Complex dynamics and multiple coexisting attractors in a fractional-order microscopic chemical system

2019 ◽  
Vol 228 (1) ◽  
pp. 195-207 ◽  
Author(s):  
Shaobo He ◽  
Santo Banerjee ◽  
Kehui Sun
Keyword(s):  
Fractional Order ◽  
Complex Dynamics ◽  
Chemical System ◽  
Coexisting Attractors ◽  
Multiple Coexisting Attractors

scholarly journals Multistability in a Fractional-Order Centrifugal Flywheel Governor System and Its Adaptive Control

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Bo Yan ◽  
Shaobo He ◽  
Shaojie Wang
Keyword(s):  
Fractional Order ◽  
Lyapunov Stability Theory ◽  
Adaptive Controller ◽  
Basins Of Attraction ◽  
Entropy Measure ◽  
Coexisting Attractors ◽  
Minimum Order ◽  
System Parameters ◽  
Multiple Coexisting Attractors ◽  
Derivative Order

In this paper, a 4D fractional-order centrifugal flywheel governor system is proposed. Dynamics including the multistability of the system with the variation of system parameters and the derivative order are investigated by Lyapunov exponents (LEs), bifurcation diagram, phase portrait, entropy measure, and basins of attraction, numerically. It shows that the minimum order for chaos of the fractional-order centrifugal flywheel governor system is q = 0.97, and the system has rich dynamics and produces multiple coexisting attractors. Moreover, the system is controlled by introducing the adaptive controller which is proved by the Lyapunov stability theory. Numerical analysis results verify the effectiveness of the proposed method.

scholarly journals Network Dynamics of a Fractional-Order Phase-Locked Loop with Infinite Coexisting Attractors

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Anitha Karthikeyan ◽  
Karthikeyan Rajagopal
Keyword(s):  
Fractional Order ◽  
Large Scale ◽  
Equilibrium Points ◽  
Network Dynamics ◽  
Phase Locked Loop ◽  
Coexisting Attractors ◽  
Third Order ◽  
Large Scale Networks ◽  
The Stability ◽  
Multiple Coexisting Attractors

We have investigated a fractional-order phase-locked loop characterised by a third-order differential equation. The integer-order mathematical model of the phase-locked loop (PLL) is first converted to fractional order using the Caputo-Fabrizio method. The stability of the equilibrium points is discussed in detail in both parameter and fractional-order domain. The proposed fractional-order phase-locked loop (FOPLL) model shows multiple coexisting attractors which was not discussed in the earlier literature of PLL. The significance of these infinite coexisting attractors is that they exist in the operation region of the PLL between [−π,π] which increases the complexity of operation of the PLLs. Mainly when such FOPLLs are used in large-scale networks, the synchronisation of the FOPLLs becomes complicated and will result in unstable control conditions. Hence, studying the network dynamics of such FOPLLs is significant which motivates us to investigate the synchronisation phenomenon of the FOPLLs constructed in a square network. We could show that, because of the multiple coexisting attractors, the FOPLLs show various synchronisation phenomena, and more importantly in the chaotic region for lower fractional-order values, we could show that the FOPLLs are synchronised and this finding is very useful to completely analyse the FOPLL networks in high-frequency operations.

Chaos and coexisting attractors in glucose-insulin regulatory system with incommensurate fractional-order derivatives

2021 ◽  
Vol 143 ◽  
pp. 110575
Author(s):  
Nadjette Debbouche ◽  
A. Othman Almatroud ◽  
Adel Ouannas ◽  
Iqbal M. Batiha
Keyword(s):  
Fractional Order ◽  
Regulatory System ◽  
Coexisting Attractors

Various Types of Coexisting Attractors in a New 4D Autonomous Chaotic System

2017 ◽  
Vol 27 (09) ◽  
pp. 1750142 ◽  
Author(s):  
Qiang Lai ◽  
Akif Akgul ◽  
Xiao-Wen Zhao ◽  
Huiqin Pei
Keyword(s):  
Numerical Simulation ◽  
Dynamic Behavior ◽  
Chaotic System ◽  
Limit Cycles ◽  
Electronic Circuit ◽  
Strange Attractors ◽  
Bifurcation Diagrams ◽  
Coexisting Attractors ◽  
Signum Function ◽  
Multiple Coexisting Attractors

