Infinite Process of Forward and Backward Bifurcations in the Logistic Equation with Two Delays

2019 ◽  
Vol 22 (4) ◽  
pp. 407-412 ◽  
Author(s):  
Ilia Kashchenko ◽  
Sergey Kaschenko
Keyword(s):  
Normal Form ◽  
Small Parameter ◽  
Infinite Number ◽  
Dynamic Effect ◽  
Logistic Equation ◽  
Biological Processes ◽  
Large Parameter ◽  
Main Assumption ◽  
Two Delays ◽  
Equation With Delay

Logistic equation with delay play important role in modelling of various biological processes. In this paper we study the behaviour of solutions of a logistic equation with two delays in a small neighbourhood of equilibrium. The main assumption is that Malthusian coefficient is large, so problem is singular perturbed. To study the local dynamics near points of bifurcation an analogues of normal form was constructed. Its coefficients depends on special bounded discontinues function, which takes all its values infinite number of times when large parameter increases to infinity. It is shown that the system under study has such dynamic effect as infinite process of direct and inverse bifurcations as the small parameter tends to zero.

scholarly journals Bifurcation Analysis in Delayed Nicholson Equation With Harvest Term

2021 ◽  
Author(s):  
Yuying Liu ◽  
Junjie Wei
Keyword(s):  
Normal Form ◽  
Mathematical Modelling ◽  
Normal Forms ◽  
Center Manifolds ◽  
Open Problems ◽  
Two Delays ◽  
The Stability ◽  
Two Parameters ◽  
Equation With Delay ◽  
Two Parameter

Abstract In this paper, we investigate a delayed Nicholson equation with delay harvesting term which was proposed in open problems and conjectures formulated by Berezansky et al. (Applied Mathematical Modelling 34 (2010) 1405). The stability switching curves by taking two delays as parameters are obtained via the method introduced by An et al.(J. Differential Equations 266 (2019) 7073). The existence of Hopf singularity on a two-parameter plane is determined by the varying direction of two parameters. Furthermore, the normal form near the Hopf singularity is derived via applying the center manifolds theory and normal forms method of FDEs. Finally, some numerical simulations are carried out to illustrate the theoretical conclusions.

scholarly journals Solving a Problem of Rotary Motion for a Heavy Solid Using the Large Parameter Method

2020 ◽  
Vol 2020 ◽  
pp. 1-7 ◽  
Author(s):  
A. I. Ismail
Keyword(s):  
Angular Velocity ◽  
Small Parameter ◽  
Initial Conditions ◽  
Geometric Interpretation ◽  
Rigid Bodies ◽  
Parameter Method ◽  
Large Parameter ◽  
Small Parameter Method ◽  
Rotational Motions ◽  
The Stability

The small parameter method was applied for solving many rotational motions of heavy solids, rigid bodies, and gyroscopes for different problems which classify them according to certain initial conditions on moments of inertia and initial angular velocity components. For achieving the small parameter method, the authors have assumed that the initial angular velocity is sufficiently large. In this work, it is assumed that the initial angular velocity is sufficiently small to achieve the large parameter instead of the small one. In this manner, a lot of energy used for making the motion initially is saved. The obtained analytical periodic solutions are represented graphically using a computer program to show the geometric periodicity of the obtained solutions in some interval of time. In the end, the geometric interpretation of the stability of a motion is given.

scholarly journals Finite-dimensional maps describing the dynamics of a logistic equation with delay

2019 ◽  
Vol 487 (6) ◽  
pp. 611-616
Author(s):  
S. D. Glyzin ◽  
S. A. Kashchenko
Keyword(s):  
Numerical Simulation ◽  
Equilibrium State ◽  
Stable Equilibrium ◽  
Dynamic Properties ◽  
Logistic Equation ◽  
Mathematical Ecology ◽  
Stable Equilibrium State ◽  
Finite Dimensional ◽  
Equation With Delay ◽  
Dynamics Of Populations

This article discusses a family of maps that are used in the numerical simulation of a logistic equation with delay. This equation and presented maps are widely used in problems of mathematical ecology as models of the dynamics of populations. The paper compares the dynamic properties of the trajectories of these mappings and the original equation with delay. It is shown that the behavior of the solutions of maps can be quite complicated, while the logistic equation with delay has only a stable equilibrium state or cycle.

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