Hopf bifurcation in Caputo-Hadamard fractional order differential system

Fractals ◽  
2021 ◽  
Author(s):  
Mohamed Doubbi Bounoua ◽  
Chuntao Yin
Keyword(s):  
Hopf Bifurcation ◽  
Fractional Order ◽  
Differential System

scholarly journals Control Scheme for a Fractional-Order Chaotic Genesio-Tesi Model

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-15 ◽  
Author(s):  
Changjin Xu ◽  
Peiluan Li ◽  
Maoxin Liao ◽  
Zixin Liu ◽  
Qimei Xiao ◽  
...  
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Fractional Order ◽  
Characteristic Equation ◽  
Chaotic Systems ◽  
Chaotic Behavior ◽  
Feedback Controllers ◽  
Control Scheme ◽  
The Stability ◽  
Time Delayed Feedback

In this paper, based on the earlier research, a new fractional-order chaotic Genesio-Tesi model is established. The chaotic phenomenon of the fractional-order chaotic Genesio-Tesi model is controlled by designing two suitable time-delayed feedback controllers. With the aid of Laplace transform, we obtain the characteristic equation of the controlled chaotic Genesio-Tesi model. Then by regarding the time delay as the bifurcation parameter and analyzing the characteristic equation, some new sufficient criteria to guarantee the stability and the existence of Hopf bifurcation for the controlled fractional-order chaotic Genesio-Tesi model are derived. The research shows that when time delay remains in some interval, the equilibrium point of the controlled chaotic Genesio-Tesi model is stable and a Hopf bifurcation will happen when the time delay crosses a critical value. The effect of the time delay on the stability and the existence of Hopf bifurcation for the controlled fractional-order chaotic Genesio-Tesi model is shown. At last, computer simulations check the rationalization of the obtained theoretical prediction. The derived key results in this paper play an important role in controlling the chaotic behavior of many other differential chaotic systems.

scholarly journals Complexity and Hopf Bifurcation Analysis on a Kind of Fractional-Order IS-LM Macroeconomic System

2016 ◽  
Vol 26 (11) ◽  
pp. 1650181 ◽  
Author(s):  
Junhai Ma ◽  
Wenbo Ren
Keyword(s):  
Hopf Bifurcation ◽  
Fractional Order ◽  
Dynamic Characteristics ◽  
Fractional Order System ◽  
Order System ◽  
Complex Dynamic ◽  
Macroeconomic Regulation And Control ◽  
And Control ◽  
System Order ◽  
Macroeconomic Environment

On the basis of our previous research, we deepen and complete a kind of macroeconomics IS-LM model with fractional-order calculus theory, which is a good reflection on the memory characteristics of economic variables, we also focus on the influence of the variables on the real system, and improve the analysis capabilities of the traditional economic models to suit the actual macroeconomic environment. The conditions of Hopf bifurcation in fractional-order system models are briefly demonstrated, and the fractional order when Hopf bifurcation occurs is calculated, showing the inherent complex dynamic characteristics of the system. With numerical simulation, bifurcation, strange attractor, limit cycle, waveform and other complex dynamic characteristics are given; and the order condition is obtained with respect to time. We find that the system order has an important influence on the running state of the system. The system has a periodic motion when the order meets the conditions of Hopf bifurcation; the fractional-order system gradually stabilizes with the change of the order and parameters while the corresponding integer-order system diverges. This study has certain significance to policy-making about macroeconomic regulation and control.

scholarly journals On the integrability and the zero-Hopf bifurcation of a Chen–Wang differential system

2015 ◽  
Vol 80 (1-2) ◽  
pp. 353-361 ◽  
Author(s):  
Jaume Llibre ◽  
Regilene D. S. Oliveira ◽  
Claudia Valls
Keyword(s):  
Hopf Bifurcation ◽  
Differential System

scholarly journals Hopf Bifurcations of a Stochastic Fractional-Order Van der Pol System

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Xiaojun Liu ◽  
Ling Hong ◽  
Lixin Yang
Keyword(s):  
Hopf Bifurcation ◽  
Fractional Order ◽  
Stochastic System ◽  
Critical Parameter ◽  
Random Parameter ◽  
Order System ◽  
Hopf Bifurcations ◽  
Van Der Pol ◽  
Existence Conditions ◽  
Theoretical Results

The Hopf bifurcation of a fractional-order Van der Pol (VDP for short) system with a random parameter is investigated. Firstly, the Chebyshev polynomial approximation is applied to study the stochastic fractional-order system. Based on the method, the stochastic system is reduced to the equivalent deterministic one, and then the responses of the stochastic system can be obtained by numerical methods. Then, according to the existence conditions of Hopf bifurcation, the critical parameter value of the bifurcation is obtained by theoretical analysis. Then, numerical simulations are carried out to verify the theoretical results.

scholarly journals Positive Solutions of a Singular Nonlocal Fractional Order Differential System via Schauder’s Fixed Point Theorem

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Xinguang Zhang ◽  
Cuiling Mao ◽  
Yonghong Wu ◽  
Hua Su
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Positive Solutions ◽  
Fractional Order ◽  
Differential System ◽  
Point Theorem ◽  
Schauder’S Fixed Point Theorem ◽  
Schauder's Fixed Point Theorem ◽  
Existence Of Positive Solutions ◽  
Two Parameters

We establish the existence of positive solutions to a class of singular nonlocal fractional order differential system depending on two parameters. Our methods are based on Schauder’s fixed point theorem.

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