Polynomial Differential Systems Research Articles

New Family of Centers of Planar Polynomial Differential Systems of Arbitrary Even Degree

The problem of distinguishing between a focus and a center is one of the classical problems in the qualitative theory of planar differential systems. In this paper, we provide a new family of centers of polynomial differential systems of arbitrary even degree. Moreover, we classify the global phase portraits in the Poincare disc of the centers of this family having degree 2, 4, and 6.

Phase portraits of integrable quadratic systems with an invariant parabola and an invariant straight line

We classify the phase portraits of the quadratic polynomial differential systems having an invariant parabola, an invariant straight line, and a Darboux first integral produced by these two invariant curves.

Planar Semi-quasi Homogeneous Polynomial Differential Systems with a Given Degree

This paper study the planar semi-quasi homogeneous polynomial differential systems (PSQHPDS), which can be regarded as a generalization of semi-homogeneous and of quasi-homogeneous systems. By using the algebraic skills, several important properties of PSQHPDS are derived and are employed to establish an algorithm for obtaining all the explicit expressions of PSQHPDS with a given degree. As an application of this algorithm, we research the center problem of quadratic and cubic PSQHPDS. The nonexistence of the quadratic center is proved and the canonical form of cubic center is found.

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.

Limit Cycles for a Discontinuous Quintic Polynomial Differential System

In this article, we study the maximum number of limit cycles for a discontinuous quintic differential system. Using the first-order averaging method, we explain how limit cycles can bifurcate from the period annulus around the center of the considered system when it is perturbed inside a class of discontinuous quintic polynomial differential systems. Our results show that the lower bound and the upper bound of the number of limit cycles, 8 and 10 respectively, that can bifurcate from the period annulus around the center.

LIMIT CYCLES FOR TWO CLASSES OF PLANAR POLYNOMIAL DIFFERENTIAL SYSTEMS WITH UNIFORM ISOCHRONOUS CENTERS

In this article, we study the maximum number of limit cycles for two classes of planar polynomial differential systems with uniform isochronous centers. Using the first-order averaging method, we analyze how many limit cycles can bifurcate from the period solutions surrounding the centers of the considered systems when they are perturbed inside the class of homogeneous polynomial differential systems of the same degree. We show that the maximum number of limit cycles, <i>m</i> and <i>m</i>+1, that can bifurcate from the period solutions surrounding the centers for the two classes of differential systems of degree 2<i>m</i> and degree 2<i>m</i>+1, respectively. Both of the bounds can be reached for all <i>m</i>.

Cyclicity of &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ (1,3) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-switching FF type equilibria

Hilbert's 16th Problem suggests a concern to the cyclicity of planar polynomial differential systems, but it is known that a key step to the answer is finding the cyclicity of center-focus equilibria of polynomial differential systems (even of order 2 or 3). Correspondingly, the same question for polynomial discontinuous differential systems is also interesting. Recently, it was proved that the cyclicity of $ (1, 2) $-switching FF type equilibria is at least 5. In this paper we prove that the cyclicity of $ (1, 3) $-switching FF type equilibria with homogeneous cubic nonlinearities is at least 3.

Centers of discontinuous piecewise smooth quasi–homogeneous polynomial differential systems

In this paper we investigate the center problem for the discontinuous piecewise smooth quasi–homogeneous but non–homogeneous polynomial differential systems. First, we provide sufficient and necessary conditions for the existence of a center in the discontinuous piecewise smooth quasi–homogeneous polynomial differential systems. Moreover, these centers are global, and the period function of their periodic orbits is monotonic. Second, we characterize the centers of the discontinuous piecewise smooth quasi–homogeneous cubic and quartic polynomial differential systems.

On the Number of Limit Cycles of Discontinuous Liénard Polynomial Differential Systems

In this paper, we investigate the number of limit cycles for two classes of discontinuous Liénard polynomial perturbed differential systems. By the second-order averaging theorem of discontinuous differential equations, we provide several criteria on the lower upper bounds for the maximum number of limit cycles. The results show that the second-order averaging theorem of discontinuous differential equations can predict more limit cycles than the first-order one.

