scholarly journals An Efficient Method for Hopf Bifurcation Control in Fractional-Order Neuron Model

IEEE Access ◽  
2019 ◽  
Vol 7 ◽  
pp. 77490-77498 ◽  
Author(s):  
Shaolong Chen ◽  
Yuan Zou ◽  
Xudong Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Fractional Order ◽  
Efficient Method ◽  
Neuron Model ◽  
Bifurcation Control ◽  
Order Neuron

scholarly journals Bifurcation Control of a Delayed Fractional Mosaic Disease Model for Jatropha curcas with Farming Awareness

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-16
Author(s):  
Shouzong Liu ◽  
Mingzhan Huang ◽  
Juan Wang
Keyword(s):  
Hopf Bifurcation ◽  
Fractional Order ◽  
Jatropha Curcas ◽  
Mosaic Disease ◽  
Bifurcation Control ◽  
Feedback Gain ◽  
Infection Model ◽  
Feedback Delay ◽  
Execution Delay ◽  
Farming Awareness

In this paper, the bifurcation control of a fractional-order mosaic virus infection model for Jatropha curcas with farming awareness and an execution delay is investigated. By analyzing the associated characteristic equation, Hopf bifurcation induced by the execution delay is studied for the uncontrolled system. Then, a time-delayed controller is introduced to control the occurrence of Hopf bifurcation. Our study implies that bifurcation dynamics is significantly affected by the change of the fractional order, the feedback gain and the extended feedback delay provided that the other parameters are fixed. A series of numerical simulations is performed, which not only verifies our theoretical results but also reveals some specific features. Numerically, we find that the Hopf bifurcation gradually occurs in advance with the increase of the fractional order, and there exist extreme points for the feedback gain and the extended feedback delay which can minimize the bifurcation value.

scholarly journals On the reasonability of linearized approximation and Hopf bifurcation control for a fractional-order delay Bhalekar–Gejji chaotic system

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Jianping Shi ◽  
Liyuan Ruan
Keyword(s):  
Hopf Bifurcation ◽  
Fractional Order ◽  
Chaotic System ◽  
Delayed Feedback ◽  
Feedback Controller ◽  
Bifurcation Control ◽  
Delay Systems ◽  
Feedback Gain ◽  
Stability Of Equilibrium ◽  
Delayed Feedback Controller

Abstract In this paper, we study the reasonability of linearized approximation and Hopf bifurcation control for a fractional-order delay Bhalekar–Gejji (BG) chaotic system. Since the current study on Hopf bifurcation for fractional-order delay systems is carried out on the basis of analyses for stability of equilibrium of its linearized approximation system, it is necessary to verify the reasonability of linearized approximation. Through Laplace transformation, we first illustrate the equivalence of stability of equilibrium for a fractional-order delay Bhalekar–Gejji chaotic system and its linearized approximation system under an appropriate prior assumption. This semianalytically verifies the reasonability of linearized approximation from the viewpoint of stability. Then we theoretically explore the relationship between the time delay and Hopf bifurcation of such a system. By introducing the delayed feedback controller into the proposed system, the influence of the feedback gain changes on Hopf bifurcation is also investigated. The obtained results indicate that the stability domain can be effectively controlled by the proposed delayed feedback controller. Moreover, numerical simulations are made to verify the validity of the theoretical results.

scholarly journals Vibrational Resonance and Electrical Activity Behavior of a Fractional-Order FitzHugh–Nagumo Neuron System

Mathematics ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 87
Author(s):  
Jia-Wei Mao ◽  
Dong-Liang Hu
Keyword(s):  
Time Delay ◽  
Fractional Order ◽  
Neuron Model ◽  
Low Frequency ◽  
Order Model ◽  
Vibrational Resonance ◽  
Frequency Signal ◽  
Neuron System ◽  
Order Neuron ◽  
Activity Behavior

Making use of the numerical simulation method, the phenomenon of vibrational resonance and electrical activity behavior of a fractional-order FitzHugh–Nagumo neuron system excited by two-frequency periodic signals are investigated. Based on the definition and properties of the Caputo fractional derivative, the fractional L1 algorithm is applied to numerically simulate the phenomenon of vibrational resonance in the neuron system. Compared with the integer-order neuron model, the fractional-order neuron model can relax the requirement for the amplitude of the high-frequency signal and induce the phenomenon of vibrational resonance by selecting the appropriate fractional exponent. By introducing the time-delay feedback, it can be found that the vibrational resonance will occur with periods in the fractional-order neuron system, i.e., the amplitude of the low-frequency response periodically changes with the time-delay feedback. The weak low-frequency signal in the system can be significantly enhanced by selecting the appropriate time-delay parameter and the fractional exponent. In addition, the original integer-order model is extended to the fractional-order model, and the neuron system will exhibit rich dynamical behaviors, which provide a broader understanding of the neuron system.

scholarly journals Firing Activities in Fractional-Order Hindmarsh-Rose Neuron with Multistable Memristor as Autapse

2021 ◽  
Author(s):  
Zhijun Li ◽  
WenQiang Xie ◽  
Jinfang Zeng ◽  
Yicheng Zeng
Keyword(s):  
Fractional Order ◽  
Neuron Model ◽  
Hysteresis Loops ◽  
Active Regions ◽  
Firing Frequency ◽  
Initial Value ◽  
Firing Patterns ◽  
Firing Behavior ◽  
Order Neuron ◽  
Current Coupling

Abstract Compared with integer order neurons, fractional-order neuron model can more accurately describe the firing behavior of biological neurons. Considering the fact that memristors have the characteristics similar to biological synapses, a fractional-order multistable memristor is firstly proposed in this study. It is verified that the fractional-order memristor has multiple local active regions and multiple stable hysteresis loops, and the influence of fractional order on its nonvolatility is also revealed. Then by considering the fractional-order memristor as an autapse of HR neuron model, a fractional-order memristive neuron model is developed. The effects of the initial value, external excitation current, coupling strength and fractional order on the firing behavior are discussed by time series, phase diagrams, Lyapunov exponents and inter spike interval (ISI) bifurcation diagrams. Three coexisting firing patterns, including irregulate A-periodic bursting, A-periodic bursting and chaotic bursting, dependent on the memristor initial values are observed. It is also revealed that the fractional order can not only induce the transition of firing patterns, but also change the firing frequency of the neuron. Finally, a neuron circuit with variable fractional order is designed to verify the numerical simulations.

Stability and Hopf Bifurcation of Fractional-Order Complex-Valued Single Neuron Model with Time Delay

2017 ◽  
Vol 27 (13) ◽  
pp. 1750209 ◽  
Author(s):  
Zhen Wang ◽  
Xiaohong Wang ◽  
Yuxia Li ◽  
Xia Huang
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Fractional Order ◽  
Single Neuron ◽  
Neuron Model ◽  
Sufficient Conditions ◽  
Order Complex ◽  
The Stability ◽  
Complex Valued ◽  
Theoretical Results

In this paper, the problems of stability and Hopf bifurcation in a class of fractional-order complex-valued single neuron model with time delay are addressed. With the help of the stability theory of fractional-order differential equations and Laplace transforms, several new sufficient conditions, which ensure the stability of the system are derived. Taking the time delay as the bifurcation parameter, Hopf bifurcation is investigated and the critical value of the time delay for the occurrence of Hopf bifurcation is determined. Finally, two representative numerical examples are given to show the effectiveness of the theoretical results.

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