Two-dimensional System Of Differential Equations Research Articles

Dynamics of a Reduced System Connected to the Investigation of an Infinite Network of Identical Theta Neurons

We consider an infinite network of identical theta neurons, all-to-all coupled by instantaneous synapses. Using the Watanabe–Strogatz Ansatz, the mathematical model of this infinite network is reduced to a two-dimensional system of differential equations. We determine the number of equilibria of this reduced system with respect to two characteristic parameters. Furthermore, we discuss the stability properties of each equilibrium and the possible bifurcations that may take place. As a result, the occurrence of exotic higher codimension bifurcations involving a degenerate center is also unveiled. Numerical results are also presented to illustrate complex dynamic behaviour in the reduced system.

Incommensurate conformable-type three-dimensional Lotka–Volterra model: discretization, stability, and bifurcation

The classic Lotka–Volterra model is a two-dimensional system of differential equations used to model population dynamics among two-species: a predator and its prey. In this article, we consider a modified three-dimensional fractional-order Lotka–Volterra system that models population dynamics among three-species: a predator, an omnivore and their mutual prey. Biologically speaking, population models with a discrete and continuous structure often provide richer dynamics than either discrete or continuous models, so we first discretize the model while keeping one time-continuous dependent variable in each equation. Then, we analyze the stability and bifurcation near the equilibria. The results demonstrated that the dynamic behaviors of the discretized model are sensitive to the fractional-order parameter and discretization parameter. Finally, numerical simulations are performed to explain and validate the findings, and the maximum Lyapunov exponents is computed to confirm the presence of chaotic behavior in the studied model.

Student Group Dynamic Model Based on Understanding in Mathematics Subjects

This study discusses the interaction of students with a mathematical modeling point of view. This interaction involves students who understand and do not understand mathematics subject matter. The interaction process between groups is modeled in a two-dimensional system of differential equations. Variable A is the percentage of students who understand the material, and variable B is the percentage of students who do not understand the material. The dynamic analysis results obtained by one trivial equilibrium point and three non-trivial equilibrium points exist with several conditions. Based on the stability analysis of the non-trivial equilibrium point, it is found that the conditions without students do not understand mathematics subject matter. This condition is the goal of this study, which involves interaction between students; it can increase the learning process's success.

Conservation Strategy for Palm Groves: Optimal Chemical Control Model for Red Palm Weevil, Rhynchophorus ferrugineus

Rhynchophorus ferrugineus (Olivier, 1790) is an invasive pest species that constitutes one of the most important problems around the Mediterranean region and has been responsible for the loss of over 100,000 palm trees with an estimated annual cost of EUR several hundred million since its introduction into Europe. Methodological approaches of conservation ecology, such as multidisciplinary modelling, also apply in the management of cultural landscapes concerning ornamental plants, such as palm trees of the area. In this paper, we propose a dynamic model for the control of the red palm weevil, contributing in this way to the sustainability of an existing cultural landscape. The primary data set collected is a sample from the density-time function of a two-cohort pest population. This data set suggests a bimodal analytic description. If, from this data set, we calculate a sample from the accumulated density-time function (the integral of the density-time function), it displays a double sigmoid function (with two inflections). A good candidate for the analytical description of the latter is the sum of two logistic functions. As for the dynamic description of the process, a two-dimensional system of differential equations can be obtained, where the solution’s second component provides the analytical description of the original density-time function for the two-cohort population. Since the two-cohort waves appear in all three cycle stages, this reasoning applies to the subpopulations of larvae, pupae and adults. The model fitting is always performed using the SimFit package. On this basis, a mathematical model is proposed, which is sufficiently versatile to be of help in the control of this pest species in other geographical areas.

Global stability and periodicity in a glucose-insulin regulation model with a single delay

A two-dimensional system of differential equations with delay modelling the glucose-insulin interaction processes in the human body is considered. Sufficient conditions are derived for the unique positive equilibrium in the system to be globally asymptotically stable. They are given in terms of the global attractivity of the fixed point in a limiting interval map. The existence of slowly oscillating periodic solutions is shown in the case when the equilibrium is unstable. The mathematical results are supported by extensive numerical simulations. It is deduced that typical behaviour in the system is the convergence to either a stable periodic solution or to the unique stable equilibrium. The coexistence of several periodic solutions together with the stable equilibrium is demonstrated as a possibility.

