Nonlinear Dynamics in Complex Systems via Fractals and Fractional Calculus

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

Deadline for manuscript submissions: closed (20 January 2023) | Viewed by 19666

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Special Issue Editor

Special Issue Information

Dear Colleagues,

Nowadays, advances in the knowledge of nonlinear dynamical systems and processes as well as their unified repercussions allow us to include some typical complex phenomena taking place in nature, from nanoscale to galactic scale, in a unitary comprehensive manner. After all, any of these systems called generic dynamical systems, chaotic systems or fractal systems have something essential in common and can be considered to belong to the same class of complex phenomena, discussed here. The available physical, biological and financial data and technological (mechanical or electronic devices) complex systems can be managed by the same conceptual approach, both analytically and through a computer simulation, using effective nonlinear dynamics methods. Currently, the utilization of fractional-order partial differential equations in real physical systems is commonly encountered in the fields of theoretical science and engineering applications. This means that the productive, efficacious computational tools required for analytical and numerical estimations of such physical models, and our reliance on their development in referenced works, are welcome. Chaotic instabilities in the mathematical physics theory, fractal-type spatiotemporal behaviors in the field theory, nonlinear dynamic processes in plasma complex structures, fractional calculus and novel algorithms to solve fractional-order derivatives of classic problems are expected.

Prof. Dr. Viorel-Puiu Paun
Guest Editor

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Keywords

  • chaotic systems
  • fractal systems
  • fractal-type field theory
  • fractal analysis
  • fractional calculus
  • fractional-order derivatives algorithms
  • fractional derivatives neural networks
  • image processing
  • fractional diffusion
  • nonlinear dynamics
  • time series method
  • diffusion process
  • control theory
  • mathematical modeling

Published Papers (13 papers)

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Editorial

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5 pages, 225 KiB  
Editorial
Special Issue: Nonlinear Dynamics in Complex Systems via Fractals and Fractional Calculus
Fractal Fract. 2023, 7(5), 412; https://doi.org/10.3390/fractalfract7050412 - 20 May 2023
Viewed by 695
Abstract
Advances in our knowledge of nonlinear dynamical networks, systems and processes (as well as their unified repercussions) currently allow us to study many typical complex phenomena taking place in nature, from the nanoscale to the extra-galactic scale, in an comprehensive manner [...] Full article

