Bifurcation and comparison of a discrete-time Hindmarsh-Rose model

Mapping Intimacies ◽

10.22541/au.161324877.71133890/v1 ◽

2021 ◽

Author(s):

Yue Li ◽

Hongjun Cao

Keyword(s):

Discrete Time ◽

Continuous System ◽

Stability And Convergence ◽

Euler Scheme ◽

Step Size ◽

Time Model ◽

Discrete Time Model ◽

Dynamical Behaviors ◽

Nsfd Scheme ◽

Forward Euler

In this paper, a discrete-time Hindmarsh-Rose model is obtained by a nonstandard finite difference (NSFD) scheme. Bifurcation behaviors between the model obtained by the forward Euler scheme and the model obtained by the NSFD scheme are compared. Through analytical and numerical comparisons, much more bifurcations and dynamical behaviors can be obtained and preserved by using the NSFD scheme, in which the integral step size can be chosen larger relatively due to its better stability and convergence than those in the forward Euler scheme. It means that the discretetime model obtained by the NSFD scheme is closer to the original continuous system than the discrete-time model obtained by the forward Euler scheme. These confirmed results can at least guarantee true available numerical results to investigate complex neuron dynamical systems.

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Fractional-Order Calculus in Antimicrobial Resistance and Waning Vaccination

Advanced Applications of Fractional Differential Operators to Science and Technology - Advances in Computer and Electrical Engineering ◽

10.4018/978-1-7998-3122-8.ch001 ◽

2020 ◽

pp. 1-20

Author(s):

E. Ahmed ◽

Ahmed Ezzat Mohamed Matouk

Keyword(s):

Antimicrobial Resistance ◽

Fractional Order ◽

Discrete Time ◽

Local Stability ◽

Chaotic Attractors ◽

Time Model ◽

Fractional Order Calculus ◽

Discrete Time Model ◽

Dynamical Behaviors ◽

Numerical Tools

In this chapter, a simple model for competition between drug resistant and drug sensitive bacteria is given. So, a model of antimicrobial resistance (AMR) and waning vaccination is presented. The model's steady states are obtained. The conditions of local stability of the equilibria are also derived via the fractional Routh-Hurwitz criterion. A discretization scheme has also been applied to the proposed fractional-order model to enhance the model's adequacy and accuracy of describing natural phenomena. So, dynamical behaviors of the resulting discrete-time model are studied such as local stability, existence of Neimark-Sacker, and flip bifurcations. Furthermore, existence of chaos in the discrete-time model is proved using the theorem given by Marotto. Chaotic attractors and routes to chaos are also depicted via various numerical tools.

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A Method of Making a Discrete-Time Model for Continuous System with Dead Time

Transactions of the Society of Instrument and Control Engineers ◽

10.9746/sicetr1965.19.846 ◽

1983 ◽

Vol 19 (10) ◽

pp. 846-848

Author(s):

Junya SHIMAZAKI

Keyword(s):

Discrete Time ◽

Dead Time ◽

Continuous System ◽

Time Model ◽

Discrete Time Model

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Qualitative properties and bifurcations of discrete-time Bazykin–Berezovskaya predator–prey model

International Journal of Biomathematics ◽

10.1142/s1793524520500400 ◽

2020 ◽

Vol 13 (06) ◽

pp. 2050040

Author(s):

A. A. Elsadany ◽

Qamar Din ◽

S. M. Salman

Keyword(s):

Discrete Time ◽

Allee Effect ◽

Initial Conditions ◽

Qualitative Properties ◽

Step Size ◽

Time Model ◽

Predator Prey ◽

Discrete Time Model ◽

Predator Prey Model ◽

Prey Model

The positive connection between the total individual fitness and population density is called the demographic Allee effect. A demographic Allee effect with a critical population size or density is strong Allee effect. In this paper, discrete counterpart of Bazykin–Berezovskaya predator–prey model is introduced with strong Allee effects. The steady states of the model, the existence and local stability are examined. Moreover, proposed discrete-time Bazykin–Berezovskaya predator–prey is obtained via implementation of piecewise constant method for differential equations. This model is compared with its continuous counterpart by applying higher-order implicit Runge–Kutta method (IRK) with very small step size. The comparison yields that discrete-time model has sensitive dependence on initial conditions. By implementing center manifold theorem and bifurcation theory, we derive the conditions under which the discrete-time model exhibits flip and Niemark–Sacker bifurcations. Moreover, numerical simulations are provided to validate the theoretical results.

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