scholarly journals Neimark-Sacker Bifurcation and Chaos Control in a Fractional-Order Plant-Herbivore Model

2017 ◽  
Vol 2017 ◽  
pp. 1-15 ◽  
Author(s):  
Qamar Din ◽  
A. A. Elsadany ◽  
Hammad Khalil
Keyword(s):  
Numerical Simulations ◽  
Chaos Control ◽  
Hybrid Control ◽  
Chaotic Behavior ◽  
Control Strategies ◽  
Pole Placement ◽  
Positive Equilibrium ◽  
Local Asymptotic Stability ◽  
Positive Equilibrium Point ◽  
Plant Herbivore

This work is related to dynamics of a discrete-time 3-dimensional plant-herbivore model. We investigate existence and uniqueness of positive equilibrium and parametric conditions for local asymptotic stability of positive equilibrium point of this model. Moreover, it is also proved that the system undergoes Neimark-Sacker bifurcation for positive equilibrium with the help of an explicit criterion for Neimark-Sacker bifurcation. The chaos control in the model is discussed through implementation of two feedback control strategies, that is, pole-placement technique and hybrid control methodology. Finally, numerical simulations are provided to illustrate theoretical results. These results of numerical simulations demonstrate chaotic long-term behavior over a broad range of parameters. The computation of the maximum Lyapunov exponents confirms the presence of chaotic behavior in the model.

Chaos Control for a Fractional-Order Jerk System via Time Delay Feedback Controller and Mixed Controller

2021 ◽  
Vol 5 (4) ◽  
pp. 257
Author(s):  
Changjin Xu ◽  
Maoxin Liao ◽  
Peiluan Li ◽  
Lingyun Yao ◽  
Qiwen Qin ◽  
...  
Keyword(s):  
Time Delay ◽  
Fractional Order ◽  
Chaos Control ◽  
Chaotic Behavior ◽  
Control Strategies ◽  
Feedback Controller ◽  
Research Approach ◽  
Controlling Chaos ◽  
The Stability ◽  
Jerk System

In this study, we propose a novel fractional-order Jerk system. Experiments show that, under some suitable parameters, the fractional-order Jerk system displays a chaotic phenomenon. In order to suppress the chaotic behavior of the fractional-order Jerk system, we design two control strategies. Firstly, we design an appropriate time delay feedback controller to suppress the chaos of the fractional-order Jerk system. The delay-independent stability and bifurcation conditions are established. Secondly, we design a suitable mixed controller, which includes a time delay feedback controller and a fractional-order PDσ controller, to eliminate the chaos of the fractional-order Jerk system. The sufficient condition ensuring the stability and the creation of Hopf bifurcation for the fractional-order controlled Jerk system is derived. Finally, computer simulations are executed to verify the feasibility of the designed controllers. The derived results of this study are absolutely new and possess potential application value in controlling chaos in physics. Moreover, the research approach also enriches the chaos control theory of fractional-order dynamical system.

Qualitative analysis of a chemostat model with inhibitory exponential substrate uptake and a time delay

2014 ◽  
Vol 07 (04) ◽  
pp. 1450045 ◽  
Author(s):  
Qinglai Dong ◽  
Wanbiao Ma
Keyword(s):  
Time Delay ◽  
Asymptotic Stability ◽  
Qualitative Analysis ◽  
Numerical Simulations ◽  
Global Asymptotic Stability ◽  
Positive Equilibrium ◽  
Substrate Uptake ◽  
Sufficient Condition ◽  
Chemostat Model ◽  
Local Asymptotic Stability

In this paper, we consider a simple chemostat model with inhibitory exponential substrate uptake and a time delay. A detailed qualitative analysis about existence and boundedness of its solutions and the local asymptotic stability of its equilibria are carried out. Using Lyapunov–LaSalle invariance principle, we show that the washout equilibrium is global asymptotic stability for any time delay. Using the fluctuation lemma, the sufficient condition of the global asymptotic stability of the positive equilibrium [Formula: see text] is obtained. Numerical simulations are also performed to illustrate the results.

