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Nonlinear Dynamical Behaviors in Complex Systems

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A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Complexity".

Deadline for manuscript submissions: closed (15 April 2024) | Viewed by 20140

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Special Issue Editors

Prof. Dr. Shaoting Tang

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Guest Editor

Institute of Artificial Intelligence, Beihang University, Beijing 100191, China
Interests: complex systems; nonlinear dynamics; network science; brain science; artificial intelligence; evolutionary dynamics

Dr. Xin Wang

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Guest Editor

Institute of Artificial Intelligence, Beihang University, Beijing 100191, China
Interests: network science, nonlinear dynamics; evolutionary game theory; opinion dynamics; collective intelligence; brain networks

Dr. Longzhao Liu

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Guest Editor

Institute of Artificial Intelligence, Beihang University, Beijing 100191, China
Interests: complex networks; nonlinear dynamics; mathematical modeling; diffusion process; collective behavior

Special Issue Information

Dear Colleagues,

Most real-world systems, such as social, economic, ecological, biological and brain systems, are complex systems with rich emergence phenomena such as phase transition, cooperation, oscillation, synchronization, etc. Such complex dynamic features are commonly believed to be caused by the nonlinearity and randomicity at multi-scales. The unveiling principles and effects of nonlinear interactions play a pivotal role in understanding the dynamic behaviors of complex systems, which always attracts great attention in diverse fields. Take some recent advances as examples; higher-order interactions induce bi-stable phenomena in diffusion processes and imply an effective controlling strategy. The dynamic mechanisms of confirmation bias and selective exposure lead to shifts from group consistency to echo chamber/opinion polarization. The coupling between game dynamics and environmental feedback results in oscillating or periodic evolutions. Other examples include ecological diversity, brain dynamics, biological networks, etc.

Nowadays, modeling and analyzing the nonlinear dynamical mechanisms in complex systems are still important challenging tasks, which are the basis for further predicting and controlling the system behaviors. This Special Issue invites all contributions, including original research, reviews and perspective articles that address any of these issues.

Prof. Dr. Shaoting Tang
Dr. Xin Wang
Dr. Longzhao Liu
Guest Editors

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Keywords

complex systems
nonlinear dynamics
complex networks
contagion process
evolutionary game theory
opinion dynamics
agent-based modeling
brain networks
coupled dynamics
higher-order dynamics

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Published Papers (11 papers)

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Research

18 pages, 6928 KiB

Open AccessArticle

Streamflow Prediction Using Complex Networks

by Abdul Wajed Farhat, B. Deepthi and Bellie Sivakumar

Entropy 2024, 26(7), 609; https://doi.org/10.3390/e26070609 - 18 Jul 2024

Viewed by 1668

Abstract

The reliable prediction of streamflow is crucial for various water resources, environmental, and ecosystem applications. The current study employs a complex networks-based approach for the prediction of streamflow. The approach consists of three major steps: (1) the formation of a network using streamflow time series; (2) the calculation of the clustering coefficient (CC) as a network measure; and (3) the use of a clustering coefficient-based nearest neighbor search procedure for streamflow prediction. For network construction, each timestep is considered as a node and the existence of link between any node pair is identified based on the difference (distance) between the streamflow values of the nodes. Different distance threshold values are used to identify the critical distance threshold to form the network. The complex networks-based approach is implemented for the prediction of daily streamflow at 142 stations in the contiguous United States. The prediction accuracy is quantified using three statistical measures: correlation coefficient (R), normalized root mean square error (NRMSE), and Nash–Sutcliffe efficiency (NSE). The influence of the number of neighbors on the prediction accuracy is also investigated. The results, obtained with the critical distance threshold, reveal that the clustering coefficients for the 142 stations range from 0.799 to 0.999. Overall, the prediction approach yields reasonably good results for all 142 stations, with R values ranging from 0.05 to 0.99, NRMSE values ranging from 0.1 to 12.3, and the NSE values ranging from −0.89 to 0.99. An attempt is also made to examine the relationship between prediction accuracy and the catchment characteristics/streamflow statistical properties (drainage area, mean flow, coefficient of variation of flow). The results suggest that the prediction accuracy does not have much of a relationship with the drainage area and the mean streamflow values, but with the coefficient of variation of flow. The outcomes from this study are certainly promising regarding the application of complex networks-based concepts for the prediction of streamflow (and other hydrologic) time series. Full article

(This article belongs to the Special Issue Nonlinear Dynamical Behaviors in Complex Systems)

