Bifurcation analysis of an ecological model with nonlinear state-dependent feedback control by Poincaré map defined in phase set

2021 ◽  
pp. 106212
Author(s):  
Qianqian Zhang ◽  
Sanyi Tang
Keyword(s):  
Feedback Control ◽  
Bifurcation Analysis ◽  
Poincaré Map ◽  
Ecological Model ◽  
Poincare Map ◽  
State Dependent ◽  
Nonlinear State

scholarly journals The Dynamics of a Predator-Prey System with State-Dependent Feedback Control

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Hunki Baek
Keyword(s):  
Feedback Control ◽  
Periodic Solutions ◽  
Poincaré Map ◽  
Sufficient Conditions ◽  
Period Doubling ◽  
Poincare Map ◽  
Predator Prey ◽  
Prey System ◽  
State Dependent ◽  
Fold Bifurcation

A Lotka-Volterra-type predator-prey system with state-dependent feedback control is investigated in both theoretical and numerical ways. Using the Poincaré map and the analogue of the Poincaré criterion, the sufficient conditions for the existence and stability of semitrivial periodic solutions and positive periodic solutions are obtained. In addition, we show that there is no positive periodic solution with period greater than and equal to three under some conditions. The qualitative analysis shows that the positive period-one solution bifurcates from the semitrivial solution through a fold bifurcation. Numerical simulations to substantiate our theoretical results are provided. Also, the bifurcation diagrams of solutions are illustrated by using the Poincaré map, and it is shown that the chaotic solutions take place via a cascade of period-doubling bifurcations.

scholarly journals Nonlinear state-dependent feedback control of a pest-natural enemy system

2018 ◽  
Vol 94 (3) ◽  
pp. 2243-2263 ◽  
Author(s):  
Yuan Tian ◽  
Sanyi Tang ◽  
Robert A. Cheke
Keyword(s):  
Feedback Control ◽  
Natural Enemy ◽  
State Dependent ◽  
Nonlinear State

scholarly journals Dynamic Complexity of a Phytoplankton-Fish Model with the Impulsive Feedback Control by means of Poincaré Map

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Dezhao Li ◽  
Yu Liu ◽  
Huidong Cheng
Keyword(s):  
Periodic Solution ◽  
Feedback Control ◽  
Poincaré Map ◽  
Dynamic Change ◽  
Sufficient Conditions ◽  
Reference Value ◽  
Poincare Map ◽  
Dynamic Complexity ◽  
Fish Model ◽  
Necessary And Sufficient

The phytoplankton-fish model for catching fish with impulsive feedback control is established in this paper. Firstly, the Poincaré map for the phytoplankton-fish model is defined, and the properties of monotonicity, continuity, differentiability, and fixed point of Poincaré map are analyzed. In particular, the continuous and discontinuous properties of Poincaré map under different conditions are discussed. Secondly, we conduct the analysis of the necessary and sufficient conditions for the existence, uniqueness, and global stability of the order-1 periodic solution of the phytoplankton-fish model and obtain the sufficient conditions for the existence of the order-kk≥2 periodic solution of the system. Numerical simulation shows the correctness of our results which show that phytoplankton and fish with the impulsive feedback control can live stably under certain conditions, and the results have certain reference value for the dynamic change of phytoplankton in aquatic ecosystems.

GENERATING HYPERCHAOS VIA STATE FEEDBACK CONTROL

2005 ◽  
Vol 15 (10) ◽  
pp. 3367-3375 ◽  
Author(s):  
YUXIA LI ◽  
WALLACE K. S. TANG ◽  
GUANRONG CHEN
Keyword(s):  
Feedback Control ◽  
Computer Simulations ◽  
Chaotic System ◽  
Bifurcation Analysis ◽  
State Feedback ◽  
Electronic Circuit ◽  
Three Dimensional ◽  
State Feedback Control ◽  
Hyperchaotic System ◽  
Nonlinear State

In this letter, a simple nonlinear state feedback controller is designed for generating hyperchaos from a three-dimensional autonomous chaotic system. The hyperchaotic system is not only demonstrated by computer simulations but also verified with bifurcation analysis, and is implemented experimentally via an electronic circuit.

