scholarly journals Dynamic Properties of the Fractional-Order Logistic Equation of Complex Variables

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
A. M. A. El-Sayed ◽  
E. Ahmed ◽  
H. A. A. El-Saka
Keyword(s):  
Dynamical System ◽  
Global Stability ◽  
Fractional Order ◽  
Existence And Uniqueness ◽  
Dynamic Properties ◽  
Equilibrium Points ◽  
Stable Solution ◽  
Complex Variables ◽  
Logistic Equation ◽  
Local And Global Stability

We study the dynamic properties (equilibrium points, local and global stability, chaos and bifurcation) of the continuous dynamical system of the logistic equation of complex variables. The existence and uniqueness of uniformly Lyapunov stable solution will be proved.

Dynamic Properties of the Predator–Prey Discontinuous Dynamical System

2012 ◽  
Vol 67 (1-2) ◽  
pp. 57-60 ◽  
Author(s):  
Ahmed M. A. El-Sayed ◽  
Mohamed E. Nasr
Keyword(s):  
Dynamical System ◽  
Global Stability ◽  
Existence And Uniqueness ◽  
Dynamic Properties ◽  
Equilibrium Points ◽  
Stable Solution ◽  
Predator Prey ◽  
Local And Global Stability ◽  
Discontinuous Dynamical System

In this work, we study the dynamic properties (equilibrium points, local and global stability, chaos and bifurcation) of the predator-prey discontinuous dynamical system. The existence and uniqueness of uniformly Lyapunov stable solution will be proved

Asymptotic Behavior of the Fractional Order three Species Prey–Predator Model

2018 ◽  
Vol 19 (7-8) ◽  
pp. 721-733 ◽  
Author(s):  
M. Sambath ◽  
P. Ramesh ◽  
K. Balachandran
Keyword(s):  
Dynamical System ◽  
Asymptotic Behavior ◽  
Global Stability ◽  
Fractional Order ◽  
Equilibrium Points ◽  
Numerical Results ◽  
Predator Prey ◽  
Local And Global Stability ◽  
Predator Prey Model ◽  
Predator Model

AbstractIn this work, we introduce fractional order predator–prey model with infected predator. First, we prove different mathematical results like existence, uniqueness, non-negativity and boundedness of the solutions of fractional order dynamical system. Further, we investigate the local and global stability of all feasible equilibrium points of the system. Numerical results are illustrated as several examples.

scholarly journals Fractional order prey-predator model with infected predators in the presence of competition and toxicity

2020 ◽  
Vol 15 ◽  
pp. 38
Author(s):  
M. R. Lemnaouar ◽  
M. Khalfaoui ◽  
Y. Louartassi ◽  
I. Tolaimate
Keyword(s):  
Global Stability ◽  
Numerical Simulations ◽  
Fractional Order ◽  
Local And Global Stability ◽  
Predator Model ◽  
Reserved Area

In this paper, we propose a fractional-order prey-predator model with reserved area in the presence of the toxicity and competition. We prove different mathematical results like existence, uniqueness, non negativity and boundedness of the solution for our model. Further, we discuss the local and global stability of these equilibria. Finally, we perform numerical simulations to prove our results.

scholarly journals Analysis of logistic equation pertaining to a new fractional derivative with non-singular kernel

2017 ◽  
Vol 9 (2) ◽  
pp. 168781401769006 ◽  
Author(s):  
Devendra Kumar ◽  
Jagdev Singh ◽  
Maysaa Al Qurashi ◽  
Dumitru Baleanu
Keyword(s):  
Numerical Simulation ◽  
Fixed Point ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Existence And Uniqueness ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Logistic Equation ◽  
Iterative Technique ◽  
Uniqueness Of The Solution

In this work, we aim to analyze the logistic equation with a new derivative of fractional order termed in Caputo–Fabrizio sense. The logistic equation describes the population growth of species. The existence of the solution is shown with the help of the fixed-point theory. A deep analysis of the existence and uniqueness of the solution is discussed. The numerical simulation is conducted with the help of the iterative technique. Some numerical simulations are also given graphically to observe the effects of the fractional order derivative on the growth of population.

scholarly journals Mathematical Model for Impact of Media on Cleanliness Drive in India

2018 ◽  
Vol 7 (1) ◽  
pp. 29-36
Author(s):  
N H Shah ◽  
J S Patel ◽  
F A Thakkar ◽  
M H Satia
Keyword(s):  
Numerical Simulation ◽  
Mathematical Model ◽  
Lyapunov Function ◽  
Global Stability ◽  
Transfer Rate ◽  
Equilibrium Points ◽  
Endemic Equilibrium ◽  
Local And Global Stability

A mathematical model on cleanliness drive in India is analysed for active cleaners and passive cleaners. Cleanliness and endemic equilibrium points are found. Local and global stability of these equilibrium points are discussed using Routh-Hurwitz criteria and Lyapunov function respectively. Impact of media (as a control) is studied on passive cleaners to become active. Numerical simulation of the model is carried out which indicates that with the help of media transfer rate to active cleaners from passive cleaners is higher.