Dulac-Cherkas Test for Determining the Exact Number of Limit Cycles of Autonomous Systems on the Cylinder

For real autonomous two-dimensional systems of differential equations with continuously differentiable right-hand sides 2π-periodic in one of the variables, we consider the problem of determining the exact number of limit cycles of the second kind on the cylinder. If the system has no equilibria, then we propose to solve this problem by two methods based on a successive two-step application of the Dulac-Cherkas test or one of its modifications, which permits determining closed transversal curves dividing the cylinder into subdomains surrounding it and such that the system has precisely one limit cycle of the second kind in each of them. The efficiency of the proposed methods is illustrated with examples of systems corresponding to the Abel equation, for which the number of limit cycles on the whole phase cylinder is established.

On the Aizerman Problem for Systems of Two Differential Equations

The stability of equilibria of systems of nonlinear ordinary differential equations is studied. Acriterion for the reducibility of a second-order linear systemto a scalar differential equation is given. Both positive definite and semidefinite Lyapunov functions are used to obtain sufficient conditions for the asymptotic stability (global stability) of second-order nonlinear differential equations. It is proved that the Aizerman problem has a positive solution with respect to the roots of the characteristic equation of two-dimensional systems of differential equations.

Stable Periodic Solutions of Periodic Systems of Differential Equations

An infinitely differentiable periodic two-dimensional system of differential equations is considered. It is assumed that there is a hyperbolic periodic solution and there exists a homoclinic solution to the periodic solution. It is shown that, for a certain type of tangency of the stable manifold and unstable manifold, any neighborhood of the nontransversal homoclinic solution contains a countable set of stable periodic solutions such that their characteristic exponents are separated from zero.

О СТАБИЛИЗАЦИИ ДИФФЕРЕНЦИАЛЬНЫХ ГИБРИДНЫХ УПРАВЛЯЕМЫХ СИСТЕМ С ОБРАТНОЙ СВЯЗЬЮ

In this paper two-dimensional systems of differential equations are considered together with their stabilization by a hybrid feedback control. A stabilizing hybrid control for an arbitrary controlled system that belongs to a certain category within two-dimensional systems is constructed as a result of this study and some stabilization proprieties of the system with the obtained hybrid control are presented.

Approximations of the Homoclinic Orbits Near a Double-Zero Bifurcation with Symmetry of Order Two

The two-dimensional system of differential equations corresponding to the normal form of the double-zero bifurcation with symmetry of order two is considered. This is a codimension two bifurcation. The associated dynamical system exhibits, among others, a homoclinic bifurcation. In this paper, we obtain second order approximations both for the curve of parametric values of homoclinic bifurcation and for the homoclinic orbits. To perform this task, we reduce first the normal form to a perturbed Hamiltonian system, using a blow-up technique. Then, by means of a perturbation method, we determine explicit first and second order approximations of the homoclinic orbits. The solutions obtained theoretically are compared with those obtained numerically for several cases. Finally, an application of the obtained results is presented.

The Approximate Solution of Some Plane Boundary Value Problems of the Moment Theory of Elasticity

We consider a two-dimensional system of differential equations of the moment theory of elasticity. The general solution of this system is represented by two arbitrary harmonic functions and solution of the Helmholtz equation. Based on the general solution, an algorithm of constructing approximate solutions of boundary value problems is developed. Using the proposed method, the approximate solutions of some problems on stress concentration on the contours of holes are constructed. The values of stress concentration coefficients obtained in the case of moment elasticity and for the classical elastic medium are compared. In the final part of the paper, we construct the approximate solution of a nonlocal problem whose exact solution is already known and compare our approximate solution with the exact one. Supposedly, the proposed method makes it possible to construct approximate solutions of quite a wide class of boundary value problems.

Analytical Homoclinic Solution of a Two-Dimensional Nonlinear System of Differential Equations

Analytical solution of the homoclinic orbit of a two-dimensional system of differential equations that describes the Hamiltonian part of the slow flow of a three-degree-of-freedom dissipative system of linear coupled oscillators with an essentially nonlinear attachment is described.

