Experimental datasets on intermittency and observability in networks of electronic oscillators. Analogue and hybrid configuration

Abstract

Synchronization phenomena are pervasive in nature and appear across a wide range of scientific areas, including physics, biology, and engineering. These phenomena describe how interacting dynamical systems tend to coordinate their behavior over time. In many cases, the transition toward a fully synchronized state is neither immediate nor continuous; instead, it consists of intermittent dynamics characterized by alternating intervals of coherent behavior, where the systems evolve in unison, and bursts of desynchronized activity. Such intermittent synchronization has been extensively observed in biological systems, particularly in ensembles of neurons, where it plays a fundamental role in processes such as information transmission and cognitive function. The data sets presented in this work originate from two distinct experimental setups involving networks of 28 chaotic electronic oscillators based on the Rössler-like system. In the first approach, the networks are constructed entirely with analog electronic components, and in the second approach, the hybrid, we use a real-time datacard for the coupling with the electronic circuits. For the two experimental setup approaches, the oscillators are interconnected in a Watts-Strogatz (WS) small-world network or an Erdős–Rényi (ER) random topology. We consider that the datasets derived from these four experiments offer valuable resources for researchers aiming to analyze and validate theoretical models of synchronization. They are suitable for systematic studies on the influence of weak linear coupling strengths in the route to synchronized states.

Specifications Table

Subject	Engineering & Materials science
Specific subject area	Nonlinear dynamics, synchronization in complex networks.
Type of data	Time series in dat file.
Data collection	Analogue Version The data were collected using a National Instruments Data Acquisition Card (DAQ), model USB-6363. The recorded signals correspond to the $\mathrm{y}$ state variable of 28 electronic Rössler-like oscillators, obtained from two distinct experiments. In the first experiment, the oscillators interact through a small-world network, with coupling implemented exclusively using analog devices. In the second experiment, the oscillators are arranged in a random network, also with purely analog implemented coupling. Each time series contains 30,000 points with 2MS/Seg sampling. Hybrid Version The experimental implementation of the network relies on a National Instruments CompactRIO (cRIO-9074) platform for the sampling we use the Data Acquisition Card (DAQ), model USB-6363. The reconfigurable FPGA within the cRIO was programmed to perform the real-time operations required to process the oscillator signals and generate the corresponding feedback according to the adjacency matrix of the network. The recorded signals correspond to the $\mathrm{y}$ state variable of 28 electronic Rössler-like oscillators, obtained from four distinct experiments. The third and fourth experiments share the same network topologies as the first and second. Specifically, the third experiment involves a small-world network with hybrid coupling, while the fourth corresponds to a random network with the same mixed analog–digital implementation. Each time series contains 500,000 points with 50kS/s sampling.
Data source location	Institution: Universidad de Guadalajara, Centro Universitario de los Lagos, Laboratorio de Metrología e Instrumentación. City, Town, Region: Lagos de Moreno, Jalisco. Country: México.
Data accessibility	Repository name: Zenodo Data identification number: 1. 14,113,045 2. 17,451,096 3. 17,451,530 Direct URL to data: 1. https://doi.org/10.5281/zenodo.14113045 2. https://doi.org/10.5281/zenodo.17451096 3. https://doi.org/10.5281/zenodo.17451530 Instructions for accessing these data: Embracing Open Science principles means ensuring transparency and reproducibility. To facilitate this, we've made all associated materials fully accessible. We strongly encourage you to engage with the complete research outputs: please access each link provided below. By visiting these repositories and downloading the data directly, you can independently verify our findings, reuse the datasets for further analysis, and contribute to the collaborative advancement of knowledge.
Related research article	Vera-Ávila, V. P., Sevilla-Escoboza, J. R., & Leyva, I. (2020). Complex networks exhibit intermittent synchronization. Chaos: An Interdisciplinary Journal of Nonlinear Science, 30(10).

1. Value of the Data

The datasets presented in this work are particularly relevant for a wide range of professionals and organizations. Potential users include researchers in nonlinear dynamics, complex systems, and network science, as well as engineers working in synchronization and control of coupled systems. Specific stakeholders include research groups in computational neuroscience, and laboratories studying neural synchronization phenomena. In engineering, power system technicians may benefit from these datasets to study synchronization stability in electrical grids. The datasets are also valuable for academic institutions and laboratories specializing in experimental nonlinear electronics, control engineering, and applied physics, enabling benchmarking and reproducibility of synchronization phenomena in real-world systems. Some of the most significant areas of application include:

• •Computational Neuroscience

  Synchronization in networks of oscillators serves as an analog model for neuronal coupling in the brain. Understanding how coherent states emerge contributes to the analysis of coordination processes in neural populations and to the study of disorders such as epilepsy and Parkinson’s disease.