An unique 4D autonomous chaotic system with signum function term is proposed in this paper. The system has four unstable equilibria and various types of coexisting attractors appear. Four-wing and four-scroll strange attractors are observed in the system and they will be broken into two coexisting butterfly attractors and two coexisting double-scroll attractors with the variation of the parameters. Numerical simulation shows that the system has various types of multiple coexisting attractors including two butterfly attractors with four limit cycles, two double-scroll attractors with a limit cycle, four single-scroll strange attractors, four limit cycles with regard to different parameters and initial values. The coexistence of the attractors is determined by the bifurcation diagrams. The chaotic and hyperchaotic properties of the attractors are verified by the Lyapunov exponents. Moreover, we present an electronic circuit to experimentally realize the dynamic behavior of the system.

scholarly journals Complex dynamics of a fractional-order SIR system in the context of COVID-19

2022 ◽  
Author(s):  
Suvankar Majee ◽  
Sayani Adak ◽  
Soovoojeet Jana ◽  
Manotosh Mandal ◽  
T. K. Kar
Keyword(s):  
Fractional Order ◽  
Complex Dynamics

scholarly journals A Nonvolatile Fractional Order Memristor Model and its Complex Dynamics

Entropy ◽  
2019 ◽  
Vol 21 (10) ◽  
pp. 955 ◽  
Author(s):  
Wu ◽  
Wang ◽  
Iu ◽  
Shen ◽  
Zhou
Keyword(s):  
Fractional Order ◽  
Complex Dynamics ◽  
Equilibrium Points ◽  
Stable State ◽  
Electrical Characteristics ◽  
Chaotic Circuit ◽  
Stable States ◽  
Road Map ◽  
Chaotic Circuits ◽  
Integral Order

It is found that the fractional order memristor model can better simulate the characteristics of memristors and that chaotic circuits based on fractional order memristors also exhibit abundant dynamic behavior. This paper proposes an active fractional order memristor model and analyzes the electrical characteristics of the memristor via Power-Off Plot and Dynamic Road Map. We find that the fractional order memristor has continually stable states and is therefore nonvolatile. We also show that the memristor can be switched from one stable state to another under the excitation of appropriate voltage pulse. The volt–ampere hysteretic curves, frequency characteristics, and active characteristics of integral order and fractional order memristors are compared and analyzed. Based on the fractional order memristor and fractional order capacitor and inductor, we construct a chaotic circuit, of which the dynamic characteristics with respect to memristor’s parameters, fractional order α, and initial values are analyzed. The chaotic circuit has an infinite number of equilibrium points with multi-stability and exhibits coexisting bifurcations and coexisting attractors. Finally, the fractional order memristor-based chaotic circuit is verified by circuit simulations and DSP experiments.

scholarly journals Complex Canard Explosion in a Fractional-Order FitzHugh–Nagumo Model

2019 ◽  
Vol 29 (08) ◽  
pp. 1950111 ◽  
Author(s):  
Mohammed-Salah Abdelouahab ◽  
René Lozi ◽  
Guanrong Chen
Keyword(s):  
Dynamical System ◽  
Fractional Order ◽  
Complex Dynamics ◽  
Search Algorithm ◽  
Two Dimensional ◽  
Mixed Mode Oscillations ◽  
The Stability ◽  
Two Parameters ◽  
Complex Phenomena ◽  
Autonomous Dynamical System

This article investigates the complex phenomena of canard explosion with mixed-mode oscillations, observed from a fractional-order FitzHugh–Nagumo (FFHN) model. To rigorously analyze the dynamics of the FFHN model, a new mathematical notion, referred to as Hopf-like bifurcation (HLB), is introduced. HLB provides a precise definition for the change between a fixed point and an [Formula: see text]-asymptotically [Formula: see text]-periodic solution of the fractional-order dynamical system, as well as the stability of the FFHN model and the appearance of the HLB. The existence of canard oscillations in the neighborhoods of such HLB points are numerically investigated. Using a new algorithm, referred to as the global-local canard explosion search algorithm, the appearance of various patterns of solutions is revealed, with an increasing number of small-amplitude oscillations when two key parameters of the FFHN model are varied. The numbers of such oscillations versus the two parameters, respectively, are perfectly fitted using exponential functions. Finally, it is conjectured that chaos could occur in a two-dimensional fractional-order autonomous dynamical system, with the fractional order close to one. After all, the article demonstrates that the FFHN model is a very simple two-dimensional model with an incredible ability to present the complex dynamics of neurons.

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