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16 pages, 1101 KiB

Open AccessArticle

Fast Finite-Time Observer-Based Event-Triggered Consensus Control for Uncertain Nonlinear Multiagent Systems with Full-State Constraints

by Kewei Zhou and Xin Wang

Entropy 2024, 26(7), 559; https://doi.org/10.3390/e26070559 - 29 Jun 2024

Cited by 1 | Viewed by 923

Abstract

This article studies a class of uncertain nonlinear multiagent systems (MASs) with state restrictions. RBFNNs, or radial basis function neural networks, are utilized to estimate the uncertainty of the system. To approximate the unknown states and disturbances, the state observer and disturbance observer are proposed to resolve those issues. Moreover, a fast finite-time consensus control technique is suggested in order to accomplish fast finite-time stability without going against the full-state requirements. It is demonstrated that every signal could be stable and boundless, and an event-triggered controller is considered for the saving of resources. Ultimately, the simulated example demonstrates the validity of the developed approach. Full article

(This article belongs to the Special Issue Nonlinear Dynamical Behaviors in Complex Systems)

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15 pages, 3916 KiB

Open AccessArticle

Photosensitive Control and Network Synchronization of Chemical Oscillators

by Alejandro Carballosa, Ana I. Gomez-Varela, Carmen Bao-Varela, Maria Teresa Flores-Arias and Alberto P. Muñuzuri

Entropy 2024, 26(6), 475; https://doi.org/10.3390/e26060475 - 30 May 2024

Viewed by 968

Abstract

The Belousov–Zhabotinsky (BZ) reaction has long been a paradigmatic system for studying chemical oscillations. Here, we experimentally studied the synchronization control within photochemically coupled star networks of BZ oscillators. Experiments were carried out in wells performed in soda-lime glass constructed using novel laser technologies. Utilizing the inherent oscillatory nature of the BZ reaction, we engineered a star network of oscillators interconnected through photochemical inhibitory coupling. Furthermore, the experimental setup presented here could be extrapolated to more complex network architectures with both excitatory and inhibitory couplings, contributing to the fundamental understanding of synchronization in complex systems. Full article

(This article belongs to the Special Issue Nonlinear Dynamical Behaviors in Complex Systems)

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25 pages, 363 KiB

Open AccessArticle

Major Role of Multiscale Entropy Evolution in Complex Systems and Data Science

by Shahid Nawaz, Muhammad Saleem, Fedor V. Kusmartsev and Dalaver H. Anjum

Entropy 2024, 26(4), 330; https://doi.org/10.3390/e26040330 - 12 Apr 2024

Cited by 2 | Viewed by 2070

Abstract

Complex systems are prevalent in various disciplines encompassing the natural and social sciences, such as physics, biology, economics, and sociology. Leveraging data science techniques, particularly those rooted in artificial intelligence and machine learning, offers a promising avenue for comprehending the intricacies of complex systems without necessitating detailed knowledge of underlying dynamics. In this paper, we demonstrate that multiscale entropy (MSE) is pivotal in describing the steady state of complex systems. Introducing the multiscale entropy dynamics (MED) methodology, we provide a framework for dissecting system dynamics and uncovering the driving forces behind their evolution. Our investigation reveals that the MED methodology facilitates the expression of complex system dynamics through a Generalized Nonlinear Schrödinger Equation (GNSE) that thus demonstrates its potential applicability across diverse complex systems. By elucidating the entropic underpinnings of complexity, our study paves the way for a deeper understanding of dynamic phenomena. It offers insights into the behavior of complex systems across various domains. Full article

(This article belongs to the Special Issue Nonlinear Dynamical Behaviors in Complex Systems)

24 pages, 17265 KiB

Open AccessArticle

Turing–Hopf Bifurcation Analysis in a Diffusive Ratio-Dependent Predator–Prey Model with Allee Effect and Predator Harvesting

by Meiyao Chen, Yingting Xu, Jiantao Zhao and Xin Wei

Entropy 2024, 26(1), 18; https://doi.org/10.3390/e26010018 - 22 Dec 2023

Cited by 1 | Viewed by 1683

Abstract

This paper investigates the complex dynamics of a ratio-dependent predator–prey model incorporating the Allee effect in prey and predator harvesting. To explore the joint effect of the harvesting effort and diffusion on the dynamics of the system, we perform the following analyses: (a) The stability of non-negative constant steady states; (b) The sufficient conditions for the occurrence of a Hopf bifurcation, Turing bifurcation, and Turing–Hopf bifurcation; (c) The derivation of the normal form near the Turing–Hopf singularity. Moreover, we provide numerical simulations to illustrate the theoretical results. The results demonstrate that the small change in harvesting effort and the ratio of the diffusion coefficients will destabilize the constant steady states and lead to the complex spatiotemporal behaviors, including homogeneous and inhomogeneous periodic solutions and nonconstant steady states. Moreover, the numerical simulations coincide with our theoretical results. Full article

