Polynomial Differential Systems Research Articles

Global centers of a class of cubic polynomial differential systems

Global centers of a class of cubic polynomial differential systems

Global centres in a class of quintic polynomial differential systems

A centre of a differential system in the plane $ {\mathbb {R}}^2$ is an equilibrium point $p$ having a neighbourhood $U$ such that $U\setminus \{p\}$ is filled with periodic orbits. A centre $p$ is global when $ {\mathbb {R}}^2\setminus \{p\}$ is filled with periodic orbits. In general, it is a difficult problem to distinguish the centres from the foci for a given class of differential systems, and also it is difficult to distinguish the global centres inside the centres. The goal of this paper is to classify the centres and the global centres of the following class of quintic polynomial differential systems \begin{align*} \dot{x}= y,\quad \dot{y}={-}x+a_{05}\,y^5+a_{14}\,x\,y^4+a_{23}\,x^2\,y^3+a_{32}\,x^3\,y^2+a_{41}\,x^4\,y+a_{50}\,x^5, \end{align*} in the plane $ {\mathbb {R}}^2$ .

Characterization of the Riccati and Abel Polynomial Differential Systems Having Invariant Algebraic Curves

The Riccati polynomial differential systems are differential systems of the form [Formula: see text], [Formula: see text], where [Formula: see text] and [Formula: see text] for [Formula: see text] are polynomial functions. We characterize all the Riccati polynomial differential systems having an invariant algebraic curve. We show that the coefficients of the first four highest degree terms of the polynomial in the variable [Formula: see text] defining the invariant algebraic curve determine completely the Riccati differential system. A similar result is obtained for any Abel polynomial differential system.

Global centers of a family of cubic systems

AbstractConsider the family of polynomial differential systems of degree 3, or simply cubic systems $$ x' = y, \quad y' = -x + a_1 x^2 + a_2 xy + a_3 y^2 + a_4 x^3 + a_5 x^2 y + a_6 xy^2 + a_7 y^3, $$ x ′ = y , y ′ = - x + a 1 x 2 + a 2 x y + a 3 y 2 + a 4 x 3 + a 5 x 2 y + a 6 x y 2 + a 7 y 3 , in the plane $$\mathbb {R}^2$$ R 2 . An equilibrium point $$(x_0,y_0)$$ ( x 0 , y 0 ) of a planar differential system is a center if there is a neighborhood $$\mathcal {U}$$ U of $$(x_0,y_0)$$ ( x 0 , y 0 ) such that $$\mathcal {U} \backslash \{(x_0,y_0)\}$$ U \ { ( x 0 , y 0 ) } is filled with periodic orbits. When $$\mathbb {R}^2\setminus \{(x_0,y_0)\}$$ R 2 \ { ( x 0 , y 0 ) } is filled with periodic orbits, then the center is a global center. For this family of cubic systems Lloyd and Pearson characterized in Lloyd and Pearson (Comput Math Appl 60:2797–2805, 2010) when the origin of coordinates is a center. We classify which of these centers are global centers.

Solution of the Center Problem for a Class of Polynomial Differential Systems

Solution of the Center Problem for a Class of Polynomial Differential Systems

Limit cycles of septic polynomial differential systems bifurcating from the periodic annulus of cubic center

This paper focuses on investigating the maximum number of limit cycles bifurcating from the periodic orbits adapted to the cubic system given by ẋ=y−yx+a2,ẏ=−x+xx+a2,where a is a positive number with a≠1. The study specifically examines the perturbation of this system within the class of all septic polynomial differential systems. Our main result demonstrates that the first-order averaging theory associated with the perturbed system yields a maximum of twenty-two limit cycles.

Melnikov analysis of a perturbed integrable non-Hamiltonian system

This paper provides a completed Melnikov analysis of any order for a class of perturbed polynomial differential systems, where the unperturbed system is a cubic center with a straight line of singular points. The emphasis on any order Melnikov function is crucial because of two major reasons: one is that higher order Melnikov functions are not in general expressible explicitly, the other one is the essential difficulty of computing elliptic integrals due to the complexity of many iterations. The exact upper bounds of the number of limit cycles bifurcated from the period annulus under different polynomial perturbations are obtained by analyzing the algebraic structures from explicit expressions of any order Melnikov functions.

Dynamics of the Isotropic Star Differential System from the Mathematical and Physical Point of Views

The following differential quadratic polynomial differential system  dxdt=y−x, dydt=2y−yγ−12−γy−5γ−4γ−1x, when the parameter γ∈(1,2] models the structure equations of an isotropic star having a linear barotropic equation of state, being x=m(r)/r where m(r)≥0 is the mass inside the sphere of radius r of the star, y=4πr2ρ where ρ is the density of the star, and t=ln(r/R) where R is the radius of the star. First, we classify all the topologically non-equivalent phase portraits in the Poincaré disc of these quadratic polynomial differential systems for all values of the parameter γ∈R∖{1}. Second, using the information of the different phase portraits obtained we classify the possible limit values of m(r)/r and 4πr2ρ of an isotropic star when r decreases.

Configuration of planar Kolmogorov cubic polynomial differential systems with the most centers

Configuration of planar Kolmogorov cubic polynomial differential systems with the most centers

A weakened Markus–Yamabe condition for planar polynomial differential systems of degree 

AbstractFor a general autonomous planar polynomial differential system, it is difficult to find conditions that are easy to verify and which guarantee global asymptotic stability, weakening the Markus–Yamabe condition. In this paper, we provide three conditions that guarantee the global asymptotic stability for polynomial differential systems of the form $x^{\prime}=f_1(x,y)$, $y^{\prime}=f_2(x,y)$, where f1 has degree one, f2 has degree $n\ge 1$ and has degree one in the variable y. As a consequence, we provide sufficient conditions, weaker than the Markus–Yamabe conditions that guarantee the global asymptotic stability for any generalized Liénard polynomial differential system of the form $x^{\prime}=y$, $y^{\prime}=g_1(x) +y g_2(x)$ with g1 and g2 polynomials of degrees n and m, respectively.

