Stability of Non-Hyperbolic Equilibrium Point for Polynomial System of Differential Equations

In this paper, a polynomial system of plane differential equations is observed. The stability of the non-hyperbolic equilibrium point was analyzed using normal forms. The nonlinear part of the differential equation system is simplified to the maximum. Two nonlinear transformations were used to simplify first the quadratic and then the cubic part of the system of equations.

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Kestabilan Model Epidemi SEIR Dengan Laju Insidensi

Jurnal Ilmiah Poli Rekayasa ◽

10.30630/jipr.10.2.77 ◽

2015 ◽

Vol 10 (2) ◽

pp. 74

Author(s):

Roni Tri Putra ◽

Sukatik - ◽

Sri Nita

Keyword(s):

Differential Equation ◽

Differential Equations ◽

Incidence Rate ◽

Epidemic Model ◽

Existence And Uniqueness ◽

Equation System ◽

Differential Equation System ◽

Equilibrium Existence

In this paper, it will be studied stability for a SEIR epidemic model with infectious force in latent, infected and immune period with incidence rate. From the model it will be found investigated the existence and uniqueness solution of points its equilibrium. Existence solution of points equilibrium proved by show its differential equations system of equilibrium continue, and uniqueness solution of points equilibrium proved by show its differential equation system of equilibrium differentiable continue.

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A Four-Zone Model and Nonlinear Dynamic Analysis of Solution Multiplicity of Buoyancy Ventilation in Underground Building

Complexity ◽

10.1155/2020/8658797 ◽

2020 ◽

Vol 2020 ◽

pp. 1-24

Author(s):

Yuxing Wang ◽

Chunyu Wei

Keyword(s):

Differential Equations ◽

Equilibrium Point ◽

Equilibrium Points ◽

Underground Structure ◽

Unstable Equilibrium ◽

Phase Portraits ◽

Zone Model ◽

Unstable Equilibrium Point ◽

Index 2 ◽

The Stability

The solution multiplicity of natural ventilation in buildings is very important to personnel safety and ventilation design. In this paper, a four-zone model of buoyancy ventilation in typical underground building is proposed. The underground structure is divided to four zones, a differential equation is established in each zone, and therefore, there are four differential equations in the underground structure. By solving and analyzing the equilibrium points and characteristic roots of the differential equations, we analyze the stability of three scenarios and obtain the criterions to determine the stability and existence of solutions for two scenarios. According to these criterions, the multiple steady states of buoyancy ventilation in any four-zone underground buildings for different stack height ratios and the strength ratios of the heat sources can be obtained. These criteria can be used to design buoyancy ventilation or natural exhaust ventilation systems in underground buildings. Compared with the two-zone model in (Liu et al. 2020), the results of the proposed four-zone model are more consistent with CFD results in (Liu et al. 2018). In addition, the results of proposed four-zone model are more specific and more detailed in the unstable equilibrium point interval. We find that the unstable equilibrium point interval is divided into two different subintervals corresponding to the saddle point of index 2 and the saddle focal equilibrium point of index 2, respectively. Finally, the phase portraits and vector field diagrams for the two scenarios are given.

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STABILITY BOUNDARY CHARACTERIZATION OF NONLINEAR AUTONOMOUS DYNAMICAL SYSTEMS IN THE PRESENCE OF A SUPERCRITICAL HOPF EQUILIBRIUM POINT

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127413501964 ◽

2013 ◽

Vol 23 (12) ◽

pp. 1350196 ◽

Cited By ~ 2

Author(s):

JOSAPHAT R. R. GOUVEIA ◽

FABÍOLO MORAES AMARAL ◽

LUÍS F. C. ALBERTO

Keyword(s):

Dynamical Systems ◽

Equilibrium Point ◽

Equilibrium Points ◽

Stability Boundary ◽

Complete Characterization ◽

Center Manifolds ◽

The Stability ◽

Hyperbolic Equilibrium ◽

Autonomous Dynamical Systems

A complete characterization of the boundary of the stability region (or area of attraction) of nonlinear autonomous dynamical systems is developed admitting the existence of a particular type of nonhyperbolic equilibrium point on the stability boundary, the supercritical Hopf equilibrium point. Under a condition of transversality, it is shown that the stability boundary is comprised of all stable manifolds of the hyperbolic equilibrium points lying on the stability boundary union with the center-stable and\or center manifolds of the type-k, k ≥ 1, supercritical Hopf equilibrium points on the stability boundary.

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Dynamics of an Innovation Diffusion Model with Time Delay

East Asian Journal on Applied Mathematics ◽

10.4208/eajam.201216.230317a ◽

2017 ◽

Vol 7 (3) ◽

pp. 455-481 ◽

Cited By ~ 8

Author(s):

Rakesh Kumar ◽

Anuj K. Sharma ◽

Kulbhushan Agnihotri

Keyword(s):

Hopf Bifurcation ◽

Time Delay ◽

Population Density ◽

Differential Equations ◽

Equilibrium Point ◽

Innovation Diffusion ◽

Critical Value ◽

Normal Form Theory ◽

Interior Equilibrium ◽

The Stability

AbstractA nonlinear mathematical model for innovation diffusion is proposed. The system of ordinary differential equations incorporates variable external influences (the cumulative density of marketing efforts), variable internal influences (the cumulative density of word of mouth) and a logistically growing human population (the variable potential consumers). The change in population density is due to various demographic processes such as intrinsic growth rate, emigration, death rate etc. Thus the problem involves two dynamic variables viz. a non-adopter population density and an adopter population density. The model is analysed qualitatively using the stability theory of differential equations, with the help of the corresponding characteristic equation of the system. The interior equilibrium point can be stable for all time delays to a critical value, beyond which the system becomes unstable and a Hopf bifurcation occurs at a second critical value. Employing normal form theory and a centre manifold theorem applicable to functional differential equations, we derive some explicit formulas determining the stability, the direction and other properties of the bifurcating periodic solutions. Our numerical simulations show that the system behaviour can become extremely complicated as the time delay increases, with a stable interior equilibrium point leading to a limit cycle with one local maximum and minimum per cycle (Hopf bifurcation), then limit cycles with more local maxima and minima per cycle, and finally chaotic solutions.

