NOLTA, IEICE
Paper

Dynamics of chaotic circuit networks
with local bridges
Yoko Uwate A 1 , Kenta Ago A 1 , and Yoshifumi Nishio A 1
1

Department of Electrical and Electronic Engineering, Tokushima University,
2-1 Minami Josanjima, Tokushima 770-8506, Japan

Received May 11, 2022; Revised November 10, 2022; Published April 1, 2023
Abstract: We investigate the influence of a local bridge on a social network of 25 coupledchaotic circuits. From synchronization phenomena of coupled chaotic circuits, we show that
coupled chaotic circuits connected at both ends of a local bridge easily break down. Throughcomputer simulations, the network is predicted to switch between global synchronization andpartial synchronization. To analyze synchronization behavior, we define the asynchronous
probability for a given time interval. Moreover, we demonstrate in experiments using real
physical circuits the same synchronization behavior.
Key Words: coupled chaotic circuits, synchronization, local bridge

1. Introduction
Complex networks have attracted a great deal of attention from various fields since the discoveryof the “small-world” network and the “scale-free” network [1–3]. In particular, how the topologicalstructure of the network influences its dynamical behaviors has recently become a topic of great importance. Moreover, synchronization phenomena on networks of coupled chaotic systems are very
interesting. Many researchers have investigated the relationship between synchronization states andnetwork structures, and confirmed that the network structure plays an important role in determining
the synchronization states [4–13]. Furthermore, [14] and [15] described the manifold associated withthe stability of the synchronous clusters composed of several coupled oscillators. Montanari and collaborators proposed a new approach to analyze cluster synchronization between the bridge oscillators
using the Kuramoto model, their results show that the proposed method is useful and this framework
can be applied to diﬀerent networks [16]. However, there are not many studies of networks of realcontinuous-time physical systems such as electrical circuits.
As one example of a social network, we focus in this study on a network composed of several
clusters connected by local bridges. The inspiration behind this network model is derived fromthe sociology theory “the strength of weak ties” by Granovetter [17]. The notion of “strength of
weak ties” has been found to play a very important role in social networks. Studies have been
performed on the transmission of information in networks [18, 19] and on the influence of humansocial networks [20–22]. Therefore, we consider that it is important to investigate the influence thisnotion has in synchronization behavior of networks composed of coupled chaotic circuits. In a largescale network, the function of the bridge between nodes may be provided locally. This kind of theThis work is licensed under a Creative Commons Attribution NonCommercial, No Derivatives 4.0 License.
c
Nonlinear Theory and Its Applications, IEICE, vol. 14, no. 2, pp. 534–546 ⃝IEICE
2023 DOI: 10.1587/nolta.14.534

Fig. 1.

Example of a bridge.

Fig. 2.

Conceptual model of a social network with local bridges.

bridge is called “local bridge”. We need to explain the keyword “bridge” before explaining the local
bridge. A bridge is an edge which provides the only direct route between the two nodes. Here, an edge
is a line connecting two nodes. An example of a bridge is shown in Fig. 1. Here, the bridge between
nodes 3 and 4 provides the only route which can flow information or influence from any contact of 3
to any contact of 4. A local bridge is also similar to a bridge, and refers to a bridge when it is notpossible to connect to other clusters without this local bridge. A local bridge is a connection between
two node, providing the shortest route that information travels from being connected to one to being
connected to the other. Local bridges diﬀer from regular edges in that they cannot directly connect
the local bridge’s endpoints after the bridge is removed and do not share a common neighbor.
In this study1 , synchronization phenomena within a 25-node coupled chaotic circuit network withlocal bridges are investigated. The nodes of the network considered in this study are represented
by electronic circuits. The reason why the electronic circuit model was used is because it is a real
physical system. In using a real physical system, we may be able to confirm that the same phenomena
observed in the computer simulations are observed in real experiments and hence prove that these
phenomena exist in the real world.
Employing our concept of the social network model with local bridges (Fig. 2), we show that any
synchronization of chaotic circuits connected to both nodes of a local bridge can be easily broken down.
From computer simulations, the network switches between a global synchronization and a partial
synchronization. The same synchronization behavior is also observed in our circuit experiments.
To analyze this synchronization, we introduce the notion of asynchronous probability over a given
time interval. Moreover, we statistically analyze the sojourn time of synchronization for each edge,
including a local bridge. Finally, the transition state of the clustering pattern is investigated.
The paper is organized as follows. In Sec. 2, we describe the social network model composed ofchaotic circuits coupled by local bridges. In Secs. 3–5, we investigate several types of synchronization
exhibited in this network, using both computer simulations and circuit experiments. In Sec. 6, thetransition state of the clustering pattern is investigated. Finally, a conclusion and future work arepresented in Sec. 7.