Stability and Bifurcation in a Logistic Model with Allee Effect and Feedback Control

2020 ◽  
Vol 30 (15) ◽  
pp. 2050231
Author(s):  
Zhenliang Zhu ◽  
Mengxin He ◽  
Zhong Li ◽  
Fengde Chen
Keyword(s):  
Feedback Control ◽  
Logistic Model ◽  
Allee Effect ◽  
Control Variable ◽  
Positive Equilibrium ◽  
Threshold Condition ◽  
Local Asymptotic Stability ◽  
Linearized System ◽  
Positive Equilibrium Point ◽  
Two Parameters

This paper aims to study the dynamic behavior of a logistic model with feedback control and Allee effect. We prove the origin of the system is always an attractor. Further, if the feedback control variable and Allee effect are big enough, the species goes extinct. According to the analysis of the Jacobian matrix of the corresponding linearized system, we obtain the threshold condition for the local asymptotic stability of the positive equilibrium point. Also, we study the occurrence of saddle-node bifurcation, supercritical and subcritical Hopf bifurcations with the change of parameter. By calculating a universal unfolding near the cusp and choosing two parameters of the system, we can draw a conclusion that the system undergoes Bogdanov–Takens bifurcation of codimension-2. Numerical simulations are carried out to confirm the feasibility of the theoretical results. Our research can be regarded as a supplement to the existing literature on the dynamics of feedback control system, since there are few results on the bifurcation in the system so far.

Chaos control of a Bose–Einstein condensate in a moving optical lattice

2016 ◽  
Vol 30 (19) ◽  
pp. 1650238 ◽  
Author(s):  
Zhiying Zhang ◽  
Xiuqin Feng ◽  
Zhihai Yao
Keyword(s):  
Optical Lattice ◽  
Chaos Control ◽  
Chaotic Behavior ◽  
Control Strategies ◽  
Lyapunov Stability Theory ◽  
Einstein Condensate ◽  
Control Parameters ◽  
Periodic Force ◽  
Bose Einstein Condensate ◽  
Bose Einstein

Chaos control of a Bose–Einstein condensate (BEC) loaded into a moving optical lattice with attractive interaction is investigated on the basis of Lyapunov stability theory. Three methods are designed to control chaos in BEC. As a controller, a bias constant, periodic force, or wavelet function feedback is added to the BEC system. Numerical simulations reveal that chaotic behavior can be well controlled to achieve periodicity by regulating control parameters. Different periodic orbits are available for different control parameters only if the maximal Lyapunov exponent of the system is negative. The abundant effect of chaotic control is also demonstrated numerically. Chaos control can be realized effectively by using our proposed control strategies.

scholarly journals Dynamical behaviors in a discrete fractional-order predator-prey system

Filomat ◽  
2018 ◽  
Vol 32 (17) ◽  
pp. 5857-5874 ◽  
Author(s):  
Yao Shi ◽  
Qiang Ma ◽  
Xiaohua Ding
Keyword(s):  
Numerical Simulations ◽  
Fixed Points ◽  
Fractional Order ◽  
Control Strategies ◽  
Flip Bifurcation ◽  
Predator Prey ◽  
Local Asymptotic Stability ◽  
Dynamical Behaviors ◽  
Prey System ◽  
Theoretical Results

This paper is related to the dynamical behaviors of a discrete-time fractional-order predatorprey model. We have investigated existence of positive fixed points and parametric conditions for local asymptotic stability of positive fixed points of this model. Moreover, it is also proved that the system undergoes Flip bifurcation and Neimark-Sacker bifurcation for positive fixed point. Various chaos control strategies are implemented for controlling the chaos due to Flip and Neimark-Sacker bifurcations. Finally, numerical simulations are provided to verify theoretical results. These results of numerical simulations demonstrate chaotic behaviors over a broad range of parameters. The computation of the maximum Lyapunov exponents confirms the presence of chaotic behaviors in the model.

Mutual interference and its effects on searching efficiency between predator–prey interaction with bifurcation analysis and chaos control

2021 ◽  
pp. 107754632110564
Author(s):  
Waqas Ishaque ◽  
Qamar Din ◽  
Muhammad Taj
Keyword(s):  
Chaos Control ◽  
Chaotic Behavior ◽  
Control Strategies ◽  
Equilibrium Points ◽  
Mutual Interference ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Predator Prey Model ◽  
Existence And Stability ◽  
Prey Interaction

In this paper, we study the dynamic of the predator–prey model based on mutual interference and its effects on searching efficiency. The parametric conditions, existence, and stability for trivial and boundary equilibrium points are studied. Also, it has shown that by applying the center manifold theorem and bifurcation theory, system undergoes Neimark–Sacker bifurcation across the neighborhood of a positive fixed point. Moreover, due to the bifurcation and chaos which objectively exist in a system, three chaos control strategies are designed and used. Moreover, to validate our theoretical and analytical discussions, numerical simulations are applied to show complex and chaotic behavior. Finally, theoretical discussions are validated with experimental field data.