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Successive approximation: A survey on stable manifold of fractional differential systems

Fractional Calculus and Applied Analysis ◽

10.1515/fca-2015-0038 ◽

2015 ◽

Vol 18 (3) ◽

Cited By ~ 9

Author(s):

Khosro Sayevand ◽

Kazem Pichaghchi

Keyword(s):

Equilibrium Point ◽

Stable Manifold ◽

Solvable Model ◽

Stability And Convergence ◽

Model Method ◽

Local Stable Manifold ◽

Fractional Differential ◽

Hyperbolic Equilibrium ◽

The Solution Existence ◽

Hyperbolic Equilibrium Point

AbstractThis paper outlines a reliable strategy to approximate the local stable manifold near a hyperbolic equilibrium point for nonlinear systems of differential equations of fractional order. Furthermore, the local behavior of these systems near a hyperbolic equilibrium point is investigated based on the fractional Hartman-Grobman theorem. The fractional derivative is described in the Caputo sense. The solution existence, uniqueness, stability and convergence of the proposed scheme is discussed. Finally, the validity and applicability of our approach is examined with the use of a solvable model method.

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Data Analysis of Physical Fitness Monitoring Based on Mathematical Models

Mathematical Problems in Engineering ◽

10.1155/2021/8353391 ◽

2021 ◽

Vol 2021 ◽

pp. 1-9

Author(s):

Cuixiang Guo ◽

Junwu Suo ◽

Chunguang Xu ◽

Xinhua Yang ◽

Liping Zhang

Keyword(s):

College Students ◽

Physical Fitness ◽

Mathematical Models ◽

Equilibrium Point ◽

Quantitative Research ◽

Equation System ◽

Differential Equation System ◽

Main Work ◽

Existence And Stability

Physical monitoring and analysis are of great significance to improve national physical fitness. The main work of this paper is that the physical health level of college students is studied and analysed by using a statistical model and mathematical model. According to the characteristics of the collected data, different mathematical models are established. Firstly, the grey correlation model is used to analyse the correlation between pull-up and other physical fitness indexes. Then, based on the classification of college students and the influence and flow law of interclass crowd, a differential equation system is established based on the LMC model. By analysing the existence and stability of the equilibrium point of the system, as well as the possible folding bifurcation and backward bifurcation at the equilibrium point, this paper makes qualitative and quantitative research on the trend of college students’ physical exercise on campus.

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A MATHEMATIC MODEL OF TWO MUTUALLY INTERACTING SPECIES WITH MORTALITY RATE FOR THE SECOND SPECIES

TROPICAL WETLAND JOURNAL ◽

10.20527/twj.v3i2.50 ◽

2017 ◽

Vol 3 (2) ◽

pp. 21-25

Author(s):

Annisa Rahayu ◽

Yuni Yulida ◽

Faisal Faisal

Keyword(s):

Mortality Rate ◽

Interaction Model ◽

Mathematic Model ◽

Equation System ◽

Differential Equation System ◽

Interacting Species ◽

Stability Model ◽

Species Population ◽

The Stability ◽

Type Of Stability

One of the interactions that occur withinthe ecosystem is the interaction of mutualism. Such mutualism interactions can be modeled into mathematical models. Reddy (2011) study suggests a model of two mutually interacting species that assumes that each species can live without its mutualism partner. In fact, not all mutual species survive without their mutualism pairs. If it is assumed that the second species lives without its mutualism partner, the first species, then the natural growth rate of the second species population will decrease (the mortality rate). The purpose of this research is to explain the model of two mutually interacting species with mortality rate for the second species, to determine the equilibrium point and the type of stability, and to simulate them with several parameters. This research was done by way of literature studies. The result of this research is the model of two mutually interacting species with mortality rate for second species modeled using Nonlinear Differential Equation System. In the model, it was obtained 3 (three) points of equilibrium, with each type and type of stability investigated. Next up from the stability, model simulations were done. Based on several simulations conducted can be seen the value of parameters and initial values affect the population growth of both species. The interaction model of two mutual species will be stable if the first species survive and the second species over time will be extinct.

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Model Epidemi Seir dengan Insidensi Standar

Jurnal Ilmiah Poli Rekayasa ◽

10.30630/jipr.12.1.37 ◽

2016 ◽

Vol 12 (1) ◽

pp. 73

Author(s):

Roni Tri Putra

Keyword(s):

Differential Equation ◽

Differential Equations ◽

Epidemic Model ◽

Existence And Uniqueness ◽

Equation System ◽

Differential Equation System ◽

Equilibrium Existence

In this paper, it will be studied stability for a SEIR epidemic model with infectious force in latent, infected and immune period with standard incidence. From the model it will be found investigated the existence and uniqueness solution of points its equilibrium. Existence solution of points equilibrium proved by show its differential equations system of equilibrium continue, and uniqueness solution of points equilibrium proved by show its differential equation system of equilibrium differentiable continue.

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