2. Network model
Figure 3(a) shows a chaotic circuit that is the three-dimensional autonomous circuit proposed by
Shinriki et al. [23, 24]. This circuit is composed of an inductor, a negative resistance, two condensers
and dual–directional diodes. This circuit generates an asymmetric attractor as shown in Fig. 3(b).
The proposed network model of 25 coupled chaotic circuits with local bridges is shown in Fig. 4.
For this study, chaotic circuits (CCn) form the nodes of the network, and resistors (R) composedof diodes form the edges. Two adjacent chaotic circuits tend to synchronize into an in-phase state.
In this model, the local bridges are 1–25, 8–9, 14–15, 15–16, and 22–23.
First, the circuit equations are given as follows:1
This study is an extension of work presented in [26] and [27]. The new results of this article are: 1) the confirmation of similar synchronization phenomena occuring in circuit experiments, and 2) the investigation of switching in
synchronization phenomena.

535

Fig. 3.

Fig. 4.










Chaotic circuit and attractor.

Network model using chaotic circuits as nodes (N = 25).

din
= v2n
dt
dv1n
1 ∑
C1
= gv1n − idn −(v1n − v1k )

dt
R

k∈S

n



 C2 dv2n = −in + idn ,
dt
L

(1)

where n = 1, 2, · · · , 25 and Sn is the set of nodes that are directly connected to node n.
We approximate the i − v characteristics of the nonlinear resistors R by a three-segment piecewiselinear function defined as:
G (v − v2n − V ) (v1n − v2n > V ) d 1nidn =(2)0(|v1n − v2n | ≤ V ) G (v − v + V ) (v − v < −V ).
d

1n

2n

1n

2n

where Gd is the slope, and V is the threshold voltage of the nonlinear resistors.
In introducing dimensionless variables (xn , yn , zn , and τ ) and parameters (α, β, γ, and δ), forwhich:
√

C2


in =
V xn , v1n = V yn , v2n = V zn



L



√
d
C2
(3)
t = LC2 τ, “ · ” =
, α=

dτ
C1


√
√
√



L
L
1
L

 β=
Gd , γ =g, δ =,
C2
C2
R C2
the normalized circuit equations are given as:

536



 ẋn = zn∑ẏn = αγyn − αf (yn − zn ) − αδ(yn − yk )


k∈Sn


żn = f (yn − zn ) − xn ,(4)

The nonlinear function f (·) corresponds to the i − v characteristics of the nonlinear resistors consisting of the diodes and is specified as follows:
β(yn − zn − 1) (yn − zn > 1)f (yn − zn ) =(5)0(|yn − zn | ≤ 1) β(y − z + 1) (y − z < −1).
n

n

n

n

For the computer simulations, we fix the parameter values as α = 0.4, β = 20, and γ = 0.5 on all
circuits. Each circuit is given diﬀerent initial values to all the others.

3. Synchronization phenomena
We next define two synchronization states to identify the notion of “global synchronization” and“partial synchronization”. In determining the synchronization state, we first measure whether the
chaotic circuits are synchronized by using the criterion.
|yn − yk | < 0.01 (k ∈ Sn ).