Research

Jump to: Editorial

18 pages, 33577 KiB  
Article
Theoretical and Experimental Designs of the Planetary Boundary Layer Dynamics through a Multifractal Theory of Motion
Fractal Fract. 2022, 6(12), 747; https://doi.org/10.3390/fractalfract6120747 - 19 Dec 2022
Cited by 2 | Viewed by 953
Abstract
The accurate determination of atmospheric temperature with telemetric platforms is an active issue, one that can also be tackled with the aid of multifractal theory to extract fundamental behaviors of the lower atmosphere, which can then be used to facilitate such determinations. Thus, [...] Read more.
The accurate determination of atmospheric temperature with telemetric platforms is an active issue, one that can also be tackled with the aid of multifractal theory to extract fundamental behaviors of the lower atmosphere, which can then be used to facilitate such determinations. Thus, in the framework of the scale relativity theory, PBL dynamics are analyzed through the aid of a multifractal hydrodynamic scenario. Considering the PBL as a complex system that is assimilated to mathematical objects of a multifractal type, its various dynamics work as a multifractal tunnel effect. Such a treatment allows one to define both a multifractal atmospheric transparency coefficient and a multifractal atmospheric reflectance coefficient. These products are then employed to create theoretical temperature profiles, which lead to correspondences with real results obtained by radiometer data (RPG-HATPRO radiometer), with favorable results. Such methods could be further used and refined in future applications to efficiently produce atmospheric temperature theoretical profiles. Full article
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25 pages, 12465 KiB  
Article
A New Variable-Boostable 3D Chaotic System with Hidden and Coexisting Attractors: Dynamical Analysis, Periodic Orbit Coding, Circuit Simulation, and Synchronization
Fractal Fract. 2022, 6(12), 740; https://doi.org/10.3390/fractalfract6120740 - 14 Dec 2022
Cited by 7 | Viewed by 1127
Abstract
The study of hidden attractors plays a very important role in the engineering applications of nonlinear dynamical systems. In this paper, a new three-dimensional (3D) chaotic system is proposed in which hidden attractors and self-excited attractors appear as the parameters change. Meanwhile, asymmetric [...] Read more.
The study of hidden attractors plays a very important role in the engineering applications of nonlinear dynamical systems. In this paper, a new three-dimensional (3D) chaotic system is proposed in which hidden attractors and self-excited attractors appear as the parameters change. Meanwhile, asymmetric coexisting attractors are also found as a result of the system symmetry. The complex dynamical behaviors of the proposed system were investigated using various tools, including time-series diagrams, Poincaré first return maps, bifurcation diagrams, and basins of attraction. Moreover, the unstable periodic orbits within a topological length of 3 in the hidden chaotic attractor were calculated systematically by the variational method, which required six letters to establish suitable symbolic dynamics. Furthermore, the practicality of the hidden attractor chaotic system was verified by circuit simulations. Finally, offset boosting control and adaptive synchronization were used to investigate the utility of the proposed chaotic system in engineering applications. Full article
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19 pages, 11070 KiB  
Article
Spatial Series and Fractal Analysis Associated with Fracture Behaviour of UO2 Ceramic Material
Fractal Fract. 2022, 6(10), 595; https://doi.org/10.3390/fractalfract6100595 - 14 Oct 2022
Cited by 2 | Viewed by 946
Abstract
SEM micrographs of the fracture surface for UO2 ceramic materials have been analysed. In this paper, we introduce some algorithms and develop a computer application based on the time-series method. Utilizing the embedding technique of phase space, the attractor is reconstructed. The [...] Read more.
SEM micrographs of the fracture surface for UO2 ceramic materials have been analysed. In this paper, we introduce some algorithms and develop a computer application based on the time-series method. Utilizing the embedding technique of phase space, the attractor is reconstructed. The fractal dimension, lacunarity, and autocorrelation dimension average value have been calculated. Full article
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22 pages, 3607 KiB  
Article
Dynamic Analysis of a Novel 3D Chaotic System with Hidden and Coexisting Attractors: Offset Boosting, Synchronization, and Circuit Realization
Fractal Fract. 2022, 6(10), 547; https://doi.org/10.3390/fractalfract6100547 - 27 Sep 2022
Cited by 7 | Viewed by 1215
Abstract
To further understand the dynamical characteristics of chaotic systems with a hidden attractor and coexisting attractors, we investigated the fundamental dynamics of a novel three-dimensional (3D) chaotic system derived by adding a simple constant term to the Yang–Chen system, which includes the bifurcation [...] Read more.
To further understand the dynamical characteristics of chaotic systems with a hidden attractor and coexisting attractors, we investigated the fundamental dynamics of a novel three-dimensional (3D) chaotic system derived by adding a simple constant term to the Yang–Chen system, which includes the bifurcation diagram, Lyapunov exponents spectrum, and basin of attraction, under different parameters. In addition, an offset boosting control method is presented to the state variable, and a numerical simulation of the system is also presented. Furthermore, the unstable cycles embedded in the hidden chaotic attractors are extracted in detail, which shows the effectiveness of the variational method and 1D symbolic dynamics. Finally, the adaptive synchronization of the novel system is successfully designed, and a circuit simulation is implemented to illustrate the flexibility and validity of the numerical results. Theoretical analysis and simulation results indicate that the new system has complex dynamical properties and can be used to facilitate engineering applications. Full article
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17 pages, 6720 KiB  
Article
Fractional Order Fuzzy Dispersion Entropy and Its Application in Bearing Fault Diagnosis