• •Electrical Power Systems

  In interconnected power grids, generator synchronization is crucial for maintaining system stability. Insights from this experiment can improve understanding of coherence loss and recovery under weak or fluctuating coupling conditions.

• •Telecommunications and Sensor Networks

  Signal synchronization is fundamental in distributed communication systems, clock synchronization, and cooperative wireless networks. Experimental data from this study can aid in developing more robust synchronization algorithms for such systems.

• •Biological and Chemical Systems

  Many biological and chemical oscillations—such as cardiac rhythms or circadian cycles—can be modeled as coupled oscillators. Studying synchronization mechanisms helps explain how stability and coordination emerge in these systems under external perturbations.

• •Nonlinear Dynamics and Control Engineering

  The experimental results provide valuable validation for master stability function-based control models, supporting the design of synchronization strategies in nonlinear and adaptive networked systems.

2. Background

The study of a network composed of 28 Rössler oscillators coupled under ER and WS topologies offers a powerful experimental framework for investigating how synchronized states emerge and stabilize in complex dynamical systems. The data obtained not only validate theoretical models of synchronization but also open new avenues for practical applications in engineering, neuroscience, and communication systems, where coordination among multiple interacting units is essential [1].

Recent advances in network science have pointed out important limitations of classical models based exclusively on pairwise interactions when describing complex collective dynamics. In particular, higher-order network frameworks, where interactions involve more than two units simultaneously, have been proposed to capture richer dynamical behaviors such as multistability and complex synchronization patterns [2].

In neuronal and oscillator networks, these studies suggest that complex phenomena such as chimera states and intermittent synchronization may arise from mechanisms that are not fully captured by purely dyadic interactions [3]. However, most experimental platforms, including the one presented in this work, are still based on pairwise coupling schemes. In this context, experimental datasets derived from controlled pairwise-coupled oscillator networks remain essential to systematically investigate the emergence of synchronization, particularly intermittent dynamics, and to provide a baseline for validating and extending theoretical models toward more complex interaction frameworks.

3. Data Description

The datasets provided correspond to the experimental implementation of networks of electronic oscillators under four distinct configurations. Each node of the networks in the four configurations consists of an electronic Rössler-like oscillator constructed exclusively from analog components, including resistors, diodes, capacitors, and operational amplifiers. The parameters of each oscillator were carefully adjusted to ensure operation within a chaotic regime. In the four configurations, from the three state variables of the nodes ($\mathrm{x},\mathrm{y},\mathrm{z}$), just one variable $\mathrm{y}$ was recorded to be computed. The four experimental realizations were conducted as follows:

• •In the first two configurations, both the network topology and the coupling between oscillators were implemented entirely using analog circuitry. The investigated topologies in these fully analog setups correspond to a random 28-node network (Erdős–Rényi model) and a small-world 28-node network (Watts–Strogatz model). The coupling between oscillators was established using electronic coupling modules, which allowed precise control of the interaction strength. To vary the coupling parameter continuously and uniformly, digital potentiometers were used, enabling repeatable adjustment across the entire experimental array.
• •In the remaining two configurations, a hybrid scheme was implemented: the individual oscillators were still realized using analog components, whereas the network topology and coupling were digitally controlled through the combined use of a CompactRIO system and a data acquisition (DAQ) card. To define the network topology, the Laplacian matrix representing the network structure was embedded into a virtual instrument (VI) developed in the LabVIEW environment. This VI continuously acquires, in real time, the signals generated by the electronic nodes through two NI-9220 analog input modules. At the same time, the coupling strength is read via an additional analog input channel on the second NI-9220 module. Once both the node signals and the coupling strength are acquired, the VI processes these data to compute the linear and diffusive coupling functions specified by the Laplacian matrix. In these hybrid configurations, the network topologies under study included a random network and a small-world network, each composed of 28 nodes. Finally, the computed coupling signals are summed and fed back to the electronic nodes through a pair of NI-9264 analog output modules integrated into the CompactRIO system.

