Bifurcation, Chaos, and Fractals in Fractional-Order Electrical and Electronic Systems

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "Complexity".

Deadline for manuscript submissions: 25 October 2024 | Viewed by 5354

Special Issue Editors


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Guest Editor
State Key Laboratory of Electrical Insulation and Power Equipment, School of Electrical Engineering, Xi’an Jiaotong University, Xi’an 710049, China
Interests: fractional calculus and its applications in electrical and electronic engineering; bifurcation and chaos in electrical and electronic engineering
School of Automation and Electronic Information, Xiangtan University, Xiangtan 411105, China
Interests: fractional calculus and its applications; complex dynamic properties of nonlinear systems; memristor and memristor neural networks; complex networks
Special Issues, Collections and Topics in MDPI journals

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Guest Editor
School of Automation and Information Engineering, Xi’an University of Technology, Xi’an 710048, China
Interests: analysis and control of complex nonlinear behaviors in electrical and electronic engineering

Special Issue Information

Dear Colleagues,

Fractional calculus is the extension and generalization of integer calculus; additionally, compared with integer calculus, fractional calculus has superiority as it is able to more accurately describe models, can more easily improve control performance, and has more freedom in designing controllers. At present, with the rapid development of engineering technology, practical electrical and electronic systems are becoming more and more complex, to the point where most of them should be described by using fractional calculus; thus, the requirements for control performance are also getting higher and higher. Meanwhile, a fractional-order controller can guarantee the stable operation of a complex practical electrical and electronic system due to its outstanding advantages. Therefore, the construct of the fractional-order model and the design of the fractional-order controller has become a current hot topic. However, fractional-order electrical and electronic systems have complex dynamical properties, among them, bifurcation, chaos, and fractals are typical nonlinear phenomena and will have an important effect on the system performance. Thus, it is necessary to reveal the underlying mechanism of the occurrence of these typical nonlinear phenomena and design a controller to make these typical nonlinear phenomena disappear.

This Special Issue aims to focus on bifurcation, chaos, and fractals in fractional-order electrical and electronic systems and their control; continuous/discrete modeling and stability analysis of fractional-order electrical and electronic systems; multi-timescale and entropy analysis of fractional-order electrical and electronic systems; and optimization of the control accuracy for fractional-order electrical and electronic systems. Improvements and applications of fractional calculus in electrical and electronic systems are also required.

Dr. Faqiang Wang
Dr. Shaobo He
Dr. Hongbo Cao
Guest Editors

Manuscript Submission Information

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Keywords

  • bifurcation and its control
  • chaos and its control
  • fractals and their control
  • fractional-order inductor/capacitor
  • fractional-order memristor/meminductor/memcapacitor
  • fractional-order circuits and systems
  • fractional-order electrical and electronic systems
  • multi-timescale and entropy analysis
  • fractional calculus applications

Published Papers (5 papers)

