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.

Simplified analysis of a stochastic system

1979 ◽  
Vol 44 (2) ◽  
pp. 328-339
Author(s):  
Vladimír Herles
Keyword(s):  
Statistical Analysis ◽  
Dynamic Behavior ◽  
Stochastic System ◽  
Additive Noise ◽  
Random Parameter ◽  
Systematic Difference ◽  
Order System ◽  
Constant Parameter ◽  
Random Parameters ◽  
Simplified Analysis

Contradictious results published by different authors about the dynamics of systems with random parameters have been examined. Statistical analysis of the simple 1st order system proves that the random parameter can cause a systematic difference in the dynamic behavior that cannot be (in general) described by the usual constant-parameter model with the additive noise at the output.

Adaptive Synchronization of a Stochastic Fractional-Order System

2015 ◽  
Vol 733 ◽  
pp. 939-942
Author(s):  
Xiao Jun Liu
Keyword(s):  
Numerical Simulations ◽  
Fractional Order ◽  
Stochastic System ◽  
Fractional Order System ◽  
Adaptive Synchronization ◽  
Order System ◽  
Unknown Parameters ◽  
Adaptive Laws

In this paper, adaptive synchronization of a stochastic fractional-order system with unknown parameters is studied. Firstly, the stochastic system is reduced into the equivalent deterministic one with Laguerre approximation. Then, the synchronization for the system is realized by designing appropriate controllers and adaptive laws of the unknown parameters. Numerical simulations are carried out to demonstrate the effectiveness of the controllers and laws.

Analysis of a Nonlinear System with a Random Parameter

2014 ◽  
Vol 721 ◽  
pp. 366-369
Author(s):  
Hong Gang Dang ◽  
Xiao Ya Yang ◽  
Wan Sheng He
Keyword(s):  
Nonlinear System ◽  
Fractional Order ◽  
Approximation Method ◽  
Lorenz System ◽  
Laguerre Polynomial ◽  
Random Parameter ◽  
Order System ◽  
Order Complex ◽  
Ensemble Mean ◽  
Complex Lorenz System

In this paper, a nonlinear system with random parameter, which is called stochastic fractional-order complex Lorenz system, is investigated. The Laguerre polynomial approximation method is used to study the system. Then, the stochastic fractional-order system is reduced into the equivalent deterministic one with Laguerre approximation. The ensemble mean and sample responses of the stochastic system can be obtained.

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 Controlling hopf bifurcations: Discrete-time systems

2000 ◽  
Vol 5 (1) ◽  
pp. 29-33 ◽  
Author(s):  
Guanrong Chen ◽  
Jin-quing Fang ◽  
Yiguang Hong ◽  
Huashu Qin
Keyword(s):  
Hopf Bifurcation ◽  
Computer Simulations ◽  
Discrete Time ◽  
State Feedback ◽  
Bifurcation Control ◽  
Hopf Bifurcations ◽  
Control Task ◽  
Discrete Time Systems ◽  
Time Systems ◽  
Theoretical Results

Bifurcation control has attracted increasing attention in recent years. A simple and unified state-feedback methodology is developed in this paper for Hopf bifurcation control for discrete-time systems. The control task can be either shifting an existing Hopf bifurcation or creating a new Hopf bifurcation. Some computer simulations are included to illustrate the methodology and to verify the theoretical results.

Stability and Hopf bifurcation analysis of a biotic resource enrichment on a prey predator population in fractional-order system

2021 ◽  
Author(s):  
Abiodun Ezekiel Owoyemi ◽  
Ibrahim Mohammed Sulaiman ◽  
Salisu Sadiya Muhammad ◽  
Oluwatayo Olatunde Oni ◽  
Olukayode Williams Okedokun
Keyword(s):  
Hopf Bifurcation ◽  
Fractional Order ◽  
Bifurcation Analysis ◽  
Fractional Order System ◽  
Order System ◽  
Predator Population ◽  
Biotic Resource

Persistence, Turing Instability and Hopf Bifurcation in a Diffusive Plankton System with Delay and Quadratic Closure

2016 ◽  
Vol 26 (03) ◽  
pp. 1650047 ◽  
Author(s):  
Jiantao Zhao ◽  
Junjie Wei
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Sufficient Conditions ◽  
Reaction Diffusion ◽  
Turing Instability ◽  
Positive Equilibrium ◽  
Hopf Bifurcations ◽  
System With Delay ◽  
The Stability ◽  
Theoretical Results

A reaction–diffusion plankton system with delay and quadratic closure term is investigated to study the interactions between phytoplankton and zooplankton. Sufficient conditions independent of diffusion and delay are obtained for the persistence of the system. Our conclusions show that diffusion can induce Turing instability, delay can influence the stability of the positive equilibrium and induce Hopf bifurcations to occur. The computational formulas which determine the properties of bifurcating periodic solutions are given by calculating the normal form on the center manifold, and some numerical simulations are carried out for illustrating the theoretical results.

scholarly journals Period-Doubling Bifurcation of Stochastic Fractional-Order Duffing System via Chebyshev Polynomial Approximation

2017 ◽  
Vol 2017 ◽  
pp. 1-12
Author(s):  
Youming Lei ◽  
Yanyan Wang
Keyword(s):  
Fractional Order ◽  
Chebyshev Polynomial ◽  
Polynomial Approximation ◽  
Period Doubling ◽  
Random Parameter ◽  
Fractional Order System ◽  
Period Doubling Bifurcation ◽  
Order System ◽  
Integer Order ◽  
Chebyshev Polynomial Approximation

Fractional-order calculus is more competent than integer-order one when modeling systems with properties of nonlocality and memory effect. And many real world problems related to uncertainties can be modeled with stochastic fractional-order systems with random parameters. Therefore, it is necessary to analyze the dynamical behaviors in those systems concerning both memory and uncertainties. The period-doubling bifurcation of stochastic fractional-order Duffing (SFOD for short) system with a bounded random parameter subject to harmonic excitation is studied in this paper. Firstly, Chebyshev polynomial approximation in conjunction with the predictor-corrector approach is used to numerically solve the SFOD system that can be reduced to the equivalent deterministic system. Then, the global and local analysis of period-doubling bifurcation are presented, respectively. It is shown that both the fractional-order and the intensity of the random parameter can be taken as bifurcation parameters, which are peculiar to the stochastic fractional-order system, comparing with the stochastic integer-order system or the deterministic fractional-order system. Moreover, the Chebyshev polynomial approximation is proved to be an effective approach for studying the period-doubling bifurcation of the SFOD system.

scholarly journals Bifurcation Based-Delay Feedback Control Strategy for a Fractional-Order Two-Prey One-Predator System

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-13 ◽  
Author(s):  
Shuai Li ◽  
Chengdai Huang ◽  
Xinyu Song
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Fractional Order ◽  
Feedback Gain ◽  
Controlled System ◽  
Delay Feedback Control ◽  
Feedback Control Strategy ◽  
System With Time Delay ◽  
Theoretical Results ◽  
Predator System

The issue of bifurcation control for a novel fractional-order two-prey and one-predator system with time delay is dealt with in this paper. Firstly, the characteristic equation is investigated by picking time delay as the bifurcation parameter, and some conditions for the appearance of Hopf bifurcation are obtained. It is shown that time delay can give rise to periodic oscillations and each order has an important impact on the occurrence of Hopf bifurcation for the controlled system. Then, it is illustrated that the control result is obviously influenced by the feedback gain. It is also noted that the inception of the bifurcation can be postponed if the feedback gain decreases. Finally, two simulation examples are carried out to verify the chief theoretical results.

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