scholarly journals PHASE BEHAVIOUR OF FIRST - ORDER SYSTEMS

2018 ◽  
Vol 16 (2) ◽  
pp. 131-137 ◽  
Author(s):  
Kaloyan Yankov
Keyword(s):  
Differential Equations ◽  
Response Curve ◽  
Phase Plane ◽  
Nonlinear Differential Equations ◽  
Equilibrium Points ◽  
Phase Behaviour ◽  
First Order ◽  
Phase Plane Method ◽  
The Stability ◽  
First Order Differential Equations

The phase-plane method gives possibility to study the stability of systems described by linear and nonlinear differential equations. The article is devoted to the capabilities of MathCad for analysis of first order differential equations. An algorithm is proposed and Mathcad's specific operators for the construction and analysis of phase trajectories are described. Approaches for calculation of equilibrium points and determination the type of bifurcation in function of parameter are described. The proposed algorithm is applied to the dose-response curve of the antibiotic tubazid.

Asymptotics for third-order nonlinear differential equations: Non-oscillatory and oscillatory cases

2021 ◽  
pp. 1-19
Author(s):  
Calogero Vetro ◽  
Dariusz Wardowski
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Order Differential Equation ◽  
Nonlinear Differential Equations ◽  
Oscillatory Behavior ◽  
Third Order ◽  
First Order ◽  
Third Order Differential Equation ◽  
Comparison Technique ◽  
First Order Differential Equations

We discuss a third-order differential equation, involving a general form of nonlinearity. We obtain results describing how suitable coefficient functions determine the asymptotic and (non-)oscillatory behavior of solutions. We use comparison technique with first-order differential equations together with the Kusano–Naito’s and Philos’ approaches.

PHASE-PLANE STUDY USING WOLFRAM MATHEMATICA

2018 ◽  
pp. 149-153
Author(s):  
Krum Videnov
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Nonlinear Differential Equation ◽  
Phase Plane ◽  
Plane Method ◽  
First Order ◽  
Specialized Software ◽  
Wolfram Mathematica ◽  
Phase Plane Method

In this paper, the capabilities of the specialized software Wolfram Mathematica for investigating processes described with differential equations are discussed. The aim is to create procedures and algorithms in Mathematica environment for study and analysis of systems and processes using the Phase-plane method. The proposed algorithm has been experimented to evaluate a nonlinear differential equation of first order.

DYNAMICS OF NUMERICAL METHODS FOR COSYMMETRIC ORDINARY DIFFERENTIAL EQUATIONS

2001 ◽  
Vol 11 (09) ◽  
pp. 2339-2357 ◽  
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.

Generalization of the Krylov-Bogoliubov Method for Nonlinear Oscillators

2015 ◽  
Vol 801 ◽  
pp. 3-11 ◽  
Author(s):  
Livija Cveticanin
Keyword(s):  
Approximate Solution ◽  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Nonlinear Oscillators ◽  
Original Method ◽  
Strong Nonlinearity ◽  
First Order ◽  
Numerical Example ◽  
The Difference ◽  
First Order Differential Equations

In this paper the Krylov-Bogoliubov method for solving nonlinear oscillators is considered. Based on the original method, developed for the oscillator with small nonlinearity, a generalization is made to oscillators with strong nonlinearity. After rewriting the equation into two first order differential equations, the averaging procedure is introduced. Truly nonlinear differential equations are investigated where the linear term does not exist nor the linearization of the equation is possible. Solution is assumed in the form of the Ateb-function. After averaging the approximate solution for the oscillator is obtained. A numerical example is tested. It is shown that the difference between the analytical approximate solution and the exact numerical one is negligible.

scholarly journals Stability analysis for first-order nonlinear differential equations with three-point boundary conditions

2020 ◽  
Vol 2020 (1) ◽  
pp. 40-52
Author(s):  
Kamala E. Ismayilova
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Nonlinear Differential Equations ◽  
Point Boundary ◽  
Uniqueness Of Solutions ◽  
First Order ◽  
Fixed Point Principle ◽  
The Stability ◽  
The Given ◽  
Stability Results

AbstractIn the present paper, we study a system of nonlinear differential equations with three-point boundary conditions. The given original problem is reduced to the equivalent integral equations using Green function. Several theorems are proved concerning the existence and uniqueness of solutions to the boundary value problems for the first order nonlinear system of ordinary differential equations with three-point boundary conditions. The uniqueness theorem is proved by Banach fixed point principle, and the existence theorem is based on Schafer’s theorem. Then, we describe different types of Ulam stability: Ulam-Hyers stability, generalized Ulam-Hyers stability. We discuss the stability results providing suitable example.