(6)

If Eq. (5) is satisfied, the chaotic circuits are synchronized; otherwise, they are not. For all 41 edges
in the network model, we apply this synchronization criterion to all pairs of chaotic circuits connectedby a edge at any specified time. If all chaotic circuits are synchronized, we define the network as
being in “global synchronization”, otherwise the network is in “partial synchronization”.
Figure 5 shows a typical example of the synchronization states observed in the network model. The
vertical axis denotes the voltage diﬀerence between the two chaotic circuits, and the horizontal axis
represents the computer simulation time. Numbers in red mark the voltages of the local bridges (8–9,
14–15, 15–16, 22–23, and 1–25).
We display all 41 voltages diﬀerences of the bridges linking the 25 chaotic circuits in the network.
When, two chaotic circuits are synchronized, the voltage diﬀerence should be almost zero as for bridge
19–20, whereas, if two chaotic circuits are not synchronized, the voltage diﬀerence is large as exists forthe five local bridges (8–9, 14–15, 15–16, 22–23, and 1–25). The blue box in the diagram corresponds
to “global synchronization”, whereas the red box shows one example of “partial synchronization”.
The green boxes correspond to the results from the circuit experiment. We confirm that the network
switches between the two synchronization states, and synchronizations of chaotic circuits connected
via the local bridges are easily broken down compared with the others. The switching mechanism is
aﬀected by three attributes; coupling strength, chaotic behavior, and network structure.
Figure 6 shows the setup of the actual circuit experiment and the coupling pattern in the network.
If we change the network structure or coupling strength, we change the pattern (Fig. 6(b)).
The circuit experimental results of the chaotic attractor, time waveform and phase diﬀerence waveform are shown in Figs. 7 and 8. The parameters for this circuit experiment are set to L = 20 mH,
C1 = 100 nF, C2 = 40 nF, and g = 1 mS. The coupling resistances in the network are fixed at
R = 3.0 kΩ. From this figure, we confirm that the results obtained from this circuit experimentmatch well with simulation results at least qualitatively.

4. Statistical analysis
Next, we fixed the time interval at (τ = 10, 000 and the time-step at h = 0.01), and we statisticallyanalyzed the synchronization phenomena observed from the 25 coupled chaotic circuits with four
local bridges. We then investigated the coupling strength dependence of global synchronization (see.
Fig. 9). We checked whether all edges were synchronized during the given time interval. We confirmed

537

Fig. 5. Voltage diﬀerence waveform obtained through computer simulations(δ = 1.0). Blue box: global synchronization, Red box: partial synchronization,
Green box: circuit experiment result (Fig. 8).

Fig. 6.

Photos of the circuit setup.

that by increasing the coupling strength δ, the network easily attains the global synchronization state.
In particular, value of δ = within 1.6–1.7 enable the network to reach global synchronization rapidly
with high probability.
From this point forward, we choose a coupling strength of δ = 1.0, and we analyze the synchronization behavior more statistically with a focus of the edges. Figure 10 shows all edges stored bytheir asynchronous probability in order from highest to lowest, along with the number of switching538

Fig. 7. Attractor and time waveform obtained from the circuit experiment.
(a) Chaotic attractor (Horizontal axis: [0.5 V/div], Vertical axis: [0.5 V/div]).
(b) Time waveform (Horizontal axis: [0.5 ms/div], Vertical axis: [1.0 V/div]).

Fig. 8. Comparison of phase diﬀerence waveforms obtained from computer
simulations and circuit experiments. (a) Computer simulations (Green box in
Fig. 5). (b) Circuit experiments (Horizontal axis: [2 ms/div], Vertical axis:[5 V/div]).

Fig. 9.