Minimal sets in monotone and concave skew-product semiflows II: Two-dimensional systems of differential equations

Recurrent nonautonomous two-dimensional systems of differential equations of ordinary, finite delay and reaction–diffusion types given by cooperative and concave vector fields define monotone and concave skew-product semiflows, whose dynamics is analyzed in this paper. A complete description of all the minimal sets in a very interesting dynamical situation is provided, and some criteria to determine its possible existence and the area on which they lie are given. Suitable examples prove the optimality of the results.

Minimal sets in monotone and sublinear skew-product semiflows II: Two-dimensional systems of differential equations

Monotone and sublinear skew-product semiflows defined from recurrent non-autonomous differential equations of ordinary, finite-delay and reaction–diffusion types are analyzed. A complete description of all the minimal sets is provided, and the optimality of the results is shown by means of suitable examples. Some significative differences with the autonomous case are established.

Linearizability conditions for a cubic system

We obtain the necessary and sufficient conditions for linearizability of an eight-parameter family of two-dimensional system of differential equations in the form of linear canonical saddle perturbed by polynomials with four quadratic and four cubic terms.

Parallel navigation for reaching a moving goal by a mobile robot

In this paper, we present a method for robot navigation toward a moving object with unknown maneuvers. Our strategy is based on the integration of the robot and the target kinematics equations with geometric rules. The tracking problem is modeled in polar coordinates using a two-dimensional system of differential equations. The control law is then derived based on this model. Our approach consists of a rendezvous course, which means that the robot reaches the moving goal without following its path. In the presence of obstacles, two navigation modes are integrated, namely the tracking and the obstacle-avoidance modes. To confirm our theoretical results, the navigation strategy is illustrated using an extensive simulation for different scenarios.

A rheological model for the description of boulder impacts on granular strata

This paper considers the mechanics of rock boulders impacting on granular strata. The study is aimed at improving the numerical methods devoted both to describing boulder trajectories along slopes and designing preventive structures in mountain regions. The problem is analysed by using a simplified approach inspired by the lumped mass method. This is based on (a) the macro-element concept, (b) the definition of generalised stresses and displacements, and (c) delayed plasticity theory. Both vertical and inclined impacts on horizontal strata and vertical impacts on inclined slopes are discussed. The numerical results are obtained by using a finite difference numerical discretisation to integrate, over time, a coupled system of two-dimensional differential equations. Most of the input data requested by the model are basic and geotechnically meaningful, and comparison of the numerical results with the experimental data seems to be quite promising.

A rheological model for the description of boulder impacts on granular strata

This paper considers the mechanics of rock boulders impacting on granular strata. The study is aimed at improving the numerical methods devoted both to describing boulder trajectories along slopes and designing preventive structures in mountain regions. The problem is analysed by using a simplified approach inspired by the lumped mass method. This is based on (a) the macro-element concept, (b) the definition of generalised stresses and displacements, and (c) delayed plasticity theory. Both vertical and inclined impacts on horizontal strata and vertical impacts on inclined slopes are discussed. The numerical results are obtained by using a finite difference numerical discretisation to integrate, over time, a coupled system of two-dimensional differential equations. Most of the input data requested by the model are basic and geotechnically meaningful, and comparison of the numerical results with the experimental data seems to be quite promising.

Delay in decision making causes oscillation

In this paper, a mathematical model will be studied and developed that describes the spread of information in a human population. This model, consisting of a two-dimensional system of differential equations, has been motivated by attempts to explain the behaviour of individuals (students) who may choose between two alternatives with little and more risk. We show that if there is no delay in the decision making of the individuals the system has a globally asymptotically stable equilibrium point. Then we assume that after having received some communication there is a delay in decision making. This delay is regarded as the lag due to a meditative attitude. The most important observation is that as the delay is increased the originally asymptotic stable interior equilibrium loses its stability and at a certain critical value a Hopf bifurcation takes place: a small amplitude periodic solution arises.

Forced Oscillations in a Two-Dimensional Cone

A two-dimensional system of differential equations in a cone and a sufficient condition for the existence of its periodic solution are investigated. Such problems are encountered in the study of forced oscillations in electrical circuits containing diodes.