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.

scholarly journals Epidemic Dynamics of a Fractional-Order SIS Infectious Network Model

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Na Liu ◽  
Yunliu Li ◽  
Junwei Sun ◽  
Jie Fang ◽  
Peng Liu
Keyword(s):  
Fractional Order ◽  
Network Model ◽  
Large Scale ◽  
Equilibrium Points ◽  
Transmission Dynamics ◽  
Economic Losses ◽  
Epidemic Dynamics ◽  
Order Coefficient ◽  
The Stability ◽  
And Control

Outbreak and large-scale of the infectious diseases have caused enormous economic losses to all countries in the world. Constructing a network model which could reflect the transmission dynamics of the epidemics and investigating their transmission laws have a significant meaning in the precaution and control of the epidemics. In this article, a fractional-order SIS epidemic network model is proposed. First, an expression of the basic reproduction number is deduced. Second, applying the Lyapunov function, the stability of the equilibrium points about the infectious model is analyzed in detail. Finally, an example is present to verify the theoretical analysis. Furthermore, on account of the fractional-order coefficient, its influence on the transmission dynamics is also exhibited.

scholarly journals A Fractional-Order Discrete Noninvertible Map of Cubic Type: Dynamics, Control, and Synchronization

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-21
Author(s):  
Xiaojun Liu ◽  
Ling Hong ◽  
Lixin Yang ◽  
Dafeng Tang
Keyword(s):  
Fractional Order ◽  
Initial Conditions ◽  
Dimensional Space ◽  
Equilibrium Points ◽  
Three Dimensional ◽  
State Variables ◽  
Discrete Maps ◽  
The Stability ◽  
Control And Synchronization ◽  
Simultaneous Variation

In this paper, a new fractional-order discrete noninvertible map of cubic type is presented. Firstly, the stability of the equilibrium points for the map is examined. Secondly, the dynamics of the map with two different initial conditions is studied by numerical simulation when a parameter or a derivative order is varied. A series of attractors are displayed in various forms of periodic and chaotic ones. Furthermore, bifurcations with the simultaneous variation of both a parameter and the order are also analyzed in the three-dimensional space. Interior crises are found in the map as a parameter or an order varies. Thirdly, based on the stability theory of fractional-order discrete maps, a stabilization controller is proposed to control the chaos of the map and the asymptotic convergence of the state variables is determined. Finally, the synchronization between the proposed map and a fractional-order discrete Loren map is investigated. Numerical simulations are used to verify the effectiveness of the designed synchronization controllers.

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 Oxytocin modulates the intrinsic dynamics between attention-related large scale networks

2018 ◽  
Author(s):  
Fei Xin ◽  
Feng Zhou ◽  
Xinqi Zhou ◽  
Xiaole Ma ◽  
Yayuan Geng ◽  
...  
Keyword(s):  
Attentional Control ◽  
Large Scale ◽  
Functional Integration ◽  
Network Dynamics ◽  
Resting State Fmri ◽  
Neural Systems ◽  
Functional Dynamics ◽  
Fmri Study ◽  
N 93 ◽  
Large Scale Networks

AbstractAttention and salience processing have been linked to the intrinsic between- and within-network dynamics of large scale networks engaged in internal (default mode network, DN) and external attention allocation (dorsal attention, DAN, salience network, SN). The central oxytocin (OXT) system appears ideally organized to modulate widely distributed neural systems and to regulate the switch between internal attention and salient stimuli in the environment. The current randomized placebo (PLC) controlled between-subject pharmacological resting-state fMRI study in N = 187 (OXT, n = 94; n = 93; single-dose intranasal administration) healthy male and female participants employed an independent component analysis (ICA) approach to determine the modulatory effects of OXT on the within- and between-network dynamics of the DAN-SN-DN triple network system. OXT increased the functional integration between subsystems within SN and DN and increased functional segregation of the DN with the SN and DAN engaged in attentional control. Whereas no sex differences were observed, OXT effects on the DN-SN interaction were modulated by autism traits. Together, the findings suggest that OXT may facilitate efficient attentional allocation towards social cues by modulating the intrinsic functional dynamics between DN components engaged in social processing and large-scale networks involved in external attentional demands (SN, DAN).

scholarly journals Chaos synchronization of a fractional nonautonomous system

2014 ◽  
Vol 1 (1) ◽  
Author(s):  
Zakia Hammouch ◽  
Toufik Mekkaoui
Keyword(s):  
Dynamic Behavior ◽  
Fractional Order ◽  
Chaotic System ◽  
Stability Criterion ◽  
Chaos Synchronization ◽  
Phase Locked Loop ◽  
Numerical Results ◽  
The Other ◽  
Fractional Systems ◽  
The Stability

AbstractIn this paper we investigate the dynamic behavior of a nonautonomous fractional-order biological system.With the stability criterion of active nonlinear fractional systems, the synchronization of the studied chaotic system is obtained. On the other hand, using a Phase-Locked-Loop (PLL) analogy we synchronize the same system. The numerical results demonstrate the efiectiveness of the proposed methods

scholarly journals Dynamics Analysis and Control of a Five-Term Fractional-Order System

2018 ◽  
Vol 2018 ◽  
pp. 1-10
Author(s):  
Li-xin Yang ◽  
Xiao-jun Liu
Keyword(s):  
Fractional Order ◽  
Equilibrium Points ◽  
Fractional Order System ◽  
Phase Portraits ◽  
Order System ◽  
Bifurcation Diagrams ◽  
Compound Structure ◽  
Dynamical Properties ◽  
The Stability ◽  
And Control

This paper proposes a new fractional-order chaotic system with five terms. Firstly, basic dynamical properties of the fractional-order system are investigated in terms of the stability of equilibrium points, Jacobian matrices theoretically. Furthermore, rich dynamics with interesting characteristics are demonstrated by phase portraits, bifurcation diagrams numerically. Besides, the control problem of the new fractional-order system is discussed via numerical simulations. Our results demonstrate that the new fractional-order system has compound structure.