Coupling strength dependence of global synchronization.

between synchronization and asynchronous for each edge. Among all the edges, the local bridges
occupy the top five place in asynchronous probability. In addition, they undergo a large number ofstate switching compared with the other edges.
Figure 11 shows the percentage frequency distribution of the number of synchronized edges during
the given time interval. There are 41 edges in the network model in Fig. 4. The vertical axis
shows the ratio of number of edges which are synchronized. Namely, “0” denotes that there are no
synchronized edges (fully asynchronous). While, “41” denotes that all edges are synchronized (global
synchronization). For the frequency distribution, various partial synchronizations exist within the
network. Of note in the frequency distribution is the fact that global synchronization is the second
highest in occurrence, implying that happens very often.
Next, we target node 15 as being located between two local bridges (14–15 and 15–16). This
constitutes a special node because it represents a cluster consisting of only one circuit. Therefore, aninvestigation of the state of this node 15 in this network is important. Table I lists the four states that
depend on the synchronization of node 15. Figure 12 shows a pie chart of the percentage occurrence539

Fig. 10.
1.0).

Asynchronous probability and the number of state switching (δ =Fig. 11.
1.0).

Frequency distribution of the number of synchronized edges (δ =Table I.

Four states that depend on the synchronization of node 15.

State name

Synchronization patterns

StatesA
B
CD(14) (15) (16)(14 15 16)(14) (15 16)(14 15) (16)The three chaotic circuits are NOT synchronized.
The three chaotic circuits are synchronized.
Chaotic circuits 15 and 16 are synchronized.
Chaotic circuits 14 and 15 are synchronized.

Fig. 12. Frequency percentage of the states that depend on the synchronization of node 15.

of the states involving node 15 during the given time interval. Among the four states, state A occurs
the most whereas state B is notably second highest. Therefore, node 15 can be easily isolated.
We focused next on the sojourn time of synchronization for each edge including the local bridges,
more specifically the frequency distribution of synchronization for six diﬀerent intervals of sojourntime (Fig. 13). The six small slots labelled (1, · · · ,6) along the horizontal axes identify equally-spaced540

Table II.

Ranges of Slots in Fig. 13.

Slot

Sojourn time (τs )Slot

Sojourn time (τs )

1
2
3

τs <1
1 ≤τs < 2
2 ≤τs < 345
6

3 ≤τs <44 ≤τs <5τs ≥5Fig. 13. Histograms of the frequency distribution of sojourn times of edges
in partial synchronization states; τs is the sojourn time-step.

sojourn time interval (see Table II). From Fig. 13(a), the histograms for the local bridges are very
similar. With the percentage for Slot 1 being predominantly high, the synchronization of local bridges
disappears quite rapidly. In contrast, from Fig. 13(b), the histograms of various edges show that thesojourn time in partial synchronization states for nodes of a cluster is longer than that for local
bridges. In particular, the sojourn times of synchronization for edge 23–25 that are located between
two local bridges (22–23 and 1–25), are similar to those of local bridges. From this result, edge 23–25
behaves like a local bridge.

5. Partial synchronization
We next consider partial synchronization in network of five clusters (nodes 1–8, 9–14, 15, 16–22, and
23–25) connected by only local bridges (see Fig. 14(a)).
We investigated statistically the partial synchronization in each cluster of the network employing
the above definition of synchronization. The time interval is fixed at τ = 10, 000 with a time-steph = 0.01. The coupling strength is set to δ = 1.0. Figure 14(b) shows the partial synchronization
probability of each cluster during the specified time interval. We suppose that the partial synchronization probabilities of these clusters depend on the network topology. Topological features in complex
networks can be evaluated using three typical structural metrics (degree, clustering coeﬃcient andpath length). The degree specifies the number of edges connecting a node. The clustering coeﬃcient
spefifies the ratio of the number of actual links between neighbors of a node to the number of possible
links between those neighbors. The path length is defined as the shortest path in the network between
two nodes.
The properties of each cluster (see Table III) include the number of nodes N , the number ofedges E, and the average degree k, along with Avg-CC. Avg-CC denotes the average of the clustering
coeﬃcient and Avg-PL denotes the average of path length. We assume that the distribution outcomesfor synchronization probability of each cluster (Fig. 14(b)) depend on the three structural metrics.
Cluster 16–22 appears with the highest probability among the four clusters and exhibits the largest
values for structural metrics k and Avg-CC of the four clusters. In addition, the path length of thiscluster 16–22 is the second shortest. We therefore infer that the the partial synchronization probabilityof each cluster can determine the group strength characterized by its clustering coeﬃcient and path
length.
We analyzed the sojourn time of partial synchronization for each cluster. Figure 15 shows thepercentage distribution of six sojourn time slots for various clusters. The six slots are specified ranges541

Fig. 14.

