Structure Identification of Fractional-Order Dynamical Network with Different Orders

Topology structure and system parameters have a great influence on the dynamical behavior of dynamical networks. However, they are sometimes unknown or uncertain in advance. How to effectively identify them has been investigated in various network models, from integer-order networks to fractional-order networks with the same order. In the real world, many systems consist of subsystems with different fractional orders. Therefore, the structure identification of a dynamical network with different fractional orders is investigated in this paper. Through designing proper adaptive controllers and parameter updating laws, two network estimators are well constructed. One is for identifying only the unknown topology structure. The other is for identifying both the unknown topology structure and system parameters. Based on the Lyapunov function method and the stability theory of fractional-order dynamical systems, the theoretical results are analytically proved. The effectiveness is verified by three numerical examples as well. In addition, the designed estimators have a good performance in monitoring switching topology. From the practical viewpoint, the designed estimators can be used to monitor the change of current and voltage in the fractional-order circuit systems.

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Structure Identification of Fractional-order Complex-variable Network With Complex Coupling

10.21203/rs.3.rs-221279/v1 ◽

2021 ◽

Author(s):

Mingcong Zhou ◽

Zhaoyan Wu

Keyword(s):

Fractional Order ◽

Complex Variable ◽

Topological Structure ◽

Dynamical Behavior ◽

Critical Issue ◽

Structure Identification ◽

Order Complex ◽

System Parameters ◽

Lyapunov Function Method ◽

Theoretical Results

Abstract Fractional-order complex-variable dynamical network with complex coupling is considered in this paper. The topological structures and system parameters are assumed to be unknown. As we know, the topological structure and system parameters play a key role on the dynamical behavior of complex network. Thus, how to effectively identify them is a critical issue for better studying the network. Through designing proper controllers and updating laws, two corresponding network estimators are constructed. Based on the Lyapunov function method and Gronwall-Bellman integral inequality, the results are analytically derived. Finally, two numerical examples are performed to illustrate the feasibility of the theoretical results.

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Fractional-Order Discrete-Time SIR Epidemic Model with Vaccination: Chaos and Complexity

Mathematics ◽

10.3390/math10020165 ◽

2022 ◽

Vol 10 (2) ◽

pp. 165

Author(s):

Zai-Yin He ◽

Abderrahmane Abbes ◽

Hadi Jahanshahi ◽

Naif D. Alotaibi ◽

Ye Wang

Keyword(s):

Fractional Order ◽

Discrete Time ◽

Epidemic Model ◽

Chaotic Behavior ◽

Dynamical Behavior ◽

Maximum Lyapunov Exponent ◽

Sir Epidemic Model ◽

Sir Epidemic ◽

Reasonable Range ◽

Fractional Orders

This research presents a new fractional-order discrete-time susceptible-infected-recovered (SIR) epidemic model with vaccination. The dynamical behavior of the suggested model is examined analytically and numerically. Through using phase attractors, bifurcation diagrams, maximum Lyapunov exponent and the 0−1 test, it is verified that the newly introduced fractional discrete SIR epidemic model vaccination with both commensurate and incommensurate fractional orders has chaotic behavior. The discrete fractional model gives more complex dynamics for incommensurate fractional orders compared to commensurate fractional orders. The reasonable range of commensurate fractional orders is between γ = 0.8712 and γ = 1, while the reasonable range of incommensurate fractional orders is between γ2 = 0.77 and γ2 = 1. Furthermore, the complexity analysis is performed using approximate entropy (ApEn) and C0 complexity to confirm the existence of chaos. Finally, simulations were carried out on MATLAB to verify the efficacy of the given findings.

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Fractional-Order 2 × n RLC Circuit Network

Journal of Circuits System and Computers ◽

10.1142/s021812661550142x ◽

2015 ◽

Vol 24 (09) ◽

pp. 1550142 ◽

Cited By ~ 4

Author(s):

Rui Zhou ◽

Diyi Chen ◽

Herbert H. C. Iu

Keyword(s):

Numerical Simulations ◽

Fractional Order ◽

System Parameters ◽

Differential Equation Models ◽

Rlc Circuit ◽

The Matrix ◽

Mathematical Analyses ◽

New Phenomena ◽

Order Domain ◽

Fractional Orders

This paper introduces new fundamentals of the 2 × n RLC circuit network in the fractional-order domain. First, we derive the three general formulae of the equivalent impedances of the circuit network by using the matrix transform methods and constructing the differential equation models in three different cases. Moreover, we systematically study the effects of the system parameters on the impedence characteristics in the three different cases. Specifically, the new phenomena and laws are presented by the results of the numerical simulations, which are impossible in the conventional cases. Finally, a comparative sensitivity analysis about the three cases with respect to the fractional orders for the fractional-order circuit network is carried out in detail. Mathematical analyses and numerical simulations are included to validate the study.

