Polynomial Differential Systems Of Degree Research Articles

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.

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.

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

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.

Phase portraits of some polynomial differential systems with maximal multiplicity of the line at the infinity

The present study delves into the investigation of phase portraits of polynomial differential systems, which are systems of differential equations of the form $\frac{dx}{dt} = P(x,y), \frac{dy}{dt} = Q(x,y)$, where $x$ and $y$ are the dependent variables and $t$ is the independent variable. The functions $P(x,y)$ and $Q(x,y)$ are polynomials in $x$ and $y$. The main objective of this research is to obtain the phase portraits of polynomial differential systems of degree $n\in \{ 3, 4, 5\}$ and having an invariant straight line at the infinity of maximal multiplicity.

On Reeb components of nonsingular polynomial differential systems on the real plane

Denote by r(n) the maximal number of Reeb components that a nonsingular polynomial differential system of degree n on the real plane can have. It is known that r(0)=r(1)=0, r(2)=r(3)=2 and n−1≤r(n)≤n for n≥4. In this paper we prove that r(n)=n for all n≥4. This completely solves, for nonsingular systems, a problem posed by Chicone and Tian in 1982. We prove our result by explicitly constructing nonsingular Hamiltonian systems of degree n presenting n Reeb components for any n≥4.

Fifteen Limit Cycles Bifurcating from a Perturbed Cubic Center

In this work, we study the bifurcation of limit cycles from the period annulus surrounding the origin of a class of cubic polynomial differential systems; when they are perturbed inside the class of all polynomial differential systems of degree six, we obtain at most fifteenth limit cycles by using the averaging theory of first order.

The Number of Limit Cycles Bifurcating from a Quadratic Reversible Center

By using the first order Melnikov function method with multiple parameters presented in [Han & Xiong, 2014], we prove that [Formula: see text] limit cycles can bifurcate from the quadratic reversible center [Formula: see text] under [Formula: see text]th degree polynomial perturbations for [Formula: see text]. Our result in this paper improves the existing lower bound on the maximal number of limit cycles bifurcating from quadratic reversible centers inside the polynomial differential systems of degree [Formula: see text] which is 4 (resp., [Formula: see text]) when [Formula: see text] (resp., [Formula: see text]).

On a class of polynomial differential systems of degree 4: phase portraits and limit cycles

In this paper we characterize the phase portraits in the Poincaré disc of the class of polynomial differential systems of the form \begin{equation*} \dot{x}=-y,\qquad \dot{y} =x+ax^{4}+bx^{2}y^{2}+cy^{4}, \end{equation*} with $a^2+b^2+c^2\neq0$, which are symmetric with respect to the $x$-axis. Such systems have a center at the origin of coordinates. Moreover, using the averaging theory of five order, we study the number of limit cycles which can bifurcate from the period annulus of this center when it is perturbed inside the class of all polynomial differential systems of degree $4$.

Dynamics and Darboux Integrability of the D2 Polynomial Vector Fields of Degree 2 in $\mathbb {R}^{3}$

We characterize the Darboux integrability and the global dynamics of the 3-dimensional polynomial differential systems of degree 2 which are invariant under the D2 symmetric group, which have been partially studied previously by several authors.

Configurations of the Topological Indices of the Planar Polynomial Differential Systems of Degree (2, m)

Using the Euler–Jacobi formula there is a relation between the singular points of a polynomial vector field and their topological indices. Using this formula we obtain the configuration of the singular points together with their topological indices for the polynomial differential systems \(\dot{x}=P(x,y)\), \(\dot{y} =Q(x,y)\) with degree of P equal to 2 and degree of Q equal to m when these systems have 2m finite singular points.