An analytical approach of applying the Lyapunov direct method to polynomial differential systems with discrete time delays

The Lyapunov direct method is a common and effective tool for discussing the global stability of dynamical systems. In this paper, we propose an analytical approach to apply this method for the global stability of the positive equilibrium of a polynomial differential system with discrete time delays, which includes how to construct an applicable Lyapunov functional and to verify the negative (semi-)definiteness of its derivative. Here the linear combination of the Volterra-type functions/functionals and their integrals plays an important role. Moreover, when the delayed system also has a boundary equilibrium, we show that the Lyapunov functional for the global stability of the positive equilibrium can be reformulated to establish the global stability of the boundary equilibrium. As an illustration, we apply this approach to a virus infection model with two delays.

Limit cycles of polynomial differential systems of degree 1 and 2 on the cylinder

Limit cycles of polynomial differential systems of degree 1 and 2 on the cylinder

Polynomial differential systems with hyperbolic limit cycles

One of the main problems of the qualitative theory of the planar differential equations is the study their limit cycles. In this paper given an algebraic curve of degree n we characterize the planar polynomial differential systems of degree greater or equal than n which admit the ovals of the algebraic curve as hyperbolic limit cycles.

Dynamics of the polynomial differential systems

In general, there are several free parameters. By using a method introduced in a previous paper,  we obtain a sequence of algebraic a proximations to the bifurcationsets,in the parameter space

Phase Portraits of Families VII and VIII of the Quadratic Systems

The quadratic polynomial differential systems in a plane are the easiest nonlinear differential systems. They have been studied intensively due to their nonlinearity and the large number of applications. These systems can be classified into ten classes. Here, we provide all topologically different phase portraits in the Poincaré disc of two of these classes.

Some families of quadratic systems with at most one limit cycle

The work of Chicone and Shafer published in 1982 together with the work of Bamon published in 1986 proved that any polynomial differential system of degree two has finitely many limit cycles. But the problem remains open of providing a uniform upper bound for the maximum number of limit cycles that a polynomial differential system of degree two can have, i.e. the second part of the 16th Hilbert problem restricted to the polynomial differential systems of degree two remains open. Here we present six subclasses of polynomial differential systems of degree two for which we can prove that an upper bound for their maximum number of limit cycles is one.

Symmetric Phase Portraits of Homogeneous Polynomial Hamiltonian Systems of Degree 1, 2, 3, 4, and 5 with Finitely Many Equilibria

Roughly speaking, the Poincaré disc D2 is the closed disc centered at the origin of the coordinates of R2, where the whole of R2 is identified with the interior of D2 and the circle of the boundary of D2 is identified with the infinity of R2, because in the plane R2, we can go to infinity in as many directions as points have the circle. The phase portraits of the quadratic Hamiltonian systems in the Poincaré disc were classified in 1994. Since then, no new interesting classes of Hamiltonian systems have been classified on the Poincaré disc. In this paper, we determine the phase portraits in the Poincaré disc of five classes of homogeneous Hamiltonian polynomial differential systems of degrees 1, 2, 3, 4, and 5 with finitely many equilibria. Moreover, all these phase portraits are symmetric with respect to the origin of coordinates. We showed that these polynomial differential systems exhibit precisely 2, 2, 3, 3, and 4 topologically distinct phase portraits in the Poincaré disc. Of course, the new results are for the homogeneous Hamiltonian polynomial differential systems of degrees 3, 4, and 5. The tools used here for obtaining these phase portraits also work for obtaining any phase portrait of a homogeneous Hamiltonian polynomial differential system of arbitrary degree.

Limit cycles obtained by perturbing a degenerate center

This paper deals with the maximum number of limit cycles bifurcating from the degenerate centre \[ \dot{x}=-y(3x^2+y^2),\: \dot{y}=x(x^2-y^2), \] when we perturb it inside a class of all homogeneous polynomial differential systems of degree \(5\). Using averaging theory of second order, we show that, at most, five limit cycles are produced from the periodic orbits surrounding the degenerate centre under quintic perturbation. In addition, we provide six examples that give rise to exactly \(5, 4, 3, 2, 1\) and \(0\) limit cycles.

On the Local Stability of a Certain Class of Polynomial Differential System

We state and prove a condition for the local stability of a certain class of two dimensional system of polynomial differential equations. We give some examples of polynomial differential systems of equations to demonstrate that this local stability condition established for the trivial equilibrium point (0,0) is quite sharp and compare our result with the well known Lyapunov local stability criterion (Lyapunov's second method).

Phase portraits of two classes of quadratic differential systems exhibiting as solutions two cubic algebraic curves

Abstract The classification of the phase portraits is one of the classical and difficult problems in the qualitative theory of polynomial differential systems in R 2 {{\mathbb{R}}}^{2} , particularly for quadratic systems. Even with the hundreds of studies on the topology of real planar quadratic vector fields, fully characterizing their phase portraits is still a difficult problem. This paper is devoted to classifying the phase portraits of two polynomial vector fields with two usual invariant algebraic curves, by investigating the geometric solutions within the Poincaré disc. One can notice that these systems yield 26 topologically different phase portraits.