CHAOS IN DIFFUSIONLESS LORENZ SYSTEM WITH A FRACTIONAL ORDER AND ITS CONTROL

2012 ◽  
Vol 22 (04) ◽  
pp. 1250088 ◽  
Author(s):  
YONG XU ◽  
RENCAI GU ◽  
HUIQING ZHANG ◽  
DONGXI LI
Keyword(s):  
Feedback Control ◽  
Control Problem ◽  
Fractional Order ◽  
Lorenz System ◽  
Equilibrium Points ◽  
Control Technique ◽  
Order System ◽  
System Parameters ◽  
Simulation Results ◽  
The Stability

This paper aims to investigate the phenomenon of Diffusionless Lorenz system with fractional-order. We discuss the stability of equilibrium points of the fractional-order system theoretically, and analyze the chaotic behaviors and typical bifurcations numerically. We find rich dynamics in fractional-order Diffusionless Lorenz system with appropriate fractional order and system parameters. Besides, the control problem of fractional-order Diffusionless Lorenz system is examined using feedback control technique, and simulation results show the effectiveness of the method.

scholarly journals Coexisting Oscillation and Extreme Multistability for a Memcapacitor-Based Circuit

2017 ◽  
Vol 2017 ◽  
pp. 1-13 ◽  
Author(s):  
Guangyi Wang ◽  
Chuanbao Shi ◽  
Xiaowei Wang ◽  
Fang Yuan
Keyword(s):  
Limit Cycle ◽  
Infinite Number ◽  
Chaotic Attractor ◽  
Electronic Circuit ◽  
Equilibrium Points ◽  
Experimental Result ◽  
Coexisting Attractors ◽  
Extreme Multistability ◽  
Fix Point ◽  
The Stability

The coexisting oscillations are observed with a memcapacitor-based circuit that consists of two linear inductors, two linear resistors, and an active nonlinear charge-controlled memcapacitor. We analyze the dynamics of this circuit and find that it owns an infinite number of equilibrium points and coexisting attractors, which means extreme multistability arises. Furthermore, we also show the stability of the infinite many equilibria and analyze the coexistence of fix point, limit cycle, and chaotic attractor in detail. Finally, an experimental result of the proposed oscillator via an analog electronic circuit is given.

Antimonotonicity, Bifurcation and Multistability in the Vallis Model for El Niño

2019 ◽  
Vol 29 (03) ◽  
pp. 1950032 ◽  
Author(s):  
Karthikeyan Rajagopal ◽  
Sajad Jafari ◽  
Viet-Thanh Pham ◽  
Zhouchao Wei ◽  
Durairaj Premraj ◽  
...  
Keyword(s):  
Hopf Bifurcation ◽  
El Niño ◽  
El Nino ◽  
Equilibrium Points ◽  
Period Doubling ◽  
Period Doubling Bifurcation ◽  
Coexisting Attractors ◽  
The Stability ◽  
First Time ◽  
Parameter Condition

In this paper, the well-known Vallis model for El Niño is analyzed for the parameter condition [Formula: see text]. The conditions for the stability of the equilibrium points are derived. The condition for Hopf bifurcation occurring in the system for [Formula: see text] and [Formula: see text] are investigated. The multistability feature of the Vallis model when [Formula: see text] is explained with forward and backward continuation bifurcation plots and with the coexisting attractors. The creation of period doubling followed by their annihilation via inverse period-doubling bifurcation known as antimonotonicity occurrence in the Vallis model for [Formula: see text] is presented for the first time in the literature.

scholarly journals A study on the AH1N1/09 influenza transmission model with the fractional Caputo–Fabrizio derivative

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Shahram Rezapour ◽  
Hakimeh Mohammadi
Keyword(s):  
Fractional Order ◽  
Fixed Point Theory ◽  
Equilibrium Points ◽  
Point Theory ◽  
Euler Method ◽  
Transmission Model ◽  
Order Model ◽  
Influenza Transmission ◽  
The Stability ◽  
Disease Free

Abstract We study the SEIR epidemic model for the spread of AH1N1 influenza using the Caputo–Fabrizio fractional-order derivative. The reproduction number of system and equilibrium points are calculated, and the stability of the disease-free equilibrium point is investigated. We prove the existence of solution for the model by using fixed point theory. Using the fractional Euler method, we get an approximate solution to the model. In the numerical section, we present a simulation to examine the system, in which we calculate equilibrium points of the system and examine the behavior of the resulting functions at the equilibrium points. By calculating the results of the model for different fractional order, we examine the effect of the derivative order on the behavior of the resulting functions and obtained numerical values. We also calculate the results of the integer-order model and examine their differences with the results of the fractional-order model.

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