scholarly journals The Impulsive Model with Pest Density and Its Change Rate Dependent Feedback Control

2020 ◽  
Vol 2020 ◽  
pp. 1-20 ◽  
Author(s):  
Ihsan Ullah Khan ◽  
Sanyi Tang
Keyword(s):  
Feedback Control ◽  
Periodic Solutions ◽  
Poincaré Map ◽  
Dynamical Behavior ◽  
Sufficient Conditions ◽  
Control Measures ◽  
Poincare Map ◽  
Change Rate ◽  
Action Threshold ◽  
Pest Density

The idea of action threshold depends on the pest density and its change rate is more general and furthermore can produce new modelling techniques related to integrated pest management (IPM) as compared with those that appeared in earlier studies, which definitely bring challenges to analytical analysis and generate new ideas to the state control measures. Keeping this in mind, using the strategies of IPM, we develop a prey-predator system with action threshold depending on the pest density and its change rate, and study its dynamical behavior. We develop new criteria guaranteeing the existence, uniqueness, local and global stability of order-1 periodic solutions. Applying the properties of Lambert W function, the Poincaré map is portrayed for the exact phase set, which is helpful to provide the sufficient conditions for the existence and stability of the interior order-1 periodic solutions and boundary order-1 periodic solution, also confirmed by numerical simulations. It is studied in detail that how and under what conditions the fixed point of Poincaré map and its stability are affected by the newly introduced action threshold. The analytical methods developed in this paper will be very beneficial to study other generalized models with state-dependent feedback control.

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 The State-Dependent Impulsive Model with Action Threshold Depending on the Pest Density and Its Changing Rate

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-15 ◽  
Author(s):  
Ihsan Ullah Khan ◽  
Sanyi Tang ◽  
Biao Tang
Keyword(s):  
Periodic Solutions ◽  
Integrated Control ◽  
Dynamical Behavior ◽  
Analytical Techniques ◽  
The State ◽  
Control Measures ◽  
Lambert W Function ◽  
Action Threshold ◽  
Pest Density ◽  
Changing Rate

Whether the integrated control measures are applied or not depends not only on the current density of pest population, but also on its current growth rate, and this undoubtedly brings challenges and new ideas to the state control measures that only rely on the pest density. To address this, utilizing the tactics of IPM, we constructed a Lotka-Volterra predator-prey system with action threshold depending on the pest density and its changing rate and examined its dynamical behavior. We present new criteria guaranteeing the existence, uniqueness, and global stability of periodic solutions. With the help of Lambert W function, the Poincaré map is constructed for the phase set, which can help us to provide the satisfactory conditions for the existence and stability of the semitrivial periodic solution and interior order-1 periodic solutions. Furthermore, the existence of order-2 and nonexistence of order-k(k≥3) periodic solutions are discussed. The idea of action threshold depending on the pest density and its changing rate is more general and can generate new remarkable directions as well compared with those represented in earlier studies. The analytical techniques developed in this paper can play a significant role in analyzing the impulsive models with complex phase set or impulsive set.

BIFURCATION ANALYSIS OF CURRENT COUPLED BVP OSCILLATORS

2007 ◽  
Vol 17 (03) ◽  
pp. 837-850 ◽  
Author(s):  
SHIGEKI TSUJI ◽  
TETSUSHI UETA ◽  
HIROSHI KAWAKAMI
Keyword(s):  
Periodic Solutions ◽  
Poincaré Map ◽  
Dynamical Behavior ◽  
Coupled Systems ◽  
Poincare Map ◽  
Circuit Implementation ◽  
Coupling Structure ◽  
Bifurcation Phenomena ◽  
Junctional Coupling ◽  
Current Coupling

The Bonhöffer–van der Pol (BVP) oscillator is a simple circuit implementation describing neuronal dynamics. Lately the diffusive coupling structure of neurons attracts much attention since the existence of the gap-junctional coupling has been confirmed in the brain. Such coupling is easily realized by linear resistors for the circuit implementation, however, there are not enough investigations about diffusively coupled BVP oscillators, even a couple of BVP oscillators. We have considered several types of coupling structure between two BVP oscillators, and discussed their dynamical behavior in preceding works. In this paper, we treat a simple structure called current coupling and study their dynamical properties by the bifurcation theory. We investigate various bifurcation phenomena by computing some bifurcation diagrams in two cases, symmetrically and asymmetrically coupled systems. In symmetrically coupled systems, although all internal elements of two oscillators are the same, we obtain in-phase, anti-phase solution and some chaotic attractors. Moreover, we show that two quasi-periodic solutions disappear simultaneously by the homoclinic bifurcation on the Poincaré map, and that a large quasi-periodic solution is generated by the coalescence of these quasi-periodic solutions, but it disappears by the heteroclinic bifurcation on the Poincaré map. In the other case, we confirm the existence a conspicuous chaotic attractor in the laboratory experiments.

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.

scholarly journals Poincaré Map and Periodic Solutions of First-Order Impulsive Differential Equations on Moebius Stripe

2013 ◽  
Vol 2013 ◽  
pp. 1-11
Author(s):  
Yefeng He ◽  
Yepeng Xing
Keyword(s):  
Periodic Orbit ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Poincaré Map ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
Poincare Map ◽  
First Order ◽  
Existence And Stability ◽  
Stability And Bifurcations

This paper is mainly concerned with the existence, stability, and bifurcations of periodic solutions of a certain scalar impulsive differential equations on Moebius stripe. Some sufficient conditions are obtained to ensure the existence and stability of one-side periodic orbit and two-side periodic orbit of impulsive differential equations on Moebius stripe by employing displacement functions. Furthermore, double-periodic bifurcation is also studied by using Poincaré map.