Under the influence of crowding effects: Stability, bifurcation and chaos control for a discrete-time predator–prey model

2019 ◽  
Vol 12 (04) ◽  
pp. 1950044 ◽  
Author(s):  
Muhammad Aqib Abbasi ◽  
Qamar Din
Keyword(s):  
Steady State ◽  
Numerical Simulations ◽  
Chaos Control ◽  
Chaotic Behavior ◽  
Control Technique ◽  
Qualitative Behavior ◽  
Flip Bifurcation ◽  
Predator Prey Model ◽  
Positive Steady State ◽  
Crowding Effects

The interaction between predators and preys exhibits more complicated behavior under the influence of crowding effects. By taking into account the crowding effects, the qualitative behavior of a prey–predator model is investigated. Particularly, we examine the boundedness as well as existence and uniqueness of positive steady-state and stability analysis of the unique positive steady-state. Moreover, it is also proved that the system undergoes Hopf bifurcation and flip bifurcation with the help of bifurcation theory. Moreover, a chaos control technique is proposed for controlling chaos under the influence of bifurcations. Finally, numerical simulations are provided to illustrate the theoretical results. These results of numerical simulations demonstrate chaotic long-term behavior over a broad range of parameters. The presence of chaotic behavior in the model is confirmed by computing maximum Lyapunov exponents.

scholarly journals Supercritical Neimark–Sacker Bifurcation and Hybrid Control in a Discrete-Time Glycolytic Oscillator Model

2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
A. Q. Khan ◽  
E. Abdullah ◽  
Tarek F. Ibrahim
Keyword(s):  
Discrete Time ◽  
Hybrid Control ◽  
Oscillator Model ◽  
Positive Equilibrium ◽  
Time Model ◽  
Discrete Time Model ◽  
Prime Period ◽  
Dynamical Properties ◽  
Positive Equilibrium Point ◽  
Theoretical Results

We study the local dynamical properties, Neimark–Sacker bifurcation, and hybrid control in a glycolytic oscillator model in the interior of ℝ+2. It is proved that, for all parametric values, Pxy+α/β+α2,α is the unique positive equilibrium point of the glycolytic oscillator model. Further local dynamical properties along with different topological classifications about the equilibrium Pxy+α/β+α2,α have been investigated by employing the method of linearization. Existence of prime period and periodic points of the model under consideration are also investigated. It is proved that, about the fixed point Pxy+α/β+α2,α, the discrete-time glycolytic oscillator model undergoes no bifurcation, except Neimark–Sacker bifurcation. A further hybrid control strategy is applied to control Neimark–Sacker bifurcation in the discrete-time model. Finally, theoretical results are verified numerically.

Complex Dynamics of a Discrete-Time Prey–Predator System with Leslie Type: Stability, Bifurcation Analyses and Chaos

2020 ◽  
Vol 30 (10) ◽  
pp. 2050149
Author(s):  
Pinar Baydemir ◽  
Huseyin Merdan ◽  
Esra Karaoglu ◽  
Gokce Sucu
Keyword(s):  
Numerical Simulations ◽  
Equilibrium Point ◽  
Discrete Time ◽  
Chaotic Behavior ◽  
Positive Equilibrium ◽  
Flip Bifurcation ◽  
Step Size ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Predator System

Dynamic behavior of a discrete-time prey–predator system with Leslie type is analyzed. The discrete mathematical model was obtained by applying the forward Euler scheme to its continuous-time counterpart. First, the local stability conditions of equilibrium point of this system are determined. Then, the conditions of existence for flip bifurcation and Neimark–Sacker bifurcation arising from this positive equilibrium point are investigated. More specifically, by choosing integral step size as a bifurcation parameter, these bifurcations are driven via center manifold theorem and normal form theory. Finally, numerical simulations are performed to support and extend the theoretical results. Analytical results show that an integral step size has a significant role on the dynamics of a discrete system. Numerical simulations support that enlarging the integral step size causes chaotic behavior.

CHAOS CONTROL OF A MODIFIED COUPLED DYNAMOS SYSTEM

2007 ◽  
Vol 21 (26) ◽  
pp. 4593-4610 ◽  
Author(s):  
XING-YUAN WANG ◽  
XIANG-JUN WU
Keyword(s):  
Numerical Simulations ◽  
Periodic Orbits ◽  
Limit Cycles ◽  
Chaos Control ◽  
Chaotic Behavior ◽  
Direct Method ◽  
Equilibrium Points ◽  
Unstable Equilibrium ◽  
Unstable Periodic Orbits ◽  
Lyapunov Direct Method

This paper studies the problem of controlling the chaotic behavior of a modified coupled dynamos system. Two different methods, feedback and non-feedback methods, are used to control chaos in the modified coupled dynamos system. Based on the Lyapunov direct method and Routh–Hurwitz criterion, the conditions suppressing chaos to unstable equilibrium points or unstable periodic orbits (limit cycles) are discussed, and they are also proved theoretically. Numerical simulations show the effectiveness of the two different methods.

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