Fractal Fract. 2022, 6(10), 544; https://doi.org/10.3390/fractalfract6100544 - 26 Sep 2022
Cited by 36 | Viewed by 1744
Abstract
Fuzzy dispersion entropy (FuzzDE) is a very recently proposed non-linear dynamical indicator, which combines the advantages of both dispersion entropy (DE) and fuzzy entropy (FuzzEn) to detect dynamic changes in a time series. However, FuzzDE only reflects the information of the original signal [...] Read more.
Fuzzy dispersion entropy (FuzzDE) is a very recently proposed non-linear dynamical indicator, which combines the advantages of both dispersion entropy (DE) and fuzzy entropy (FuzzEn) to detect dynamic changes in a time series. However, FuzzDE only reflects the information of the original signal and is not very sensitive to dynamic changes. To address these drawbacks, we introduce fractional order calculation on the basis of FuzzDE, propose FuzzDEα, and use it as a feature for the signal analysis and fault diagnosis of bearings. In addition, we also introduce other fractional order entropies, including fractional order DE (DEα), fractional order permutation entropy (PEα) and fractional order fluctuation-based DE (FDEα), and propose a mixed features extraction diagnosis method. Both simulated as well as real-world experimental results demonstrate that the FuzzDEα at different fractional orders is more sensitive to changes in the dynamics of the time series, and the proposed mixed features bearing fault diagnosis method achieves 100% recognition rate at just triple features, among which, the mixed feature combinations with the highest recognition rates all have FuzzDEα, and FuzzDEα also appears most frequently. Full article
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21 pages, 4125 KiB  
Article
Global Bifurcation Behaviors and Control in a Class of Bilateral MEMS Resonators
Fractal Fract. 2022, 6(10), 538; https://doi.org/10.3390/fractalfract6100538 - 23 Sep 2022
Cited by 3 | Viewed by 1140
Abstract
The investigation of global bifurcation behaviors the vibrating structures of micro-electromechanical systems (MEMS) has received substantial attention. This paper considers the vibrating system of a typical bilateral MEMS resonator containing fractional functions and multiple potential wells. By introducing new variations, the Melnikov method [...] Read more.
The investigation of global bifurcation behaviors the vibrating structures of micro-electromechanical systems (MEMS) has received substantial attention. This paper considers the vibrating system of a typical bilateral MEMS resonator containing fractional functions and multiple potential wells. By introducing new variations, the Melnikov method is applied to derive the critical conditions for global bifurcations. By engaging in the fractal erosion of safe basin to depict the phenomenon pull-in instability intuitively, the point-mapping approach is used to present numerical simulations which are in close agreement with the analytical prediction, showing the validity of the analysis. It is found that chaos and pull-in instability, two initial-sensitive phenomena of MEMS resonators, can be due to homoclinic bifurcation and heteroclinic bifurcation, respectively. On this basis, two types of delayed feedback are proposed to control the complex dynamics successively. Their control mechanisms and effect are then studied. It follows that under a positive gain coefficient, delayed position feedback and delayed velocity feedback can both reduce pull-in instability; nevertheless, to suppress chaos, only the former can be effective. The results may have some potential value in broadening the application fields of global bifurcation theory and improving the performance reliability of capacitive MEMS devices. Full article
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16 pages, 8558 KiB  
Article
Minkowski’s Loop Fractal Antenna Dedicated to Sixth Generation (6G) Communication
Fractal Fract. 2022, 6(7), 402; https://doi.org/10.3390/fractalfract6070402 - 21 Jul 2022
Cited by 5 | Viewed by 1477
Abstract
In this study, we will discuss the engineering construction of a special sixth generation (6G) antenna, based on the fractal called Minkowski’s loop. The antenna has the shape of this known fractal, set at four iterations, to obtain maximum performance. The frequency bands [...] Read more.
In this study, we will discuss the engineering construction of a special sixth generation (6G) antenna, based on the fractal called Minkowski’s loop. The antenna has the shape of this known fractal, set at four iterations, to obtain maximum performance. The frequency bands for which this 6G fractal antenna was designed in the current paper are 170 GHz to 260 GHz (WR-4) and 110 GHz to 170 GHz (WR-6), respectively. The three resonant frequencies, optimally used, are equal to 140 GHz (WR-6) for the first, 182 GHz (WR-4) for the second and 191 GHz (WR-4) for the third. For these frequencies the electromagnetic behaviors of fractal antennas and their graphical representations are highlighted. Full article
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17 pages, 4354 KiB  
Article
Particle Swarm Optimization Fractional Slope Entropy: A New Time Series Complexity Indicator for Bearing Fault Diagnosis
Fractal Fract. 2022, 6(7), 345; https://doi.org/10.3390/fractalfract6070345 - 21 Jun 2022
Cited by 36 | Viewed by 2106
Abstract
Slope entropy (SlEn) is a time series complexity indicator proposed in recent years, which has shown excellent performance in the fields of medical and hydroacoustics. In order to improve the ability of SlEn to distinguish different types of signals and solve the problem [...] Read more.
Slope entropy (SlEn) is a time series complexity indicator proposed in recent years, which has shown excellent performance in the fields of medical and hydroacoustics. In order to improve the ability of SlEn to distinguish different types of signals and solve the problem of two threshold parameters selection, a new time series complexity indicator on the basis of SlEn is proposed by introducing fractional calculus and combining particle swarm optimization (PSO), named PSO fractional SlEn (PSO-FrSlEn). Then we apply PSO-FrSlEn to the field of fault diagnosis and propose a single feature extraction method and a double feature extraction method for rolling bearing fault based on PSO-FrSlEn. The experimental results illustrated that only PSO-FrSlEn can classify 10 kinds of bearing signals with 100% classification accuracy by using double features, which is at least 4% higher than the classification accuracies of the other four fractional entropies. Full article
