trajectories of that subset stay in the chaotic regime, the larger
gets the likelihood of additional units joining this subset via the action of the coupling. In turn, as the subset
of desynchronized units grows, the mutual disturbances among its units give rise to an even lower likelihood
of escaping the saddle’s neighborhood. As a consequence, after a critical number of units oscillate close to the
chaotic saddle, the state of complete desynchronization becomes accessible. This mechanism is discussed in
detail in reference [27] for a symmetric network of continuous-time units.
6
J.Phys.Complex. 2(2021) 035002 (10pp) E S Medeiros et al
4. State-dependent vulnerability
Remarkably, the presence of the chaotic saddle λin the state-space of the discrete-time units composing the
network described in equation (1) has consequences not only for perturbations at t=0. In fact, even after
the whole network is completely synchronized in the period-5 attractor, perturbations in the different states
of this solution may still desynchronize the system. Similar state-dependent phenomena have been recently
reported in the vulnerability of synchronization in a network of continuous-time units [28], in the excitability
of a limit-cycle [32], and in the survivability of the solitary states in a multiplex network [33].
To demonstrate this state-dependent vulnerability in an ER network of discrete-time units, we first consider
thespace–timediagramsoffigure7. In these diagrams, all units are initialized such that they reach the period-
5 attractor. Hence, the synchronization manifold follows the trajectory specified by vt
S=(xt
S,yt
S). Specifically,
we define the synchronization manifold Sat t=0 by attributing the IC (0.3, 0.2) to all network units. After
transients, when the system reaches the attractor, we apply an external perturbation at any of the five states vt
S
belonging to the attractor of the synchronized high-dimensional system. The perturbation is defined as a devi-
ation Δt
i=(Δxt
i,Δyt
i) and applied directly to the dynamics of a specific unit iexactly in that moment when
this is at the state vt
S. This is illustrated in figure 7(a), where we apply the perturbation Δ1522
1=(0.05, −0.2)
at the state v1522
Scorresponding to the iteration at t=1522 time steps and the unit i=1. The color code in the
figureindicatesthesynchronizationerrorEt
i=1/Dij∈Divt
i−vt
jwith i=1, ...,N. We observe that after a
very small time interval with desynchronized behavior right after t=1522 time steps the system returns to the
synchronized configuration. This result is emphasized by the snapshot of the dynamics of the whole network at
t=5000 time steps showing Et
i=0infigure7(b). Now, we apply the same perturbation Δ1524
1=(0.05, −0.2)
at the same unit i=1 but at a different state of the period-5 attractor, namely at the time t=1524 steps. As
an outcome, in figure 7(c), we observe the entire network approaching a completely desynchronized behavior.
This observation is supported by the snapshot at t=5000 time steps shown in figure 7(d).
The sensitivity observed in figure 7suggests a state-dependent vulnerability of the synchronized state.
To clarify this behavior, we show in figure 8(a) each of the xt
S-components during one cycle of the syn-
chronized period-5 orbit using a color code indicating the state-dependent vulnerability to the perturbation
Δt
1=(0.05, −0.2). In blue are the states at which the system returns to synchronization after Δt
1is applied,
the safe sets. The red points indicate the states v1523
Sand v1524
Sat which the same perturbation Δt
1leads the
whole network to desynchronization. These make up the unsafe set. In figure 8(b), we show a projection of the
synchronized high-dimensional state space, (xt
S,yt
S) onto the two-dimensional state space of unit i=1oscil-
lating in the period-5 orbit, the red dots show the region occupied by the unsafe set. In both figures 8(a) and
(b), the synchronization is assessed by the global synchronization error Zdefined in equation (2).
To shed more light on the role of the amplitude of the perturbations Δt
iplayed in such state-dependent
vulnerability, we investigate the sets of perturbations in x–ystate where (Δxt
i,Δyt
i) stands for the amplitude of
each component of the perturbation. In figure 9(a), we obtain such a diagram for the perturbation Δ1522
1.The
color code denotes the global synchronization error Z. Specifically, blue color indicates perturbation regions
for which the network synchronizes (Z=0) after the perturbation (t=1522 time steps), the safe set denoted
by B1522
S. The gray color indicates regions for which the perturbation at t=1522 time steps leads the entire
network to desynchronization (Z=1), the unsafe set denoted by B1522
λ. Regions in white indicate perturbations
for which the solution converges to the attractor at infinity. In figure 9(b), we obtain the same diagram for the
perturbation applied in a different state, at t=1524 time steps. Despite visual similarities between the diagram
obtained for both states, internally, the structures are significantly different. This indicates that a whole set of
perturbations that are safe when applied in a given state have a different outcome in another state. We illustrate
this phenomenon in figure 9(c) by estimating the intersection I=B1522
S∩B
1524
Sbetween the safe sets obtained
for t=1522 and t=1524 time steps.