Partial synchronization.

Table III.

Properties of each cluster.

ClusterN

Ek

Avg-CC

Avg-PL

1–8
9–14
16–22
23–25

8
6
7
3

12
9
12
3

3.25
3.17
3.43
2.67

0.60
0.50
0.66
0.56

1.79
1.57
1.52
1.00

Fig. 15. Synchronization probability distribution of sojourn time slots forclusters of various sizes in a network.

of sojourn times (see Table II). The cluster size equals the number of nodes in each cluster. From
Fig. 15(a), the probability percentage Slot 1 is dominant, indicating that all circuits are synchronized.
In other words, synchronization of the entire network tends to disappear immediately. In Figs. 15(b),(c), and (e), the histograms are very similar with the probability percentage for Slot 1 accounting
for a substantial fraction in each distribution. However, the histogram for cluster 16–22 (Fig. 15(d))
shows that Slot 3 has the highest the probability percentage among the six slots. That is the sojourntimes for synchronization of cluster 16–22 are longer than those of the other clusters and hence, thiscluster 16–22 displays a robustness within the network.

542

Table IV. Index of the local bridge state patterns (synchronous:1, asynchronous:0).

Fig. 16.

LB-1

LB-2

LB-3

LB-4

LB-5

State

0
0
0
0
0
0
0
0
...
1
1
1

0
0
0
0
0
0
0
0
...
1
1
1

0
0
0
0
1
1
1
1
...
1
1
1

0
0
1
1
0
0
1
1
...
0
1
1

0
1
0
1
0
1
0
1
...
1
0
1

Pattern-1 (5 clusters)
Pattern-2
Pattern-3
Pattern-4
Pattern-5
Pattern-6
Pattern-7
Pattern-8
Pattern-9 – 29
Pattern-30
Pattern-31
Pattern-32 (1 cluster)

Switching cluster patterns.

Fig. 17. Percentage distribution for all 32 typesof clustering patterns.

6. Determining clustering pattern based on synchronization switching
The transition state of the clusters was investigated. In the network model of Fig. 14, there are five
local bridges (LB-1 – LB-5). By examining the synchronous/asynchronous state of these local bridges,
it is possible to determine the clustering pattern among the 25 = 32 types. In computer simulations,
we fix the time interval to τ = 2, 000 and with time-step h = 0.01; the coupling strength was set
to δ = 1.0. Table IV summarizes the synchronous/asynchronous state for the five local bridges andindexing of the corresponding clustering patterns. Of the 32 patterns of chaotic circuits, Pattern-1
contains five clusters and pattern-32 contains one cluster.
Figure 16 shows the switching behavior over time for each clustering pattern represented by a rowof dots. The horizontal axis indicates the computer simulation time τ , and the vertical axis indicatesclustering pattern type. From this figure, it can be confirmed that the clustering pattern changes
irregularly.
Figure 17 shows a histogram of the percentage probability distribution for each clustering pattern
within a certain time. Pattern-1 has the highest ratio, and pattern-32 has the second highest ratio.
Patterns-9 and 2 show the high ratio of clustering pattern (see Fig. 18). We confirm that the patternwith a large number of chaotic circuits consisting clustering is easily generated.