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10 pages, 1894 KiB  
Article
Sensitivity of Uniformly Convergent Mapping Sequences in Non-Autonomous Discrete Dynamical Systems
Fractal Fract. 2022, 6(6), 319; https://doi.org/10.3390/fractalfract6060319 - 07 Jun 2022
Cited by 1 | Viewed by 1127
Abstract
Let H be a compact metric space. The metric of H is denoted by d. And let (H,f1,) be a non-autonomous discrete system where [...] Read more.
Let H be a compact metric space. The metric of H is denoted by d. And let (H,f1,) be a non-autonomous discrete system where f1,={fn}n=1 is a mapping sequence. This paper discusses infinite sensitivity, m-sensitivity, and m-cofinitely sensitivity of f1,. It is proved that, if fn(nN) are feebly open and uniformly converge to f:HH, fif=ffi for any i{1,2,}, and i=1D(fi,f)<, then (H,f) has the above sensitive property if and only if (H,f1,) has the same property where D(·,·) is the supremum metric. Full article
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24 pages, 6516 KiB  
Article
Hidden and Coexisting Attractors in a Novel 4D Hyperchaotic System with No Equilibrium Point
Fractal Fract. 2022, 6(6), 306; https://doi.org/10.3390/fractalfract6060306 - 31 May 2022
Cited by 18 | Viewed by 2016
Abstract
The investigation of chaotic systems containing hidden and coexisting attractors has attracted extensive attention. This paper presents a four-dimensional (4D) novel hyperchaotic system, evolved by adding a linear state feedback controller to a 3D chaotic system with two stable node-focus points. The proposed [...] Read more.
The investigation of chaotic systems containing hidden and coexisting attractors has attracted extensive attention. This paper presents a four-dimensional (4D) novel hyperchaotic system, evolved by adding a linear state feedback controller to a 3D chaotic system with two stable node-focus points. The proposed system has no equilibrium point or two lines of equilibria, depending on the value of the constant term. Complex dynamical behaviors such as hidden chaotic and hyperchaotic attractors and five types of coexisting attractors of the simple 4D autonomous system are investigated and discussed, and are numerically verified by analyzing phase diagrams, Poincaré maps, the Lyapunov exponent spectrum, and its bifurcation diagram. The short unstable cycles in the hyperchaotic system are systematically explored via the variational method, and symbol codings of the cycles with four letters are realized based on the topological properties of the trajectory projection on the 2D phase space. The bifurcations of the cycles are explored through a homotopy evolution approach. Finally, the novel 4D system is implemented by an analog electronic circuit and is found to be consistent with the numerical simulation results. Full article
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21 pages, 49921 KiB  
Article
Boundary Layer via Multifractal Mass Conductivity through Remote Sensing Data in Atmospheric Dynamics
Fractal Fract. 2022, 6(5), 250; https://doi.org/10.3390/fractalfract6050250 - 30 Apr 2022
Cited by 2 | Viewed by 1418
Abstract
In this manuscript, multifractal theories of motion based on scale relativity theory are considered in the description of atmospheric dynamics. It is shown that these theories have the potential to highlight nondimensional mass conduction laws that describe the propagation of atmospheric entities. Then, [...] Read more.
In this manuscript, multifractal theories of motion based on scale relativity theory are considered in the description of atmospheric dynamics. It is shown that these theories have the potential to highlight nondimensional mass conduction laws that describe the propagation of atmospheric entities. Then, using special operational procedures and harmonic mappings, these equations can be rewritten and simplified for their plotting and analysis to be performed. The inhomogeneity of these conduction phenomena is analyzed, and it is found that it can fluctuate and increase at certain fractal dimensions, leading to the conclusion that certain atmospheric structures and phenomena of either atmospheric transmission or stability can be explained by atmospheric fractal dimension inversions. Finally, this hypothesis is verified using ceilometer data throughout the atmospheric profiles. Full article
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23 pages, 5026 KiB  
Article
Dynamics, Periodic Orbit Analysis, and Circuit Implementation of a New Chaotic System with Hidden Attractor
Fractal Fract. 2022, 6(4), 190; https://doi.org/10.3390/fractalfract6040190 - 30 Mar 2022
Cited by 10 | Viewed by 2102
Abstract
Hidden attractors are associated with multistability phenomena, which have considerable application prospects in engineering. By modifying a simple three-dimensional continuous quadratic dynamical system, this paper reports a new autonomous chaotic system with two stable node-foci that can generate double-wing hidden chaotic attractors. We [...] Read more.
Hidden attractors are associated with multistability phenomena, which have considerable application prospects in engineering. By modifying a simple three-dimensional continuous quadratic dynamical system, this paper reports a new autonomous chaotic system with two stable node-foci that can generate double-wing hidden chaotic attractors. We discuss the rich dynamics of the proposed system, which have some interesting characteristics for different parameters and initial conditions, through the use of dynamic analysis tools such as the phase portrait, Lyapunov exponent spectrum, and bifurcation diagram. The topological classification of the periodic orbits of the system is investigated by a recently devised variational method. Symbolic dynamics of four and six letters are successfully established under two sets of system parameters, including hidden and self-excited chaotic attractors. The system is implemented by a corresponding analog electronic circuit to verify its realizability. Full article
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