BIFURCATION ANALYSIS AND CONTROL OF A PLANKTON–FISH MODEL WITH A DISTRIBUTION DELAY

2011 ◽  
Vol 11 (04) ◽  
pp. 857-879 ◽  
Author(s):  
XUE ZHANG ◽  
QINGLING ZHANG
Keyword(s):  
Center Manifold ◽  
Control Method ◽  
Distributed Delay ◽  
State Feedback Control ◽  
Positive Equilibrium ◽  
Center Manifold Theorem ◽  
Fish Model ◽  
Feedback Control Method ◽  
Positive Equilibrium Point ◽  
And Control

This paper studies a plankton–fish model with distributed delay in the context of marine plankton interaction together with predation by planktotrophic fish. The delay indicates that the growth of the zooplankton depends on the past density of phytoplankton. The positive equilibrium point and its local stability are investigated. Using the average time delay as bifurcation parameter, we obtain the conditions of the existence of Hopf bifurcation. Based on the normal form and center manifold theorem, stability, direction, and other properties of bifurcating periodic solutions are derived. Moreover, a state feedback control method, which can be implemented by adjusting the harvesting for zooplankton population, is proposed to drive the plankton–fish system to a steady state. Numerical simulations illustrate the effectiveness of results and the related biological implications are discussed.

scholarly journals Delay Feedback Control of the Lorenz-Like System

2018 ◽  
Vol 2018 ◽  
pp. 1-13
Author(s):  
Qin Chen ◽  
Jianguo Gao
Keyword(s):  
Hopf Bifurcation ◽  
Feedback Control ◽  
Control Method ◽  
Functional Differential Equations ◽  
Variable Parameter ◽  
Positive Equilibrium ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Feedback Control Method ◽  
Delay Feedback Control

We choose the delay as a variable parameter and investigate the Lorentz-like system with delayed feedback by using Hopf bifurcation theory and functional differential equations. The local stability of the positive equilibrium and the existence of Hopf bifurcations are obtained. After that the direction of Hopf bifurcation and stability of periodic solutions bifurcating from equilibrium is determined by using the normal form theory and center manifold theorem. In the end, some numerical simulations are employed to validate the theoretical analysis. The results show that the purpose of controlling chaos can be achieved by adjusting appropriate feedback effect strength and delay parameters. The applied delay feedback control method in this paper is general and can be applied to other nonlinear chaotic systems.

scholarly journals Chaotic Dynamics and Control of Discrete Ratio-Dependent Predator-Prey System

2017 ◽  
Vol 2017 ◽  
pp. 1-13 ◽  
Author(s):  
Sarker Md. Sohel Rana
Keyword(s):  
Chaotic Dynamics ◽  
Control Method ◽  
State Feedback Control ◽  
Phase Portraits ◽  
Predator Prey ◽  
Prey System ◽  
Dynamics And Control ◽  
Feedback Control Method ◽  
Ratio Dependent ◽  
And Control

This study examines the complexity of a discrete-time predator-prey system with ratio-dependent functional response. We establish algebraically the conditions for existence of fixed points and their stability. We show that under some parametric conditions the system passes through a bifurcation (flip or Neimark-Sacker). Numerical simulations are presented not only to justify theoretical results but also to exhibit new complex behaviors which include phase portraits, orbits of periods 9, 19, and 26, invariant closed circle, and attracting chaotic sets. Moreover, we measure numerically the Lyapunov exponents and fractal dimension to confirm the chaotic dynamics of the system. Finally, a state feedback control method is applied to control chaos which exists in the system.

scholarly journals Bifurcation Analysis in a Delayed Diffusive Leslie-Gower Model

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Shuling Yan ◽  
Xinze Lian ◽  
Weiming Wang ◽  
Youbin Wang
Keyword(s):  
Stability Analysis ◽  
Hopf Bifurcation ◽  
Normal Form ◽  
Bifurcation Analysis ◽  
Center Manifold ◽  
Neumann Boundary ◽  
Positive Equilibrium ◽  
Center Manifold Theorem ◽  
Normal Form Theorem ◽  
The Stability

We investigate a modified delayed Leslie-Gower model under homogeneous Neumann boundary conditions. We give the stability analysis of the equilibria of the model and show the existence of Hopf bifurcation at the positive equilibrium under some conditions. Furthermore, we investigate the stability and direction of bifurcating periodic orbits by using normal form theorem and the center manifold theorem.

Intervention Time in Target-Oriented Chaos Control

2021 ◽  
Vol 31 (09) ◽  
pp. 2150134
Author(s):  
Juan Segura
Keyword(s):  
Discrete Time ◽  
Chaos Control ◽  
Control Method ◽  
Single Species ◽  
Positive Equilibrium ◽  
Population Stability ◽  
Determine Parameter ◽  
And Control ◽  
Parameter Values ◽  
Proportional Feedback

The timing of interventions plays a central role in managing and exploiting biological populations. However, few studies in the literature have addressed its effect on population stability. The Seno equation is a discrete-time equation that describes the dynamics of single-species populations harvested according to the proportional feedback method at any moment between two consecutive censuses. Here we study a discrete-time equation that generalizes the Seno equation by considering the management and exploitation of populations through the target-oriented chaos control method. We investigate the combined effect of timing, targeting, and control on population stability, focusing on global stability. We prove that high enough control values create a positive equilibrium that attracts all positive solutions. We also prove that it is possible to determine parameter values to stabilize the controlled populations at any preset population size. Finally, we investigate the parameter combinations for which the management and exploitation are optimized in different scenarios.

scholarly journals Bifurcation and Chaos of a Discrete Predator-Prey Model with Crowley–Martin Functional Response Incorporating Proportional Prey Refuge

