Boundaries of Stability Domains for Equilibrium Points of Differential Equations with Parameters

ABSTRACT The microquasar GRS 1915+105 is known to exhibit a very variable X-ray emission on different time-scales and patterns. We propose a system of two ordinary differential equations, adapted from the Hindmarsh–Rose model, with two dynamical variables x(t), y(t), and an input constant parameter J0, to which we added a random white noise, whose solutions for the x(t) variable reproduce consistently the X-ray light curves of several variability classes as well as the development of low-frequency quasi-periodic oscillations (QPO). We show that changing only the value of J0, the system moves from stable to unstable solutions and the resulting light curves reproduce those of the quiescent classes like ϕ and χ, the δ class and the spiking ρ class. Moreover, we found that increasing the values of J0 the system induces high-frequency oscillations that evolve into QPO when it moves into another stable region. This system of differential equations gives then a unified view of the variability of GRS 1915+105 in term of transitions between stable and unstable states driven by a single input function J0. We also present the results of a stability analysis of the equilibrium points and some considerations on the existence of periodic solutions.

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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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Analytic expressions for the unstable manifold at equilibrium points in dynamical systems of differential equations

10.1109/cdc.1983.269759 ◽

1983 ◽

Cited By ~ 9

Author(s):

Fathi Abdel Salam ◽

Aristotle Arapostathis ◽

Pravin Varaiya

Keyword(s):

Dynamical Systems ◽

Differential Equations ◽

Unstable Manifold ◽

Equilibrium Points ◽

Systems Of Differential Equations ◽

Analytic Expressions

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Time-dependent changes in membrane excitability during glucose-induced bursting activity in pancreatic β cells

The Journal of General Physiology ◽

10.1085/jgp.201110612 ◽

2011 ◽

Vol 138 (1) ◽

pp. 39-47 ◽

Cited By ~ 23

Author(s):

Chae Young Cha ◽

Enrique Santos ◽

Akira Amano ◽

Takao Shimayoshi ◽

Akinori Noma

Keyword(s):

Differential Equations ◽

Cell Model ◽

Equilibrium Points ◽

Experimental Studies ◽

Companion Paper ◽

Membrane Excitability ◽

Β Cells ◽

Pancreatic Β Cells ◽

Single Action ◽

Functional Components

In our companion paper, the physiological functions of pancreatic β cells were analyzed with a new β-cell model by time-based integration of a set of differential equations that describe individual reaction steps or functional components based on experimental studies. In this study, we calculate steady-state solutions of these differential equations to obtain the limit cycles (LCs) as well as the equilibrium points (EPs) to make all of the time derivatives equal to zero. The sequential transitions from quiescence to burst–interburst oscillations and then to continuous firing with an increasing glucose concentration were defined objectively by the EPs or LCs for the whole set of equations. We also demonstrated that membrane excitability changed between the extremes of a single action potential mode and a stable firing mode during one cycle of bursting rhythm. Membrane excitability was determined by the EPs or LCs of the membrane subsystem, with the slow variables fixed at each time point. Details of the mode changes were expressed as functions of slowly changing variables, such as intracellular [ATP], [Ca2+], and [Na+]. In conclusion, using our model, we could suggest quantitatively the mutual interactions among multiple membrane and cytosolic factors occurring in pancreatic β cells.

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An SEIR model with infected immigrants and recovered emigrants

Advances in Difference Equations ◽

10.1186/s13662-021-03488-5 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Peter J. Witbooi

Keyword(s):

Differential Equations ◽

Ordinary Differential Equations ◽

Positive Solutions ◽

Equilibrium Points ◽

Local Population ◽

Asymptotically Stable ◽

Short Stay ◽

Seir Model ◽

Threshold Levels

AbstractWe present a deterministic SEIR model of the said form. The population in point can be considered as consisting of a local population together with a migrant subpopulation. The migrants come into the local population for a short stay. In particular, the model allows for a constant inflow of individuals into different classes and constant outflow of individuals from the R-class. The system of ordinary differential equations has positive solutions and the infected classes remain above specified threshold levels. The equilibrium points are shown to be asymptotically stable. The utility of the model is demonstrated by way of an application to measles.