4. Dataset Description

• •In the first repository, corresponding to the fully analog implementation, the adjacency matrices are provided in .dat format and are labeled as AdjER and AdjSW, representing the Erdős–Rényi and Small-World network topologies, respectively. The files TS_ER and TS_SW contain the time series associated with each configuration and are provided in compressed format.
• •The time series files are named ST_X.dat, where X ranges from 0 to 99, corresponding to each discrete step in the variation of the coupling strength between oscillators (100 steps in total). Each file contains 28 columns, representing the time evolution of the 28 oscillators in the network.
• •The second and third repositories correspond to the hybrid implementation for the random (ER) and small-world (SW) networks, respectively. In both repositories, a total of 701 time series are provided, distributed across 7 compressed files. Each step corresponds to a discrete increase in the coupling gain between nodes, controlled through a variation in the applied voltage.
• •Within each compressed file, the time series are organized following the same structure as in the first repository, ensuring consistency across all datasets.

5. Experimental Design, Materials and Methods

5.1. Analogue configuration

An array of 28 Rössler oscillators was implemented and coupled using two different network topologies: Random (Erdős–Rényi) and Small World (Watts-Strogatz). The aim was to analyze the collective dynamics and the transition to synchronization under weak coupling conditions and progressively increased up to the onset of complete synchronization.

Each oscillator was built using an electronic circuit that reproduces the Rössler-like equations in the chaotic regime. The coupling between oscillators was established using electronic coupling modules, which allowed precise control of the interaction strength. To vary the coupling parameter continuously and uniformly, digital potentiometers were used, enabling repeatable adjustment across the entire experimental array.

A general overview of the experiment setup is shown in Fig. 1. The experimental array was connected to the National Instruments Data Acquisition Card (DAQ), model USB-6363, and was monitored and controlled using LabVIEW. The control program was structured through a state machine architecture, which automated the experimental sequence, including the adjustment of coupling values, signal acquisition, and data storage. This design ensured stable operation and consistent timing throughout the experiment.

Fig. 1.

Experimental setup for the analogue implementation. The 28 ${\mathrm{v}}_{2\mathrm{i}}$ signals are fed into the data acquisition board through the analog input channels (ADC) from channel 0 to 27. To perform the coupling sweep, digital pulses (DO) are used to control the digital potentiometers (XDCP), which serve as the coupling strength ($\kappa$). The experimental setup is controlled by an interface developed using LabVIEW software.

Signal acquisition from all 28 oscillators was carried out in real time, allowing the observation of dynamic transitions from incoherent states, where oscillators behave almost independently, to the onset of synchronization as the coupling strength increased.

Each node of the network is an electronic realization of the Rössler-like oscillator [4,5], whose dynamics is given by the (1), (2), (3). These oscillators are designed to reproduce chaotic behavior using analog electronic components, enabling controlled experimental studies of nonlinear dynamics. The schematic diagram of a single Rössler-like oscillator is presented in Fig. 2. The coupling between oscillators is achieved through an electronic interface that allows the implementation of different network topologies. The corresponding circuitry used to realize the coupling scheme is shown in Fig. 3, it is important to notice that we use an operational amplifier (Op-Amp TL084) for the output of the oscillator $\mathrm{i}$ and each of its $\mathrm{j}$ neighbors, and finally, we sum the input and its neighbors.

$${\dot{\dot{v}}}_{1i}=-\frac{1}{R_1{C}_1}\left(v_{1i}+\frac{R_1}{R_2}{v}_{2i}+\frac{R_1}{R_4}{v}_{3i}\right)$$ (1)

$${\dot{\dot{v}}}_{2i}=-\frac{1}{R_6{C}_2}\left(-\frac{R_6{R}_8}{R_9{R}_7}{v}_{1i}+\left[1-\frac{R_6{R}_8}{\left(R_3+R_5\right)R_7}\right]v_{2i}-\kappa \frac{R_6}{R_{15}}\sum _{j=1}^N{A}_{ij}\left(v_{2j}-v_{2i}\right)\right)$$ (2)

$${\dot{\dot{v}}}_{3i}=-\frac{1}{R_{10}{C}_3}\left(-\frac{R_{10}}{R_{11}}G\left(v_{1i}\right)+v_{3i}\right)$$ (3)

where $v_1$, $v_2$ and $v_3$ correspond to the voltages describing the dynamical state of each electronic oscillator, $A_{ij}$ is the adjacency matrix which contains the specific structure of the network ($A_{ij}=1)$ if circuit $i$ is connected to circuit $j$, and 0 otherwise. $G\left(v_{1i}\right)$ is a piecewise nonlinear function given by:

$$G\left(v_{1i}\right)=\{\begin{array}{c}0if{v}_{1i}\le V_d+V_d\frac{R_{14}}{R_{13}}+V_{ee}\frac{R_{14}}{R_{13}}\\ \frac{R_{12}}{R_{14}}{v}_{1i}-Vee\frac{R_{12}}{R_{13}}-V_d\left(\frac{R_{12}}{R_{13}}+\frac{R_{12}}{R_{14}}\right)if{v}_{1i}>V_d+V_d\frac{R_{14}}{R_{13}}+V_{ee}\frac{R_{14}}{R_{13}}\end{array}$$

Fig. 2.