2020 ◽  
Vol 2020 ◽  
pp. 1-18 ◽  
Author(s):  
P. K. Santra ◽  
G. S. Mahapatra ◽  
G. R. Phaijoo
Keyword(s):  
Functional Response ◽  
Control Method ◽  
Equilibrium Points ◽  
Period Doubling ◽  
Phase Portraits ◽  
Prey Refuge ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Feedback Control Method ◽  
And Control

The paper investigates the dynamical behaviors of a two-species discrete predator-prey system with Crowley–Martin functional response incorporating prey refuge proportional to prey density. The existence of equilibrium points, stability of three fixed points, period-doubling bifurcation, Neimark–Sacker bifurcation, Marottos chaos, and Control Chaos are analyzed for the discrete-time domain. The time graphs, phase portraits, and bifurcation diagrams are obtained for different parameters of the model. Numerical simulations and graphics show that the discrete model exhibits rich dynamics, which also present that the system is a chaotic and complex one. This paper attempts to present a feedback control method which can stabilize chaotic orbits at an unstable equilibrium point.

Control of Oscillations of a Wing Subjected to 3D Unsteady Subsonic Aerodynamic Loads Using Piezoelectric Actuators

2012 ◽  
Author(s):  
Tahereh Mirmohammadi ◽  
Arun K. Misra ◽  
Dan Mateescu
Keyword(s):  
Control Method ◽  
Three Dimensional ◽  
Element Formulation ◽  
Sensors And Actuators ◽  
Aerodynamic Loading ◽  
Aerodynamic Model ◽  
Active Feedback ◽  
Feedback Control Method ◽  
And Control ◽  
Actuator Placement

In the recent years, using piezoelectric material as sensors and actuators has drawn significant attention in vibration analysis and control of structures. In the present paper, bonded piezoelectric sensors and actuators have been used to control the aeroelastic oscillations of a cantilever wing under the effects of three-dimensional unsteady subsonic aerodynamic loading. An aerodynamic model using a numerical panel method is developed and validated to calculate the three-dimensional unsteady aerodynamic loading and finite element formulation is applied to model the wing structure as a cantilever plate undergoing small transverse oscillations. The structural and aerodynamic models are combined to simulate the aeroelastic oscillations and interchange the data simultaneously. An active feedback control method to suppress the oscillations is presented and investigated. Finally, an analysis is performed to examine the effects of actuator placement on the wing surface in suppression of oscillations.

scholarly journals Bifurcation Analysis and Control of a Differential-Algebraic Predator-Prey Model with Allee Effect and Time Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Xue Zhang ◽  
Qing-ling Zhang
Keyword(s):  
Time Delay ◽  
Allee Effect ◽  
Control Method ◽  
Handling Time ◽  
Delayed Feedback Control ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Feedback Control Method ◽  
Predator Model ◽  
And Control

This paper studies systematically a differential-algebraic prey-predator model with time delay and Allee effect. It shows that transcritical bifurcation appears when a variation of predator handling time is taken into account. This model also exhibits singular induced bifurcation as the economic revenue increases through zero, which causes impulsive phenomenon. It can be noted that the impulsive phenomenon can be much weaker by strengthening Allee effect in numerical simulation. On the other hand, at a critical value of time delay, the model undergoes a Hopf bifurcation; that is, the increase of time delay destabilizes the model and bifurcates into small amplitude periodic solution. Moreover, a state delayed feedback control method, which can be implemented by adjusting the harvesting effort for biological populations, is proposed to drive the differential-algebraic system to a steady state. Finally, by using Matlab software, numerical simulations illustrate the effectiveness of the results.

scholarly journals Adaptive Control of Chaos in Chua's Circuit

2011 ◽  
Vol 2011 ◽  
pp. 1-14 ◽  
Author(s):  
Weiping Guo ◽  
Diantong Liu
Keyword(s):  
Feedback Control ◽  
Control Method ◽  
Equilibrium Points ◽  
Lyapunov Theory ◽  
Chua’S Circuit ◽  
Control Of Chaos ◽  
Chua's Circuit ◽  
Adaptive Technology ◽  
Feedback Control Method ◽  
And Control

A feedback control method and an adaptive feedback control method are proposed for Chua's circuit chaos system, which is a simple 3D autonomous system. The asymptotical stability is proven with Lyapunov theory for both of the proposed methods, and the system can be dragged to one of its three unstable equilibrium points respectively. Simulation results show that the proposed methods are valid, and control performance is improved through introducing adaptive technology.

scholarly journals Dynamics in a delayed diffusive cell cycle model

2018 ◽  
Vol 23 (5) ◽  
pp. 691-709
Author(s):  
Yanqin Wang ◽  
Ling Yang ◽  
Jie Yan
Keyword(s):  
Cell Cycle ◽  
Periodic Solutions ◽  
Center Manifold ◽  
Spatial Dynamics ◽  
Positive Equilibrium ◽  
Cycle Model ◽  
Center Manifold Theorem ◽  
Diffusive Model ◽  
The Stability ◽  
Theoretical Results

In this paper, we construct a delayed diffusive model to explore the spatial dynamics of cell cycle in G2/M transition. We first obtain the local stability of the unique positive equilibrium for this model, which is irrelevant to the diffusion. Then, through investigating the delay-induced Hopf bifurcation in this model, we establish the existence of spatially homogeneous and inhomogeneous bifurcating periodic solutions. Applying the normal form and center manifold theorem of functional partial differential equations, we also determine the stability and direction of these bifurcating periodic solutions. Finally, numerical simulations are presented to validate our theoretical results.

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