Linear type global centers of cubic Hamiltonian systems symmetric with respect to the x-axis

A polynomial differential system of degree 2 has no global centers (that is, centers defined in all the plane except the fixed point). In this paper we characterize the global centers of cubic Hamiltonian systems symmetric with respect to the x-axis, and such that the center has purely imaginary eigenvalues.
 For more information see https://ejde.math.txstate.edu/Volumes/2020/57/abstr.html

New lower bounds for the maximal number of inseparable leaves of nonsingular polynomial foliations of the plane

Let F be a nonsingular polynomial differential system of degree n on the real plane and denote by s(n) the maximal number of inseparable leaves that such a system can have. In this paper we prove that s(n) is at least 2n−1 for all n≥4. This improves the known lower bounds for s(n), which are 2n−4 if n≥7 or n=5, and respectively 6 and 9 if n=4 and n=6. Since it is also known that s(n)≤2n for all n≥4 and that s(0)=s(1)=0 and s(2)=s(3)=3, the problem of determining s(n) for all n is now almost solved: any improvement in lower or upper bounds will actually find the exact s(n). Our lower bounds for s(n) are attained in the class of Hamiltonian systems.

Picard–Fuchs Equation Applied to Quadratic Isochronous Systems with Two Switching Lines

The present paper is devoted to study the problem of limit cycle bifurcations for nonsmooth integrable differential systems with two perpendicular switching lines. By using the Picard–Fuchs equation, we obtain the upper bounds of the number of limit cycles bifurcating from the period annuli of the quadratic isochronous systems [Formula: see text], [Formula: see text] and [Formula: see text], [Formula: see text], when they are perturbed inside a class of all discontinuous polynomial differential systems of degree [Formula: see text]. This method can be applied to study the limit cycle bifurcations of other integrable differential systems.

The centers and their cyclicity for a class of polynomial differential systems of degree 7

We classify the global phase portraits in the Poincaré disc of the generalized Kukles systems ẋ=−y,ẏ=x+axy6+bx3y4+cx5y2+dx7,which are symmetric with respect to both axes of coordinates. Moreover using the averaging theory up to sixth order, we study the cyclicity of the center located at the origin of coordinates, i.e. how many limit cycles can bifurcate from the origin of coordinates of the previous differential system when we perturb it inside the class of all polynomial differential systems of degree 7.

Linear type global centers of linear systems with cubic homogeneous nonlinearities

A center p of a differential system in $$\mathbb {R}^2$$ is global if $$\mathbb {R}^2 \setminus \{p\}$$ is filled of periodic orbits. It is known that a polynomial differential system of degree 2 has no global centers. Here we characterize the global centers of the differential systems $$\begin{aligned} {\dot{x}} = ax + b y +P_3(x,y), \quad {\dot{y}} = c x + d y + Q_3(x,y), \end{aligned}$$ with $$P_3$$ and $$Q_3$$ homogeneous polynomials of degree 3, and such that the center has purely imaginary eigenvalues, i.e. a linear type center.

The Poincaré center‐focus problem for a class of higher order polynomial differential systems

In this paper, I have proved that for a class of polynomial differential systems of degree n + 1 (where n is an arbitrary positive integer), the composition conjecture is true. I give the sufficient and necessary conditions for these differential systems to have a center at origin point by using a different method from the previous references. By this, I can obtain all the focal values of these systems for an arbitrary n, and their expressions are succinct and beautiful. I believe that the idea and method of this article can be used to solve the center‐focus problem of more high‐order polynomial differential systems.

Periodic Orbits Bifurcating from a Nonisolated Zero–Hopf Equilibrium of Three-Dimensional Differential Systems Revisited

In this paper, we study the periodic solutions bifurcating from a nonisolated zero–Hopf equilibrium in a polynomial differential system of degree two in [Formula: see text]. More specifically, we use recent results of averaging theory to improve the conditions for the existence of one or two periodic solutions bifurcating from such a zero–Hopf equilibrium. This new result is applied for studying the periodic solutions of differential systems in [Formula: see text] having [Formula: see text]-scroll chaotic attractors.