ADAPTIVE CONTROL OF RECURRENT TRAJECTORIES BASED ON LINEARIZATION OF POINCARÉ MAP

2000 ◽  
Vol 10 (03) ◽  
pp. 621-637 ◽  
Author(s):  
ALEXANDER FRADKOV ◽  
PETER GUZENKO ◽  
ALEXEY PAVLOV
Keyword(s):  
Adaptive Control ◽  
Feedback Control ◽  
Computer Simulations ◽  
Output Feedback ◽  
Chaos Control ◽  
Poincaré Map ◽  
Control Method ◽  
Output Feedback Control ◽  
Poincare Map ◽  
Adaptive Output Feedback Control

The problem of adaptive output feedback control aimed at stabilization of a (periodic or chaotic) goal trajectory is considered. Advantages and drawbacks of chaos control method based on linearization of Poincaré map (first proposed by Ott, Grebogi and Yorke in 1990) are discussed. It is suggested that the recurrence of the goal trajectory is the key property for applicability of approach. Algorithms of adaptive control based on linearization of controlled Poincaré map and method of goal inequalities are proposed. It is shown that stabilization of recurrent trajectories is possible under additional controllability-like and observability-like conditions. Examples of stabilization of periodic and chaotic trajectories for forced brusselator and Rössler systems are studied by computer simulations.

scholarly journals Holling-Tanner Predator-Prey Model with State-Dependent Feedback Control

2018 ◽  
Vol 2018 ◽  
pp. 1-18 ◽  
Author(s):  
Jin Yang ◽  
Guangyao Tang ◽  
Sanyi Tang
Keyword(s):  
Limit Cycle ◽  
Poincaré Map ◽  
Impulsive Control ◽  
Phase Portraits ◽  
Poincare Map ◽  
Complicated Shape ◽  
Predator Prey ◽  
Predator Prey Model ◽  
State Dependent ◽  
Existence And Stability

In this paper, we propose a novel Holling-Tanner model with impulsive control and then provide a detailed qualitative analysis by using theories of impulsive dynamical systems. The Poincaré map is first constructed based on the phase portraits of the model. Then the main properties of the Poincaré map are investigated in detail which play important roles in the proofs of the existence of limit cycles, and it is concluded that the definition domain of the Poincaré map has a complicated shape with discontinuity points under certain conditions. Subsequently, the existence of the boundary order-1 limit cycle is discussed and it is shown that this limit cycle is unstable. Furthermore, the conditions for the existence and stability of an order-1 limit cycle are provided, and the existence of order-k(k≥2) limit cycle is also studied. Moreover, numerical simulations are carried out to substantiate our results. Finally, biological implications related to the mathematical results which are beneficial for successful pest control are addressed in the Conclusions section.

DEGENERATE BIFURCATIONS OF HETERODIMENSIONAL CYCLES WITH ORBIT FLIP

2013 ◽  
Vol 23 (05) ◽  
pp. 1350080 ◽  
Author(s):  
XINGBO LIU ◽  
JUNYING LIU ◽  
DEMING ZHU
Keyword(s):  
Coordinate System ◽  
Periodic Orbits ◽  
Bifurcation Analysis ◽  
Homoclinic Orbit ◽  
Poincaré Map ◽  
Local Coordinate System ◽  
Three Dimensional ◽  
Poincare Map ◽  
Orbit Flip

In this paper, nongeneric bifurcation analysis near heterodimensional cycles with orbit flip is investigated for three-dimensional systems. With the aid of a suitable local coordinate system, the Poincaré map is constructed. By means of the bifurcation equations, the existence, nonexistence, coexistence and uniqueness of homoclinic orbit, periodic orbits and the heterodimensional cycle are studied, the relevant bifurcation surfaces and their existing regions are given. Some known results are extended. An example is given to show the existence of the system which has a heterodimensional cycle with orbit flip.

scholarly journals Dynamic Analysis of a Plankton-Herbivore State-Dependent Impulsive Model With Action Threshold Depending On The Density and Its Changing Rate

2021 ◽  
Author(s):  
Wei Li ◽  
Tonghua Zhang ◽  
Yufei Wang ◽  
Huidong Cheng
Keyword(s):  
Periodic Solution ◽  
Poincaré Map ◽  
Complex Dynamics ◽  
Economic Benefits ◽  
Rate Of Change ◽  
Poincare Map ◽  
The Sustainable Development ◽  
State Dependent ◽  
Action Threshold ◽  
Impulsive Model

Abstract A plankton-herbivore state-dependent impulsive model with nonlinear impulsive functions and action threshold including population density and rate of change is proposed. Since the use of action threshold makes the model have complex phase set and pulse set, we adopt the Poincaré map as a tool to study its complex dynamics. The Poincaré map is defined on the phase set and its properties in different situations are analyzed. Furthermore, the periodic solution of model are discussed, including the existence and stability conditions of the order-1 periodic solution and the existence of the order-k (k ≥ 2) periodic solutions. Compared with the fixed threshold in the existing literature, our results show that the use of action threshold is more practical, which is conducive to the sustainable development of population and makes people obtain more economic benefits. The analysis method used in this paper can study the complex dynamics of the model more comprehensively and deeply.

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