Schematic diagram for the electronic realization of a Rössler-like oscillator used in the analogue experiment.

Fig. 3.

Electronic representation of the coupler circuit responsible for the diffusive coupling between an electronic oscillator and its neighbors.

We took the $v_2$ signal from each electronic oscillator and fed it into the coupling modules to implement diffusive coupling $\left(v_{2i}-v_{2j}\right)$ between the node $i$ and its $j$ neighbors. This variable was chosen because it provides a stable synchronization region; in the case of this oscillator, it is of type II, according to the master stability function [6,7] Table 1.

Table 1.

Components and voltage reference for the analogue experiment.

C1 = 1nF	C2 = 1nF	C3 = 1nF	k= [0–1]
R1 = 2MΩ	R2 = 200kΩ	R3 = 10kΩ	R4 = 100kΩ
R5 = 50kΩ	R6 = 5MΩ	R7 = 100kΩ	R8 = 10kΩ
R9 = 10kΩ	R10 = 100kΩ	R11 = 100kΩ	R12 = 150kΩ
R13 = 68kΩ	R14 = 10kΩ	R15 = 500kΩ	RC = 58kΩ
Vd = 0.7 V	Vee = 9 V	U1A-U1D U2A = TL084

5.2. Hybrid configuration

A complete overview of the hybrid configuration is presented in Fig. 4. In this configuration, the nodes continued to be implemented using analog electronic circuitry, while the network topology and node coupling were managed through a NI CompactRIO (cRIO-9074) embedded control and data acquisition platform. This device integrates a reconfigurable FPGA, which was specifically programmed to perform the real-time computational tasks required for signal processing and feedback generation. The FPGA was configured to continuously acquire the analog outputs of each oscillator, compute the coupling terms according to the Laplacian network topology, and subsequently deliver the corresponding feedback signals to the system. The ${\mathrm{v}}_2$ variable from each oscillator, which was continuously acquired via two NI-9220 analog input modules. The corresponding feedback signals, computed in real time by the FPGA, were delivered back to the oscillators through two NI-9264 analog output modules. Consequently, every physical oscillator was assigned a distinct pair of input and output channels, enabling direct measurement and feedback control of its ${\mathrm{v}}_2$ variable, which has high temporal resolution and minimal latency. The coupling strength among the oscillators was externally regulated using a GW-Instek GPD-2030S programmable power supply, which provided a control voltage within the range of 0 VCD - 3.5VCD, values necessary to observe the full route to the complete synchronization. This control signal was simultaneously sampled by the NI-9220 analog input module to maintain appropriate voltage scaling and consistency within the FPGA-based processing framework. By incrementally increasing the control voltage in discrete steps (0.005 VCD), the system allowed for a systematic exploration of the transition toward synchronization, enabling precise characterization of the coupling-dependent dynamical behavior [8,9].

Fig. 4.

Hybrid acquisition setup using DAQ (NI-6363) and cRIO systems. Oscillator signals ${\mathrm{v}}_{2\mathrm{i}}$ are acquired and processed in the cRIO, where coupling is implemented and controlled via an external voltage source. The coupled signals are fed back to the circuits and re-acquired by the DAQ for analysis. The entire system is managed through LabVIEW.

As we mentioned, each node of the network is an electronic version of the Rössler-like oscillator given by the (4), (5), (6). The schematic diagram with the components and the connections is shown in Fig. 5 Table 2.

$${\dot{v}}_{1i}=-\frac{1}{R_1{C}_1}\left(v_{1i}+\frac{R_1}{R_2}{v}_{2i}+\frac{R_1}{R_4}{v}_{3i}\right)$$ (4)

$${\dot{\dot{v}}}_{2i}=-\frac{1}{R_6{C}_2}\left(-\frac{R_6{R}_8}{R_9{R}_7}{v}_{1i}+\left[1-\frac{R_6{R}_8}{\left(R_3+R_5\right)R_7}\right]v_{2i}-\kappa \frac{R_6}{R_{15}}\sum _{j=1}^N{A}_{ij}\left(v_{2j}-v_{2i}\right)\right)$$ (5)

$${\dot{\dot{v}}}_{3i}=-\frac{1}{R_{10}{C}_3}\left(-\frac{R_{10}}{R_{11}}G\left(v_{1i}\right)+v_{3i}\right)$$ (6)