Dynamic Output Controllers for Exponential Stabilization of Periodic Orbits for Multidomain Hybrid Models of Robotic Locomotion

2019 ◽  
Vol 141 (12) ◽  
Author(s):  
Kaveh Akbari Hamed ◽  
Bita Safaee ◽  
Robert D. Gregg
Keyword(s):  
Periodic Orbits ◽  
Output Feedback ◽  
Poincaré Map ◽  
Optimization Problems ◽  
Sufficient Conditions ◽  
Dynamic Output Feedback ◽  
Poincare Map ◽  
Feedback Controllers ◽  
Robotic Locomotion ◽  
Dynamic Output

Abstract The primary goal of this paper is to develop an analytical framework to systematically design dynamic output feedback controllers that exponentially stabilize multidomain periodic orbits for hybrid dynamical models of robotic locomotion. We present a class of parameterized dynamic output feedback controllers such that (1) a multidomain periodic orbit is induced for the closed-loop system and (2) the orbit is invariant under the change of the controller parameters. The properties of the Poincaré map are investigated to show that the Jacobian linearization of the Poincaré map around the fixed point takes a triangular form. This demonstrates the nonlinear separation principle for hybrid periodic orbits. We then employ an iterative algorithm based on a sequence of optimization problems involving bilinear matrix inequalities to tune the controller parameters. A set of sufficient conditions for the convergence of the algorithm to stabilizing parameters is presented. Full-state stability and stability modulo yaw under dynamic output feedback control are addressed. The power of the analytical approach is ultimately demonstrated through designing a nonlinear dynamic output feedback controller for walking of a three-dimensional (3D) humanoid robot with 18 state variables and 325 controller parameters.

PERMANENCE AND UNIFORMLY ASYMPTOTICAL STABILITY OF ALMOST PERIODIC SOLUTIONS FOR A SINGLE-SPECIES MODEL WITH FEEDBACK CONTROL ON TIME SCALES

2014 ◽  
Vol 07 (01) ◽  
pp. 1450004 ◽  
Author(s):  
Yongkun Li ◽  
Li Yang ◽  
Hongtao Zhang
Keyword(s):  
Feedback Control ◽  
Periodic Solutions ◽  
Time Scales ◽  
Sufficient Conditions ◽  
Single Species ◽  
Asymptotical Stability ◽  
Almost Periodic Solutions ◽  
Almost Periodic ◽  
Uniformly Asymptotical Stability ◽  
Single Species Model

In this paper, using the time scale calculus theory, we first discuss the permanence of a single-species model with feedback control on time scales. Based on the permanence result, by the Lyapunov functional method, we establish sufficient conditions for the existence and uniformly asymptotical stability of almost periodic solutions of the considered model. Moreover, we present an illustrative example to show the effectiveness of obtained results.

scholarly journals Global Attractivity of Positive Periodic Solutions of Delay Differential Equations with Feedback Control

2007 ◽  
Vol 2007 ◽  
pp. 1-11 ◽  
Author(s):  
Zhi-Long Jin
Keyword(s):  
Feedback Control ◽  
Periodic Solutions ◽  
Global Attractivity ◽  
Sufficient Conditions ◽  
Upper And Lower Bounds ◽  
Positive Periodic Solutions ◽  
Liapunov Functionals ◽  
Delay Differential System ◽  
Several Delays ◽  
Delay Differential

By constructing suitable Liapunov functionals and estimating uniform upper and lower bounds of solutions, sufficient conditions are obtained for the global attractivity of positive periodic solutions of the delay differential system with feedback controldy/dt=y(t)F(t,y(t−τ1(t)),…,y(t−τn(t)),u(t−δ(t))),du/dt=−η(t)u(t)+a(t)y(t−σ(t)). When these results are applied to the periodic logistic equation with several delays and feedback control, some new results are obtained.

Second Order Poincaré Map to Study Dynamical Behavior of Nonsymmetrical Gyrostat Satellite

2000 ◽  
Author(s):  
K. H. Shirazi ◽  
M. H. Ghaffari-saadat
Keyword(s):  
Phase Space ◽  
Poincaré Map ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Dynamical Behavior ◽  
Two Dimensions ◽  
Second Order ◽  
Poincare Map ◽  
Runge Kutta Method ◽  
Final Space

Abstract The second order poincare’ map is described and used for investigation of the dynamical behavior of a gyrostat satellite. The normalized attitudinal equations of motion for a typical non-symmetric gyrostat satellite are considered. For different sets of initial conditions the equations simulated by Runge-Kutta method. The poincare’ section is used to dimension reduction of system phase space. By this map the dimension reduced from six to five. Using secondary map the dimension of phase space can be reduced to four and considering symmetry of phase space the final space has two dimensions that is presentable at the plane. Bifurcation in the attitudinal behavior can be demonstrated easily by the derived map.

scholarly journals A New No-Equilibrium Chaotic System and Its Topological Horseshoe Chaos

2016 ◽  
Vol 2016 ◽  
pp. 1-6
Author(s):  
Chunmei Wang ◽  
Chunhua Hu ◽  
Jingwei Han ◽  
Shijian Cang
Keyword(s):  
Numerical Simulation ◽  
Lyapunov Exponents ◽  
Chaotic System ◽  
Poincaré Map ◽  
Chaotic Behavior ◽  
Dynamical Behavior ◽  
Phase Portraits ◽  
Poincare Map ◽  
Topological Horseshoe ◽  
Simulation Techniques

A new no-equilibrium chaotic system is reported in this paper. Numerical simulation techniques, including phase portraits and Lyapunov exponents, are used to investigate its basic dynamical behavior. To confirm the chaotic behavior of this system, the existence of topological horseshoe is proven via the Poincaré map and topological horseshoe theory.

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