(This article belongs to the Special Issue Nonlinear Dynamical Behaviors in Complex Systems)

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16 pages, 8182 KiB

Open AccessArticle

Improving Robustness of High-Low-Order Coupled Networks against Malicious Attacks Based on a Simulated Annealing Algorithm

by Chengjun Zhang, Yifan Xie, Yadang Chen, Wenbin Yu, Gaofeng Xiang, Peijun Zhao and Yi Lei

Entropy 2024, 26(1), 8; https://doi.org/10.3390/e26010008 - 21 Dec 2023

Cited by 1 | Viewed by 1474

Abstract

Malicious attacks can cause significant damage to the structure and functionality of complex networks. Previous research has pointed out that the ability of networks to withstand malicious attacks becomes weaker when networks are coupled. However, traditional research on improving the robustness of networks has focused on individual low-order or higher-order networks, lacking studies on coupled networks with higher-order and low-order networks. This paper proposes a method for optimizing the robustness of coupled networks with higher-order and low-order based on a simulated annealing algorithm to address this issue. Without altering the network’s degree distribution, the method rewires the edges, taking the robustness of low-order and higher-order networks as joint optimization objectives. Making minimal changes to the network, the method effectively enhances the robustness of coupled networks. Experiments were conducted on Erdős–Rényi random networks (ER), scale-free networks (BA), and small-world networks (SW). Finally, validation was performed on various real networks. The results indicate that this method can effectively enhance the robustness of coupled networks with higher-order and low-order. Full article

(This article belongs to the Special Issue Nonlinear Dynamical Behaviors in Complex Systems)

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19 pages, 8212 KiB

Open AccessArticle

Distributed Formation Control of Multi-Robot Systems with Path Navigation via Complex Laplacian

by Xiru Wu, Rili Wu, Yuchong Zhang and Jiansheng Peng

Entropy 2023, 25(11), 1536; https://doi.org/10.3390/e25111536 - 11 Nov 2023

Cited by 2 | Viewed by 2329

Abstract

This paper focuses on the formation control of multi-robot systems with leader–follower network structure in directed topology to guide a system composed of multiple mobile robot agents to achieve global path navigation with a desired formation. A distributed linear formation control strategy based on the complex Laplacian matrix is employed, which enables the robot agents to converge into a similar formation of the desired formation, and the size and orientation of the formation are determined by the positions of two leaders. Additionally, in order to ensure that all robot agents in the formation move at a common velocity, the distributed control approach also includes a velocity consensus component. Based on the realization of similar formation control of a multi-robot system, the path navigation algorithm is combined with it to realize the global navigation of the system as a whole. Furthermore, a controller enabling the scalability of the formation size is introduced to enhance the overall maneuverability of the system in specific scenarios like narrow corridors. The simulation results demonstrate the feasibility of the proposed approach. Full article

(This article belongs to the Special Issue Nonlinear Dynamical Behaviors in Complex Systems)

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18 pages, 1464 KiB

Open AccessArticle

Recovering Power Grids Using Strategies Based on Network Metrics and Greedy Algorithms

by Fenghua Wang, Hale Cetinay, Zhidong He, Le Liu, Piet Van Mieghem and Robert E. Kooij

Entropy 2023, 25(10), 1455; https://doi.org/10.3390/e25101455 - 17 Oct 2023

Cited by 2 | Viewed by 1694

Abstract

For this study, we investigated efficient strategies for the recovery of individual links in power grids governed by the direct current (DC) power flow model, under random link failures. Our primary objective was to explore the efficacy of recovering failed links based solely on topological network metrics. In total, we considered 13 recovery strategies, which encompassed 2 strategies based on link centrality values (link betweenness and link flow betweenness), 8 strategies based on the products of node centrality values at link endpoints (degree, eigenvector, weighted eigenvector, closeness, electrical closeness, weighted electrical closeness, zeta vector, and weighted zeta vector), and 2 heuristic strategies (greedy recovery and two-step greedy recovery), in addition to the random recovery strategy. To evaluate the performance of these proposed strategies, we conducted simulations on three distinct power systems: the IEEE 30, IEEE 39, and IEEE 118 systems. Our findings revealed several key insights: Firstly, there were notable variations in the performance of the recovery strategies based on topological network metrics across different power systems. Secondly, all such strategies exhibited inferior performance when compared to the heuristic recovery strategies. Thirdly, the two-step greedy recovery strategy consistently outperformed the others, with the greedy recovery strategy ranking second. Based on our results, we conclude that relying solely on a single metric for the development of a recovery strategy is insufficient when restoring power grids following link failures. By comparison, recovery strategies employing greedy algorithms prove to be more effective choices. Full article