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Dynamic Behaviors and the Equivalent Realization of a Novel Fractional-Order Memristor-Based Chaotic Circuit

Complexity ◽

10.1155/2018/9467435 ◽

2018 ◽

Vol 2018 ◽

pp. 1-13 ◽

Cited By ~ 1

Author(s):

Ningning Yang ◽

Cheng Xu ◽

Chaojun Wu ◽

Rong Jia ◽

Chongxin Liu

Keyword(s):

Numerical Simulations ◽

Dynamic Behavior ◽

Fractional Order ◽

Equilibrium Points ◽

Dynamical Behavior ◽

Great Influence ◽

Equivalent Circuits ◽

Chaotic Circuit ◽

Undetermined Coefficient ◽

The Stability

This paper proposed a novel fractional-order memristor-based chaotic circuit. A memristive diode bridge cascaded with a fractional-order RL filter constitutes the generalized fractional-order memristor. The mathematical model of the proposed fractional-order chaotic circuit is established by extending the nonlinear capacitor and inductor in the memristive chaotic circuit to the fractional order. Detailed theoretical analysis and numerical simulations are carried out on the dynamic behavior of the proposed circuit by investigating the stability of equilibrium points and the influence of circuit parameters on bifurcations. The results show that the order of the fractional-order circuit has a great influence on the dynamical behavior of the system. The system may exhibit complicated nonlinear dynamic behavior such as bifurcation and chaos with the change of the order. The equivalent circuits of the fractional-order inductor and capacitor are also given in the paper, and the parameters of the equivalent circuits are solved by an undetermined coefficient method. Circuit simulations of the equivalent fractional-order memristive chaotic circuit are carried out in order to validate the correctness of numerical simulations and the practicability of using the integer-order equivalent circuit to substitute the fractional-order element.

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Bifurcation and Chaos in Integer and Fractional Order Two-Degree-of-Freedom Shape Memory Alloy Oscillators

Complexity ◽

10.1155/2018/8365845 ◽

2018 ◽

Vol 2018 ◽

pp. 1-9 ◽

Cited By ~ 4

Author(s):

Karthikeyan Rajagopal ◽

Anitha Karthikeyan ◽

Prakash Duraisamy ◽

Riessom Weldegiorgis

Keyword(s):

Shape Memory Alloy ◽

Shape Memory ◽

Fractional Order ◽

Dynamical Behavior ◽

Degree Of Freedom ◽

Order Model ◽

Bifurcation And Chaos ◽

Fractional Order Model ◽

Two Degree Of Freedom ◽

Fractional Orders

A two-degree-of-freedom shape memory oscillator derived using polynomial constitutive model is investigated. Periodic, quasiperiodic, chaotic, and hyperchaotic oscillations are shown by the shape memory alloy based oscillator for selected values of the operating temperatures and excitation parameters. Bifurcation plots are derived to investigate the system behavior with change in parameters. A fractional order model of the shape memory oscillator is presented and dynamical behavior of the system with fractional orders and parameters are investigated.

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A fractional order epidemic model for the simulation of outbreaks of Ebola

Advances in Difference Equations ◽

10.1186/s13662-021-03272-5 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Weiqiu Pan ◽

Tianzeng Li ◽

Safdar Ali

Keyword(s):