scholarly journals Natural Decomposition Approximation Solution for First Order Nonlinear Differential Equations

2018 ◽  
Vol 7 (4.5) ◽  
pp. 442
Author(s):  
A. Patra ◽  
T. T. Shone ◽  
B. B. Mishra
Keyword(s):  
Differential Equations ◽  
Decomposition Method ◽  
Nonlinear Differential Equations ◽  
Adomian Decomposition Method ◽  
Approximation Solution ◽  
Transform Method ◽  
First Order ◽  
Adomian Decomposition ◽  
Natural Decomposition ◽  
First Order Differential Equations

In this research paper, we propose the Natural decomposition method (NDM) to solve nonlinear first order differential equations. We compare the results obtained by NDM with the exact solutions. This method is a combination of the natural transform method and adomian decomposition method. By showing the less error one can be concluded that the NDM is easy to use and efficient.  

scholarly journals Nonlinear differential equations and algebraic systems

1978 ◽  
Vol 1 (3) ◽  
pp. 257-267
Author(s):  
Lloyd K. Williams
Keyword(s):  
Partial Differential Equations ◽  
General Solution ◽  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Asymptotic Series ◽  
Partial Differential ◽  
First Order ◽  
Variation Of Parameters ◽  
Familiar Expression ◽  
First Order Differential Equations

In this paper we obtain the general solution of scalar, first-order differential equations. The method is variation of parameters with asymptotic series and the theory of partial differential equations.The result gives us a form like a differential quotient requiring only that a limit be taken. Like the familiar expression for the solution of linear, first order, ordinary equations, it is the same in all cases.

Dynamics of Temperatures in Thermally-Coupled, Heated Rooms With PI Control

2007 ◽  
Author(s):  
Frederick W. Thwaites ◽  
Mihir Sen
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Coupled System ◽  
Time Dependent ◽  
Pi Control ◽  
Thermal Coupling ◽  
First Order ◽  
Dependent Behavior ◽  
The Stability ◽  
First Order Differential Equations

The purpose of this study is to analyze the behavior of a set of thermally-controlled rooms arranged in the form of a ring. Each room is heated and can exchange heat with its neighbors as well as with the environment. The heater in each room is PI controlled. A lumped capacitance approximation is used for the rooms leading to a system of first-order differential equations. Numerical methods are used to determine the time-dependent behavior of the coupled system. The linear stability of the system is analyzed for various parameters. The stability is found to be independent of the strength of the thermal coupling between rooms.

A model of neuronal bursting using three coupled first order differential equations

1984 ◽  
Vol 221 (1222) ◽  
pp. 87-102 ◽  
Keyword(s):  
Phase Plane ◽  
Stable Equilibrium ◽  
Equilibrium Points ◽  
Model Neuron ◽  
Recent Model ◽  
The Other ◽  
Stable Equilibrium Point ◽  
General Phenomenon ◽  
First Order ◽  
First Order Differential Equations

We describe a modification to our recent model of the action potential which introduces two additional equilibrium points. By using stability analysis we show that one of these equilibrium points is a saddle point from which there are two separatrices which divide the phase plane into two regions. In one region all phase paths approach a limit cycle and in the other all phase paths approach a stable equilibrium point. A consequence of this is that a short depolarizing current pulse will change an initially silent model neuron into one that fires repetitively. Addition of a third equation limits this firing to either an isolated burst or a depolarizing afterpotential. When steady depolarizing current was applied to this model it resulted in periodic bursting. The equations, which were initially developed to explain isolated triggered bursts, therefore provide one of the simplest models of the more general phenomenon of oscillatory burst discharge.

Jacobi stability analysis of modified Chua circuit system

2017 ◽  
Vol 14 (06) ◽  
pp. 1750089 ◽  
Author(s):  
M. K. Gupta ◽  
C. K. Yadav
Keyword(s):  
Nonlinear Dynamics ◽  
Stability Analysis ◽  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Equilibrium Points ◽  
Second Order ◽  
Geometric Properties ◽  
Chua Circuit ◽  
The Stability ◽  
Deviation Vector

In this paper, we analyze the nonlinear dynamics of the modified Chua circuit system from the viewpoint of Kosambi–Cartan–Chern (KCC) theory. We reformulate the modified Chua circuit system as a set of two second-order nonlinear differential equations and obtain five KCC-invariants which express the intrinsic geometric properties. The deviation tensor and its eigenvalues are obtained, that determine the stability of the system. We also obtain the condition for Jacobi stability and discuss the behavior of deviation vector near equilibrium points.

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