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Research

35 pages, 18605 KiB  
Article
Predator–Prey Interaction with Fear Effects: Stability, Bifurcation and Two-Parameter Analysis Incorporating Complex and Fractal Behavior
by Qamar Din, Raja Atif Naseem and Muhammad Sajjad Shabbir
Fractal Fract. 2024, 8(4), 221; https://doi.org/10.3390/fractalfract8040221 - 11 Apr 2024
Viewed by 397
Abstract
This study investigates the dynamics of predator–prey interactions with non-overlapping generations under the influence of fear effects, a crucial factor in ecological research. We propose a novel discrete-time model that addresses limitations of previous models by explicitly incorporating fear. Our primary question is: [...] Read more.
This study investigates the dynamics of predator–prey interactions with non-overlapping generations under the influence of fear effects, a crucial factor in ecological research. We propose a novel discrete-time model that addresses limitations of previous models by explicitly incorporating fear. Our primary question is: How does fear influence the stability of predator–prey populations and the potential for chaotic dynamics? We analyze the model to identify biologically relevant equilibria (fixed points) and determine the conditions for their stability. Bifurcation analysis reveals how changes in fear levels and predation rates can lead to population crashes (transcritical bifurcation) and complex population fluctuations (period-doubling and Neimark–Sacker bifurcations). Furthermore, we explore the potential for controlling chaotic behavior using established methods. Finally, two-parameter analysis employing Lyapunov exponents, spectrum, and Kaplan–Yorke dimension quantifies the chaotic dynamics of the proposed system across a range of fear and predation levels. Numerical simulations support the theoretical findings. This study offers valuable insights into the impact of fear on predator–prey dynamics and paves the way for further exploration of chaos control in ecological models. Full article
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14 pages, 2507 KiB  
Article
Phase Synchronization and Dynamic Behavior of a Novel Small Heterogeneous Coupled Network
by Mengjiao Wang, Jiwei Peng, Shaobo He, Xinan Zhang and Herbert Ho-Ching Iu
Fractal Fract. 2023, 7(11), 818; https://doi.org/10.3390/fractalfract7110818 - 13 Nov 2023
Viewed by 1749
Abstract
Studying the firing dynamics and phase synchronization behavior of heterogeneous coupled networks helps us understand the mechanism of human brain activity. In this study, we propose a novel small heterogeneous coupled network in which the 2D Hopfield neural network (HNN) and the 2D [...] Read more.
Studying the firing dynamics and phase synchronization behavior of heterogeneous coupled networks helps us understand the mechanism of human brain activity. In this study, we propose a novel small heterogeneous coupled network in which the 2D Hopfield neural network (HNN) and the 2D Hindmarsh–Rose (HR) neuron are coupled through a locally active memristor. The simulation results show that the network exhibits complex dynamic behavior and is different from the usual phase synchronization. More specifically, the membrane potential of the 2D HR neuron exhibits five stable firing modes as the coupling parameter k1 changes. In addition, it is found that in the local region of k1, the number of spikes in bursting firing increases with the increase in k1. More interestingly, the network gradually changes from synchronous to asynchronous during the increase in the coupling parameter k1 but suddenly becomes synchronous around the coupling parameter k1 = 1.96. As far as we know, this abnormal synchronization behavior is different from the existing findings. This research is inspired by the fact that the episodic synchronous abnormal firing of excitatory neurons in the hippocampus of the brain can lead to diseases such as epilepsy. This helps us further understand the mechanism of brain activity and build bionic systems. Finally, we design the simulation circuit of the network and implement it on an STM32 microcontroller. Full article
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13 pages, 1217 KiB  
Article
Research on Information Identification of Chaotic Map with Multi-Stability
by You Li and Yuexi Peng
Fractal Fract. 2023, 7(11), 811; https://doi.org/10.3390/fractalfract7110811 - 09 Nov 2023
Viewed by 893
Abstract
Influenced by the rapid development of artificial intelligence, the identification of chaotic systems with intelligent optimization algorithms has received widespread attention in recent years. This paper focuses on the intelligent information identification of chaotic maps with multi-stability properties, and an improved sparrow search [...] Read more.
Influenced by the rapid development of artificial intelligence, the identification of chaotic systems with intelligent optimization algorithms has received widespread attention in recent years. This paper focuses on the intelligent information identification of chaotic maps with multi-stability properties, and an improved sparrow search algorithm is proposed as the identification algorithm. Numerical simulations show that different initial values can lead to the same dynamic behavior, making it impossible to stably and accurately identify the initial values of multi-stability chaotic maps. An identification scheme without considering the initial values is proposed for solving this problem, and simulations demonstrate that the proposed method has the highest identification precision among seven existing intelligent algorithms and a certain degree of noise resistance. In addition, the above research reveals that chaotic systems with multi-stability may have more potential applications in fields such as secure communication. Full article
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21 pages, 2720 KiB  
Article
On Ikeda-Based Memristor Map with Commensurate and Incommensurate Fractional Orders: Bifurcation, Chaos, and Entropy
by Omar Alsayyed, Abderrahmane Abbes, Gharib Mousa Gharib, Mayada Abualhomos, Hassan Al-Tarawneh, Maha S. Al Soudi, Nabeela Abu-Alkishik, Abdallah Al-Husban and Adel Ouannas
Fractal Fract. 2023, 7(10), 728; https://doi.org/10.3390/fractalfract7100728 - 01 Oct 2023
Viewed by 864
Abstract
This paper introduces a novel fractional Ikeda-based memristor map and investigates its non-linear dynamics under commensurate and incommensurate orders using various numerical techniques, including Lyapunov exponent analysis, phase portraits, and bifurcation diagrams. The results reveal diverse and complex system behaviors arising from the [...] Read more.
This paper introduces a novel fractional Ikeda-based memristor map and investigates its non-linear dynamics under commensurate and incommensurate orders using various numerical techniques, including Lyapunov exponent analysis, phase portraits, and bifurcation diagrams. The results reveal diverse and complex system behaviors arising from the interplay of different fractional orders in the proposed map. Furthermore, the study employs the sample entropy test to quantify complexity and validate the presence of chaos. Non-linear controllers are also presented to stabilize and synchronize the model. The research emphasizes the system’s sensitivity to the fractional order parameters, leading to distinct dynamic patterns and stability regimes. The memristor-based chaotic map exhibits rich and intricate behavior, making it an interesting and important area of research. Full article
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20 pages, 11043 KiB  
Article
Complex Dynamical Characteristics of the Fractional-Order Cellular Neural Network and Its DSP Implementation
by Hongli Cao, Ran Chu and Yuanhui Cui
Fractal Fract. 2023, 7(8), 633; https://doi.org/10.3390/fractalfract7080633 - 19 Aug 2023
Viewed by 761
Abstract
A new fractional-order cellular neural network (CNN) system is solved using the Adomian decomposition method (ADM) with the hyperbolic tangent activation function in this paper. The equilibrium point is analyzed in this CNN system. The dynamical behaviors are studied as well, using a [...] Read more.
A new fractional-order cellular neural network (CNN) system is solved using the Adomian decomposition method (ADM) with the hyperbolic tangent activation function in this paper. The equilibrium point is analyzed in this CNN system. The dynamical behaviors are studied as well, using a phase diagram, bifurcation diagram, Lyapunov Exponent spectrum (LEs), and spectral entropy (SE) complexity algorithm. Changing the template parameters and the order values has an impact on the dynamical behaviors. The results indicate that rich dynamical properties exist in the system, such as hyperchaotic attractors, chaotic attractors, asymptotic periodic loops, complex coexisting attractors, and interesting state transition phenomena. In addition, the digital circuit implementation of this fractional-order CNN system is completed on a digital signal processing (DSP) platform, which proves the accuracy of ADM and the physical feasibility of the CNN system. The study in this paper offers a fundamental theory for the fractional-order CNN system as it applies to secure communication and image encryption. Full article
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