Numerical Solution ◽

Root Mean Square ◽

Fractional Order ◽

Relative Error ◽

Numerical Solutions ◽

Real Data ◽

Mean Square ◽

Order Model ◽

The Real ◽

Fractional Orders

AbstractThe Ebola outbreak in 2014 caused many infections and deaths. Some literature works have proposed some models to study Ebola virus, such as SIR, SIS, SEIR, etc. It is proved that the fractional order model can describe epidemic dynamics better than the integer order model. In this paper, we propose a fractional order Ebola system and analyze the nonnegative solution, the basic reproduction number $R_{0}$ R 0 , and the stabilities of equilibrium points for the system firstly. In many studies, the numerical solutions of some models cannot fit very well with the real data. Thus, to show the dynamics of the Ebola epidemic, the Gorenflo–Mainardi–Moretti–Paradisi scheme (GMMP) is taken to get the numerical solution of the SEIR fractional order Ebola system and the modified grid approximation method (MGAM) is used to acquire the parameters of the SEIR fractional order Ebola system. We consider that the GMMP method may lead to absurd numerical solutions, so its stability and convergence are given. Then, the new fractional orders, parameters, and the root-mean-square relative error $g(U^{*})=0.4146$ g ( U ∗ ) = 0.4146 are obtained. With the new fractional orders and parameters, the numerical solution of the SEIR fractional order Ebola system is closer to the real data than those models in other literature works. Meanwhile, we find that most of the fractional order Ebola systems have the same order. Hence, the fractional order Ebola system with different orders using the Caputo derivatives is also studied. We also adopt the MGAM algorithm to obtain the new orders, parameters, and the root-mean-square relative error which is $g(U^{*})=0.2744$ g ( U ∗ ) = 0.2744 . With the new parameters and orders, the fractional order Ebola systems with different orders fit very well with the real data.

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Hidden Extreme Multistability, Antimonotonicity and Offset Boosting Control in a Novel Fractional-Order Hyperchaotic System Without Equilibrium

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127418501675 ◽

2018 ◽

Vol 28 (13) ◽

pp. 1850167 ◽

Cited By ~ 18

Author(s):

Sen Zhang ◽

Yicheng Zeng ◽

Zhijun Li ◽

Chengyi Zhou

Keyword(s):

Fractional Order ◽

Initial Conditions ◽

State Feedback Control ◽

Hyperchaotic System ◽

Hidden Attractors ◽

System Parameters ◽

Offset Boosting ◽

Extreme Multistability ◽

Hidden Extreme Multistability ◽

New System

Recently, the notion of hidden extreme multistability and hidden attractors is very attractive in chaos theory and nonlinear dynamics. In this paper, by utilizing a simple state feedback control technique, a novel 4D fractional-order hyperchaotic system is introduced. Of particular interest is that this new system has no equilibrium, which indicates that its attractors are all hidden and thus Shil’nikov method cannot be applied to prove the existence of chaos for lacking hetero-clinic or homo-clinic orbits. Compared with other fractional-order chaotic or hyperchaotic systems, this new system possesses three unique and remarkable features: (i) The amazing and interesting phenomenon of the coexistence of infinitely many hidden attractors with respect to same system parameters and different initial conditions is observed, meaning that hidden extreme multistability arises. (ii) By varying the initial conditions and selecting appropriate system parameters, the striking phenomenon of antimonotonicity is first discovered, especially in such a fractional-order hyperchaotic system without equilibrium. (iii) An attractive special feature of the convenience of offset boosting control of the system is also revealed. The complex and rich hidden dynamic behaviors of this system are investigated by using conventional nonlinear analysis tools, including equilibrium stability, phase portraits, bifurcation diagram, Lyapunov exponents, spectral entropy complexity, and so on. Furthermore, a hardware electronic circuit is designed and implemented. The hardware experimental results and the numerical simulations of the same system on the Matlab platform are well consistent with each other, which demonstrates the feasibility of this new fractional-order hyperchaotic system.

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Analysis of Atangana–Baleanu fractional-order SEAIR epidemic model with optimal control

Advances in Difference Equations ◽

10.1186/s13662-021-03334-8 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Chernet Tuge Deressa ◽

Gemechis File Duressa

Keyword(s):

Numerical Simulation ◽

Optimal Control ◽

Fractional Order ◽

Control Strategy ◽

Epidemic Model ◽

Numerical Scheme ◽

Order Derivative ◽

Control Analysis ◽

Fractional Order Derivative ◽

Fractional Orders

AbstractWe consider a SEAIR epidemic model with Atangana–Baleanu fractional-order derivative. We approximate the solution of the model using the numerical scheme developed by Toufic and Atangana. The numerical simulation corresponding to several fractional orders shows that, as the fractional order reduces from 1, the spread of the endemic grows slower. Optimal control analysis and simulation show that the control strategy designed is operative in reducing the number of cases in different compartments. Moreover, simulating the optimal profile revealed that reducing the fractional-order from 1 leads to the need for quick starting of the application of the designed control strategy at the maximum possible level and maintaining it for the majority of the period of the pandemic.

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