Similar to the initial-state vulnerability discussed in section 3, the trajectories of the desynchronized net-
work are trapped in the neighborhood of the chaotic saddle λ. We demonstrate this result in figure 10(a) for
the perturbation Δt
1=(0.05, −0.2) applied in the dynamical states corresponding to the instants t=1523
steps (blue dots) and t=1524 steps (gray dots). The robustness of the trapping phenomenon with respect
to changes in the coupling strength σof the network is also assessed for the state-dependent vulnerability.
For that, in figure 10(b), we consider n=200 realizations of our network with different perturbation ampli-
tudes Δt
1applied in the instants t=1522 steps (green dots) and t=1524 steps (gray dots). By recording the
realizations nSin which all units are close to the period-5 attractor A, i.e. escape from the chaotic saddle, in
figure 10(b), we show the probability PS=nS/nas a function of coupling strength σ. Comparing these results
with the ones for the perturbations at t=0showninfigure6(b), we verify that for t=0theintervalofσwith
the occurrence of desynchronized behavior is slightly smaller. Additionally, in contrast to the IC vulnerability,
the occurrence of solely desynchronized behavior, i.e., PS=0, does not occur at all.
7
J.Phys.Complex. 2(2021) 035002 (10pp) E S Medeiros et al
Figure 7. Space–time plots and time snapshot of the synchronization error Et
i.Top:perturbationΔ1522
1=(0.05, −0.2) applied
to Sin the unit number 1 at the instant t=1522 steps. The network returns to the synchronized configuration. Bottom:
perturbation Δ1524
1=(0.05, −0.2) applied to Sin the unit number 1 at the instant t=1524. The network completely
desynchronizes. The network has N=25 units distributed in an ER graph, and the coupling strength is fixed at σ=0.0015.
Figure 8. (a) Projection of the high-dimensional synchronized cycle in the xt
S-component of node number 1 during one cycle of
the period-5 orbit. The blue dots indicate the safe states, while the red ones the unsafe states. (b) Projection of the synchronized
high-dimensional oscillation. The perturbation amplitude is fixed at Δt
1=(0.05, −0.2). The network has N=25 units
distributed in an ER graph, and the coupling strength is fixed at σ=0.0015.
Figure 9. Diagrams showing the global synchronization error as a function of the perturbation components (Δxt
i,Δyt
i). (a) and
(b) The blue areas correspond to perturbations leading the network to synchronization (Z=0), while the gray denotes
perturbations desynchronizing the network (Z=1). White corresponds to perturbation causing convergence to infinity.
(a) Perturbation Δ1522
1is applied at the state v1522
S. (b) Perturbation Δ1524
1is applied at the state v1524
S.ThenetworkhasN=25
units distributed in an ER graph, and the coupling strength is fixed at σ=0.0015. (c) The blue dots stand for the intersection
I=B1522
S∩B
1524
Sbetween the safe sets at t=1522 and t=1524 time steps.
The mechanism underlying the state-dependent vulnerability is the outcome of the interplay between the
network coupling and the dynamics close to the chaotic saddle. A specific perturbation applied to the same
units but at different time instants leads to different states of the network dynamics either close to the stable
manifold of the chaotic saddle in one time instant and far away from it in the next time instant. As previously
discussed in section 3, the trajectories which stay longer in the vicinity of the chaotic saddle increase the likeli-
hood of having more network units approaching the saddle via pulling by the network coupling. After a critical
number of units are following the chaotic saddle, in turn, the mutual disturbances decrease the likelihood of
escaping and provide the grounds for a persistently long desynchronized solution [28].
In conclusion, we have demonstrated that chaotic saddles present in the dynamics of single discrete time
units generate two distinct unpredictabilities to synchronized states in complex networks: (i) vulnerability of
synchronization to localized perturbations applied at the initial state. For the network structure build in both
8
J.Phys.Complex. 2(2021) 035002 (10pp) E S Medeiros et al
Figure 10. (a) Projection of the network state-space (xi×yi) of each network unit at t=5000 time steps for perturbations
Δt
1=(0.05, −0.2) desynchronizing the network. The colors indicate perturbations applied to different states corresponding to
t=1522 and t=1524 time steps. The red dots indicate the chaotic saddle λ. The coupling strength is fixed at σ=0.0015.
(b) For 200 network realizations with different perturbations Δt
1, the probability of reaching the period-5 orbit (escape from the
saddle) PSas a function of the coupling strength σ. The colors indicate perturbations applied to states at t=1522 and t=1524
time steps. The network size is fixed at N=25 (ER).
ER and BA graphs, the synchronization manifold is sensitively dependent on ICs attributed to specific network
units. By computing the single-node basin stability for units with a power-law distribution of degrees (BA),
we verify that perturbations applied to units with high degree (network hubs) reduce the relative size of the
synchronization basin, therefore, are unsafe for synchronization. (ii) Vulnerability of synchronization to per-
turbation applied at arbitrary states of its asymptotic dynamics. For an ER topology, we verify that identical
perturbations applied at different time instants, i.e. different states along the long-term trajectory produce dif-
ferent outcomes to synchronization. By inspecting large sets of perturbation amplitudes applied at two distinct
states, we observe significant changes in the inner distribution of perturbations desynchronizing the network.