where $v_1$, $v_2$ and $v_3$ correspond to the voltages describing the dynamical state of each electronic oscillator, $A_{ij}$ is the adjacency matrix which contains the specific structure of the network ($A_{ij}=1)$ if circuit $i$ is connected to circuit $j$, and 0 otherwise. $G\left(v_{1i}\right)$ is a piecewise nonlinear function given by:

$$G\left(v_{1i}\right)=\{\begin{array}{c}0if{v}_{1i}\le V_d+V_d\frac{R_{14}}{R_{13}}+V_{ee}\frac{R_{14}}{R_{13}}\\ \frac{R_{12}}{R_{14}}{v}_{1i}-Vee\frac{R_{12}}{R_{13}}-V_d\left(\frac{R_{12}}{R_{13}}+\frac{R_{12}}{R_{14}}\right)if{v}_{1i}>V_d+V_d\frac{R_{14}}{R_{13}}+V_{ee}\frac{R_{14}}{R_{13}}\end{array}$$

Fig. 5.

Schematic diagram for the electronic realization of the Rössler-like oscillator used in real-time configuration.

Table 2.

Components and voltage reference for the real-time experiment.

C1 = 10nF	C2 = 10nF	C3 = 10nF	k= [0–1]
R1 = 2MΩ	R2 = 200kΩ	R3 = 10kΩ	R4 = 100kΩ
R5 = 50kΩ	R6 = 5MΩ	R7 = 100kΩ	R8 = 10kΩ
R9 = 10kΩ	R10 = 100kΩ	R11 = 100kΩ	R12 = 150kΩ
R13 = 68kΩ	R14 = 10kΩ	R15 = 500kΩ	RC = 58kΩ
Vd = 0.7 V	Vee = 9 V	U1A-U1D U2A = TL084

As previously described, the network topology and node coupling were implemented using a combination of a CompactRIO platform and a data acquisition (DAQ) card. In this setup, the Laplacian matrix defining the network structure was integrated into a virtual instrument (VI) developed within the LabVIEW environment. This VI acquires, in real time, signals from the electronic nodes via two NI-9220 analog input modules, both configured identically to ensure consistent data acquisition. The first NI-9220 module records signals from the first fourteen oscillators via channels AI0–AI6 and AI8–AI15, while the second module acquires signals from the remaining fourteen nodes using the same channel configuration. This arrangement guarantees a balanced distribution of acquisition channels across the network. Simultaneously with this process, the coupling strength is acquired through the analog input AI15 of the second NI-9220 module. Once the signals and the coupling strength are acquired, the VI processes them to compute the linear and diffusive coupling functions defined by the Laplacian matrix. These computations determine the state differences among the nodes, which characterize the system’s dynamic interactions. Finally, the summed coupling signals are fed back to the electronic nodes through a pair of NI-9264 analog output modules in the CompactRIO, using their AO0–AO6 and AO8–AO15 channels.

Referring to the technical specifications of the modules used in the hybrid configuration. The NI-9220 module provides 16 independent analog input channels with simultaneous sampling, enabling parallel signal acquisition without temporal distortion. Each channel operates at a maximum sampling rate of 100 kS/s, ensuring accurate capture of dynamic and transient phenomena. On the other hand, the NI-9264 has 16 analog output channels that operate independently within a ± 10 V range, with a 16-bit resolution, and supports a maximum update rate of 25 kS/s per channel.

Limitations

‘Not applicable’.

Ethics Statement

The authors have read and followed the ethical requirements for publication in Data in Brief and confirm that the current work does not involve human subjects, animal experiments, or any data collected from social media platforms.

Credit Author Statement

V. P. Vera-Ávila: Methodology, Validation, Investigation, Writing (original draft). R. R. Rivera-Durón: Methodology, Validation, Investigation, Writing (original draft). J. M. Rodriguez Ornelas: Methodology, Validation, Writing (original draft). E. Vázquez-Fuentes: Methodology, Validation, Writing (original draft). I. Leyva Calleja: Methodology, Writing (review and editing), Funding. J. R. Sevilla-Escoboza: Methodology, Validation, Writing (review and editing), Supervision, Funding.

Data Availability

• ZENODODataset on complex networks exhibit intermittent synchronization (Original data)

• ZENODOExperimental datasets on intermittency in networks of electronic oscillators. Hybrid Configuration - Random (Erdös-Rényi) Topology (Original data)

• ZENODOExperimental datasets on intermittency in networks of electronic oscillators. Hybrid Configuration - Small-World (Watts-Strogatz) (Original data)