Controlling Asthma Due to Air Pollution

2020 ◽  
pp. 1-22
Author(s):  
Ankush H. Suthar ◽  
Purvi M. Pandya
Keyword(s):  
Mathematical Model ◽  
Optimal Control ◽  
Air Pollution ◽  
Control Theory ◽  
Global Stability ◽  
Optimal Control Theory ◽  
Positive Impact ◽  
Equilibrium Points ◽  
Local And Global Stability ◽  
Number Of Individuals

The health of our respiratory systems is directly affected by the atmosphere. Nowadays, eruption of respiratory disease and malfunctioning of lung due to the presence of harmful particles in the air is one of the most sever challenge. In this chapter, association between air pollution-related respiratory diseases, namely dyspnea, cough, and asthma, is analysed by constructing a mathematical model. Local and global stability of the equilibrium points is proved. Optimal control theory is applied in the model to optimize stability of the model. Applied optimal control theory contains four control variables, among which first control helps to reduce number of individuals who are exposed to air pollutants and the remaining three controls help to reduce the spread and exacerbation of asthma. The positive impact of controls on the model and intensity of asthma under the influence of dyspnea and cough is observed graphically by simulating the model.

scholarly journals Global Stability of Fractional Order Coupled Systems with Impulses via a Graphic Approach

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 744 ◽  
Author(s):  
Bei Zhang ◽  
Yonghui Xia ◽  
Lijuan Zhu ◽  
Haidong Liu ◽  
Longfei Gu
Keyword(s):  
Dynamical System ◽  
Graph Theory ◽  
Global Stability ◽  
Fractional Order ◽  
Trivial Solution ◽  
Sufficient Conditions ◽  
Fractional Order System ◽  
Coupled Systems ◽  
Order System ◽  
The Stability

Based on the graph theory and stability theory of dynamical system, this paper studies the stability of the trivial solution of a coupled fractional-order system. Some sufficient conditions are obtained to guarantee the global stability of the trivial solution. Finally, a comparison between fractional-order system and integer-order system ends the paper.

Stability analysis in prey–predator system with mixed functional response

2016 ◽  
Vol 13 (4) ◽  
pp. 364-369
Author(s):  
V. Madhusudanan ◽  
S. Vijaya
Keyword(s):  
Global Stability ◽  
Numerical Simulations ◽  
Functional Response ◽  
Equilibrium Points ◽  
Sufficient Conditions ◽  
Type I ◽  
Content Type ◽  
Local And Global Stability ◽  
Interior Equilibrium ◽  
Predator System

Purpose This paper aims to propose and analyse a two-prey–one-predator system with mixed functional response. Design/methodology/approach The predator exhibits Holling type IV functional response to one prey and Holling type I response to other. The occurrence of various positive equilibrium points with feasibility conditions are determined. The local and global stability of interior equilibrium points are examined. The boundedness of system is analysed. The sufficient conditions for persistence of the system is obtained by using Bendixson–Dulac criteria. Numerical simulations are carried out to illustrate the analytical findings. Findings The authors have derived the local and global stability condition of interior equilibrium of the system. Originality/value The authors observe that the critical values of some system parameter undergo Hopf bifurcation around some selective equilibrium. Hence, numerical simulations reveal the condition for the system to be stable and oscillatory.

LOCAL AND GLOBAL STABILITY OF SWITCHING REGULATORS

1995 ◽  
Vol 05 (03) ◽  
pp. 305-315 ◽  
Author(s):  
HENRY CHUNG ◽  
ADRIAN IOINOVICI
Keyword(s):  
Global Stability ◽  
Closed Loop ◽  
Equilibrium Points ◽  
Stability Study ◽  
Large Signal ◽  
Time Model ◽  
Local And Global Stability ◽  
Boost Regulator ◽  
Switching Regulators ◽  
Conduction Mode

A discrete-time model of closed-loop PWM regulators is derived to describe their dynamic behavior. No small-ripple approximations are required. The same model serves both local and global stability study: by discarding the nonlinear terms and using the z-transform, stability for small-signal perturbations is checked; by keeping the nonlinear terms (products of disturbances) and using the state-plane portrait, in which equilibrium points are located, stability for large-signal perturbations is studied. The theory is applied to a multiple feedback boost regulator operating in continuous conduction mode. Its local/global stability/instability for different values of the feedback gains is determined based on the new method.

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