For both vulnerability cases, we verify that the trajectories of the desynchronized network are indeed trapped
in the vicinity of the chaotic saddle. Moreover, we unveil the regimes of the network coupling strength for the
occurrence of this phenomenon. Finally, we remark that the trapping of trajectories in chaotic saddles may not
be exclusive for networks of identical units. However, further details are open to future investigations.
Acknowledgments
The authors thank Professor T T´
el and Oleh Omel’chenko for useful discussions This work was
supported by FAPESP (Processes: 2018/03211-6, 2013/26598-0, 2015/50122-0, 2017/05521-0). ESM
acknowledges support by the Deutsche Forschungsgemeinschaft (DFG, German Research Founda-
tion)—Projektnummer—163436311—SFB 910. UF acknowledges support from the Hungarian Academy of
Sciences to visit Támas T´
el at Eötvös Lorand University Budapest, Hungary. ILC acknowledges support by
CNPq (Process: 401264/2017-3). The simulations were performed at the HPC Cluster CARL, located at the
University of Oldenburg (Germany) and funded by the DFG through its Major Research Instrumentation Pro-
gram (INST 184/157-1 FUGG) and the Ministry of Science and Culture (MWK) of the Lower Saxony State,
Germany.
Data availability statement
The data that support the findings of this study are available upon reasonable request from the authors.
ORCID iDs
Everton S Medeiros https://orcid.org/0000-0001-8531-6327
References
[1] Grebogi C, Ott E and Yorke J A 1982 Phys. Rev. Lett. 48 1507
[2] Grebogi C, Ott E and Yorke J A 1983 Physica D7181
[3] Ashwin P, Buescu J and Stewart I 1996 Nonlinearity 9703
[4] McDonald S W, Grebogi C, Ott E and Yorke J A 1985 Phys. Lett. A107 51
[5] Szab´
oKG,LaiY-C,T
´
el T and Grebogi C 2000 Phys. Rev. E61 5019
[6] Robert C, Alligood K T, Ott E and Yorke J A 1998 Phys.Rev.Lett.80 4867
[7] Robert C, Alligood K T, Ott E and Yorke J A 2000 Physica D144 44
9
J.Phys.Complex. 2(2021) 035002 (10pp) E S Medeiros et al
[8] Ding M, Grebogi C, Ott E and Yorke J A 1990 Phys. Rev. A42 7025
[9] Sweet D, Ott E and Yorke J A 1999 Nature 399 315
[10] Aguirre J, Vallejo J C and Sanjuán M A 2001 Phys. Rev. E64 066208
[11] Lai Y-C and Winslow R L 1995 Phys. Rev. Lett. 74 5208
[12] Rempel E L, Chian A C-L, Macau E E N and Rosa R R 2004 Chaos 14 545
[13] Kraut S and Feudel U 2002 Phys. Rev. E66 015207
[14] Dhamala M and Lai Y-C 1999 Phys. Rev. E59 1646
[15] Schwartz I B and Triandaf I 1996 Phys. Rev. Lett. 77 4740
[16] Macau E E N and Grebogi C 1999 Phys. Rev. E59 4062
[17] Lilienkamp T and Parlitz U 2020 Chaos 30 051108
[18] Ansmann G, Lehnertz K and Feudel U 2016 Phys. Rev. X6011030
[19] P´
entek ´
A, Toroczkai Z, T´
el T, Grebogi C and Yorke J A 1995 Phys. Rev. E51 4076
[20] Eckhardt B, Schneider T M, Hof B and Westerweel J 2007 Annu. Rev. Fluid Mech. 39 447
[21] Schneider T M, Eckhardt B and Yorke J A 2007 Phys.Rev.Lett.99 034502
[22] Toroczkai Z, Károlyi G, P´
entek ´
A, T´
el T and Grebogi C 1998 Phys.Rev.Lett.80 500
[23] Vilela R D and Motter A E 2007 Phys.Rev.Lett.99 264101
[24] T´
el T, Lai Y-C and Gruiz M 2008 Int. J. Bifurcation Chaos 18 509
[25] T´
el T and Lai Y-C 2008 Phys. Rep. 460 245
[26] Lai Y-C and T´
el T 2011 Transient Chaos (Berlin: Springer)
[27] Medeiros E S, Medrano-T R O, Caldas I L and Feudel U 2018 Phys. Rev. E98 030201
[28] Medeiros E S, Medrano-T R O, Caldas I L, T´
el T and Feudel U 2019 Phys. Rev. E100 052201
[29] Barabási A-L and Albert R 1999 Science 286 509
[30] Menck P J, Heitzig J, Marwan N and Kurths J 2013 Nat. Phys. 989
[31] Mitra C, Choudhary A, Sinha S, Kurths J and Donner R V 2017 Phys. Rev. E95 032317
[32] Franovi´
c I, Omel’chenko O E and Wolfrum M 2018 Chaos 28 071105
[33] Schülen L, Janzen D A, Medeiros E S and Zakharova A 2021 Chaos Solitons Fractals 145 110670
10