7. Extension to general networks
A more general network model is considered in this final section. The network model including
local bridges (see Fig. 19) is a complex system network consisting of 47 nodes and 87 edges, whichcorrespond respectively to the chaotic circuits and coupling resistors between the chaotic circuits,
as in the previous network model. In this network model, there are eleven local bridges; 1–47, 4–5,
10–16, 11–12, 15–39, 16–21, 20–25, 23–31, 32–33, 33–34, and 39–40.
In computer simulations, we fixed the values of parameters at α = 0.4, β = 20, γ = 0.5, and δ = 1.0543

Fig. 18. Clustering patterns 9 and 2 (“–”: synchronized edges, “· · ·”: asynchronous edges).

Fig. 19.

Network model using chaotic circuits (N = 47).

Fig. 20.

Asynchronous probability (N = 47).

on all circuits. We also specified the time interval as τ = 10, 000 with a time-step of h = 0.01. Eachcircuit is given diﬀerent initial values from the rest. Figure 20 shows all edges stored in order fromhighest to lowest asynchronous probability. From this percentage distribution, the local bridges (inred) are among the gridges displaying higher asynchronous probabilities. This indicates that local
bridges with their connecting function have lower synchronization probabilities than other edges in
the network. In other words, an edge with a high asynchronous probabilities is considered most likely
to be that connects clusters. In contrast, the edges 29–31, 22–24, 6–7, 13–14, and 29–30 display astate close to perfect synchronization and these edges have common adjacent nodes. More specifically,
these edges are connected to inner clusters.
Table V summarize the network properties of detected clusters and synchronization ratio. In
general, a high density network structure is characterized by large cluster coeﬃcients and small path
lengths. With its highest synchronization rate, the network structure of cluster 21–24 is found to have544

Table V.
Cluster
sync. ratio
NEk
Avg-CC
Avg-PL

1–4
0.1687
4
5
2.50
0.58
1.17

Synchronization probabilities and network properties.

5–11
0.0624
7
13
3.57
0.77
1.38

12–15
0.2210
4
5
2.50
0.83
1.17

16–20
0.1142
5
7
2.80
0.77
1.30

21–24
0.3591
4
6
3.00
1.00
1.00

25–33
0.0781
9
17
3.56
0.54
1.72

34–39
0.0865
6
10
3.34
0.81
1.33

40–47
0.1072
8
14
3.50
0.67
1.54

1–47
0.0008
47
87
3.70
0.56
4.98

large cluster coeﬃcients and small path lengths. Therefore, it is possible that the synchronization
probabilitiy of each cluster is conjectured to be an index of community strength.

8. Conclusions
In this study, to demonstrate the eﬃcacy of circuit experiments to mimic social networks, we considered a network model composed of several clusters of chaotic oscillators connected by local bridges. We
investigated the dynamics associate with synchronization patterns in the proposed model. Through ofcomputer simulations and circuit experiments, we determined that the synchronization of the chaotic
circuits connected by local bridges is easily broken down. Additionally, we have confirmed that two
synchronization states –one global and the other partial synchronization– exist, and their switching stems from the network structure, chaos characteristics, and coupling strength between pairs ofclusters connected via local bridges.
Moreover, we statistically analyzed that the sojourn time of synchronizations of local bridges areshorter than other edges. Specifically, the local bridge almost behaves asynchronously. These phenomena show that local bridges form weak ties that promote the propagation of information. We
then focused on partial synchronization states. Partial synchronization arises almost always fromlocal bridges. In this state, significant nodes are those that tend to be isolated with links to two
local bridges in the network. The probability of partial synchronization for each cluster depends
almost entirely on three structural metics. From these results, we conclude that the local bridge issuitable for finding clusters and significant nodes in complex networks, and the probability for partial
synchronization of each cluster can be eﬀective in evaluating the community strength within complex
networks. In the final section, a more general network model was considered. In this general networkmodel, we confirmed that local bridges also perform an important role in clustering.
In future work, we shall extend the network size to that of real social networks. In addition, theperformance in controlling imperfect dynamical systems has recently been studied [25]. Considering
imperfections in physical models is important so as to account for them in real-world scenarios suchas engineered devices. A study of network models with imperfect factors is also contemplated being
considered for future work.

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