(This article belongs to the Special Issue Nonlinear Dynamical Behaviors in Complex Systems)

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16 pages, 6322 KiB

Open AccessArticle

Response Analysis of the Three-Degree-of-Freedom Vibroimpact System with an Uncertain Parameter

by Guidong Yang, Zichen Deng, Lin Du and Zicheng Lin

Entropy 2023, 25(9), 1365; https://doi.org/10.3390/e25091365 - 21 Sep 2023

Viewed by 1383

Abstract

The inherent non-smoothness of the vibroimpact system leads to complex behaviors and a strong sensitivity to parameter changes. Unfortunately, uncertainties and errors in system parameters are inevitable in mechanical engineering. Therefore, investigations of dynamical behaviors for vibroimpact systems with stochastic parameters are highly essential. The present study aims to analyze the dynamical characteristics of the three-degree-of-freedom vibroimpact system with an uncertain parameter by means of the Chebyshev polynomial approximation method. Specifically, the vibroimpact system model considered is one with unilateral constraint. Firstly, the three-degree-of-freedom vibroimpact system with an uncertain parameter is transformed into an equivalent deterministic form using the Chebyshev orthogonal approximation. Then, the ensemble means responses of the stochastic vibroimpact system are derived. Numerical simulations are performed to verify the effectiveness of the approximation method. Furthermore, the periodic and chaos motions under different system parameters are investigated, and the bifurcations of the vibroimpact system are analyzed with the Poincaré map. The results demonstrate that both the restitution coefficient and the random factor can induce the appearance of the periodic bifurcation. It is worth noting that the bifurcations fundamentally differ between the stochastic and deterministic systems. The former has a bifurcation interval, while the latter occurs at a critical point. Full article

(This article belongs to the Special Issue Nonlinear Dynamical Behaviors in Complex Systems)

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12 pages, 633 KiB

Open AccessArticle

Event-Triggered Bounded Consensus Tracking for Second-Order Nonlinear Multi-Agent Systems with Uncertainties

by Ying Ma, Chan Gu, Yungang Liu, Linzhen Yu and Wei Tang

Entropy 2023, 25(9), 1335; https://doi.org/10.3390/e25091335 - 15 Sep 2023

Cited by 3 | Viewed by 1446

Abstract

This paper is concerned with event-triggered bounded consensus tracking for a class of second-order nonlinear multi-agent systems with uncertainties (MASs). Remarkably, the considered MASs allow multiple uncertainties, including unknown control coefficients, parameterized unknown nonlinearities, uncertain external disturbances, and the leader’s control input being unknown. In this context, a new estimate-based adaptive control protocol with a triggering mechanism is proposed. We rule out Zeno behavior by testifying that the lower bound on the interval between two consecutive events is positive. It is shown that under the designed protocol, all signals caused by the closed-loop systems are bounded globally uniformly and tracking errors ultimately converge to a bounded set. The effectiveness of the devised control protocol is demonstrated through a simulation example. Full article

(This article belongs to the Special Issue Nonlinear Dynamical Behaviors in Complex Systems)

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15 pages, 1733 KiB

Open AccessArticle

Community-CL: An Enhanced Community Detection Algorithm Based on Contrastive Learning

by Zhaoci Huang, Wenzhe Xu and Xinjian Zhuo

Entropy 2023, 25(6), 864; https://doi.org/10.3390/e25060864 - 29 May 2023

Cited by 6 | Viewed by 2794

Abstract

Graph contrastive learning (GCL) has gained considerable attention as a self-supervised learning technique that has been successfully employed in various applications, such as node classification, node clustering, and link prediction. Despite its achievements, GCL has limited exploration of the community structure of graphs. This paper presents a novel online framework called Community Contrastive Learning (Community-CL) for simultaneously learning node representations and detecting communities in a network. The proposed method employs contrastive learning to minimize the difference in the latent representations of nodes and communities in different graph views. To achieve this, learnable graph augmentation views using a graph auto-encoder (GAE) are proposed, followed by a shared encoder that learns the feature matrix of the original graph and augmentation views. This joint contrastive framework enables more accurate representation learning of the network and results in more expressive embeddings than traditional community detection algorithms that solely optimize for community structure. Experimental results demonstrate that Community-CL achieves superior performance compared to state-of-the-art baselines in community detection. Specifically, the NMI of Community-CL is reported to be 0.714 (0.551) on the Amazon-Photo (Amazon-Computers) dataset, which represents a performance improvement of up to 16% compared with the best baseline. Full article

(This article belongs to the Special Issue Nonlinear Dynamical Behaviors in Complex Systems)

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