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Network Modelling on Tropical Diseases vs. Climate Change

Advances in Environmental Engineering and Green Technologies - Climate Change and Anthropogenic Impacts on Health in Tropical and Subtropical Regions ◽

10.4018/978-1-7998-2197-7.ch004 ◽

2020 ◽

pp. 64-92

Author(s):

G. Udhaya Sankar ◽

C. Ganesa Moorthy

Keyword(s):

Differential Equations ◽

Integral Equations ◽

Critical Points ◽

Blood Cells ◽

Equilibrium Points ◽

White Blood Cells ◽

Practical Method ◽

Tropical Diseases ◽

Network Systems ◽

Network Modelling

This chapter has proposed a systematic method to design mathematical models. These models have been associated with counting of white blood cells, counting of red blood cells, population of mosquitoes, and counting of foreign bodies like virus, bacteria, and parasite in a human body. Interpretations for critical points or equilibrium points have been given for network systems of differential equations associated with models. A practical method of applying these interpretations in administrating medicines to get control over diseases has been given. Order of priority in three types of critical points, namely, stable critical points, unstable critical points, and asymptotically stable critical points, has been given. Conversions of differential equations of models into integral equations and applying Picard's iteration method to solve integral equations have been explained. A step-by-step approach has been used in designing models, solving models, and interpreting solutions of models for tropical diseases.

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DYNAMICS OF NUMERICAL METHODS FOR COSYMMETRIC ORDINARY DIFFERENTIAL EQUATIONS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127401003504 ◽

2001 ◽

Vol 11 (09) ◽

pp. 2339-2357 ◽

Cited By ~ 3

Author(s):

V. N. GOVORUKHIN ◽

V. G. TSYBULIN ◽

B. KARASÖZEN

Keyword(s):

Differential Equations ◽

Ordinary Differential Equations ◽

Equilibrium Points ◽

Two Dimensions ◽

Model Systems ◽

Continuous Family ◽

First Order ◽

Spurious Solutions ◽

Overall Performance ◽

The Stability

The dynamics of numerical approximation of cosymmetric ordinary differential equations with a continuous family of equilibria is investigated. Nonconservative and Hamiltonian model systems in two dimensions are considered and these systems are integrated with several first-order Runge–Kutta methods. The preservation of symmetry and cosymmetry, the stability of equilibrium points, spurious solutions and transition to chaos are investigated by presenting analytical and numerical results. The overall performance of the methods for different parameters is discussed.

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ON ESTIMATION OF THE LINEARIZED DRIFT FOR NONLINEAR STOCHASTIC DIFFERENTIAL EQUATIONS

Stochastics and Dynamics ◽

10.1142/s0219493701000035 ◽

2001 ◽

Vol 01 (01) ◽

pp. 23-43 ◽

Cited By ~ 4

Author(s):

R. KHASMINSKII ◽

G. N. MILSTEIN

Keyword(s):

Differential Equations ◽

Equilibrium Point ◽

Stochastic Differential Equations ◽

Initial Conditions ◽

Equilibrium Points ◽

Initial Condition ◽

Nonlinear Stochastic Differential Equations

The estimation of the linearized drift for stochastic differential equations with equilibrium points is considered. It is proved that the linearized drift matrix can be estimated efficiently if the initial condition for the system is chosen close enough to the equilibrium point. Some bounds for initial conditions ensuring the asymptotical efficiency of the estimator are found.

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A Fractional-Order Epidemic Model for Bovine Babesiosis Disease and Tick Populations

Abstract and Applied Analysis ◽

10.1155/2015/729894 ◽

2015 ◽

Vol 2015 ◽

pp. 1-10 ◽

Cited By ~ 6

Author(s):

José Paulo Carvalho dos Santos ◽

Lislaine Cristina Cardoso ◽

Evandro Monteiro ◽

Nelson H. T. Lemes

Keyword(s):

Differential Equations ◽

Fractional Order ◽

Fractional Differential Equations ◽

Epidemic Model ◽

Equilibrium Points ◽

Biological Behavior ◽

Bovine Babesiosis ◽

Comparison Theory ◽

Linearization Theorem ◽

Fractional Differential

This paper shows that the epidemic model, previously proposed under ordinary differential equation theory, can be generalized to fractional order on a consistent framework of biological behavior. The domain set for the model in which all variables are restricted is established. Moreover, the existence and stability of equilibrium points are studied. We present the proof that endemic equilibrium point when reproduction numberR0>1is locally asymptotically stable. This result is achieved using the linearization theorem for fractional differential equations. The global asymptotic stability of disease-free point, whenR0<1, is also proven by comparison theory for fractional differential equations. The numeric simulations for different scenarios are carried out and data obtained are in good agreement with theoretical results, showing important insight about the use of the fractional coupled differential equations set to model babesiosis disease and tick populations.

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