Algebraic Limit Cycles Bifurcating from Algebraic Ovals of Quadratic Centers

In the integrability of polynomial differential systems it is well known that the invariant algebraic curves play a relevant role. Here we will see that they can also play an important role with respect to limit cycles.In this paper, we study quadratic polynomial systems with an algebraic periodic orbit of degree [Formula: see text] surrounding a center. We show that there exists only one family of such systems satisfying that an algebraic limit cycle of degree [Formula: see text] can bifurcate from the period annulus of the mentioned center under quadratic perturbations.

Bifurcation of limit cycles and center problem for [formula omitted] homogeneous weight systems

The quasi-homogeneous polynomial differential systems have been studied from many different points of view. In this paper, center problem and bifurcation of limit cycles for p:q homogeneous weight systems are studied. Some properties of successive function and focus values are discussed. Furthermore, the method of computing focal values is given. As an example, center problem and bifurcation of limit cycles for 2:3 homogeneous weight system are studied, three or five limit cycles in the neighborhood of origin can be obtained by different perturbations.

Liouvillian integrability of some quadratic Liénard polynomial differential systems

In this paper we characterize all the Liouvillian first integrals of some quadratic Lienard polynomial differential systems. In addition we also find Lienard polynomial differential systems which admit a transcendental first integral.

New lower bound for the Hilbert number in piecewise quadratic differential systems

We study the number of limit cycles bifurcating from a piecewise quadratic system. All the differential systems considered are piecewise in two zones separated by a straight line. We prove the existence of 16 crossing limit cycles in this class of systems. If we denote by Hp(n) the extension of the Hilbert number to degree n piecewise polynomial differential systems, then Hp(2)≥16. As fas as we are concerned, this is the best lower bound for the quadratic class. Moreover, in the studied cases, all limit cycles appear nested bifurcating from a period annulus of a isochronous quadratic center.

Invariant curves and integrability of planar $\mathcal C^r$ differential systems

We improve the known expressions of the $\mathcal {C}^r$ differential systems in the plane having a given $\mathcal {C}^{r+1}$ invariant curve, or a given $\mathcal {C}^{r+1}$ first integral. Their application to polynomial differential systems having either an invariant algebraic curve, or a first integral, also improves the known results on such systems.

An algorithm for providing the normal forms of spatial quasi-homogeneous polynomial differential systems

Quasi-homogeneous systems, and in particular those 3-dimensional, are currently a thriving line of research. But a method for obtaining all fields of this class is not yet available. The weight vectors of a quasi-homogeneous system are grouped into families. We found the maximal spatial quasi-homogeneous systems with the property of having only one family with minimum weight vector. This minimum vector is unique to the system, thus acting as identification code. We develop an algorithm that provides all normal forms of maximal 3-dimensional quasi-homogeneous systems for a given degree. All other 3-dimensional quasi-homogeneous systems can be trivially deduced from these maximal systems. We also list all the systems of this type of degree 2 using the algorithm. With this algorithm we make available to the researchers all 3-dimensional quasi-homogeneous systems.

Limit cycles of discontinuous piecewise quadratic and cubic polynomial perturbations of a linear center

We apply the averaging theory of high order for computing the limit cycles of discontinuous piecewise quadratic and cubic polynomial perturbations of a linear center. These discontinuous piecewise differential systems are formed by two either quadratic, or cubic polynomial differential systems separated by a straight line. We compute the maximum number of limit cycles of these discontinuous piecewise polynomial perturbations of the linear center, which can be obtained by using the averaging theory of order \begin{document}$ n $\end{document} for \begin{document}$ n = 1, 2, 3, 4, 5 $\end{document} . Of course these limit cycles bifurcate from the periodic orbits of the linear center. As it was expected, using the averaging theory of the same order, the results show that the discontinuous quadratic and cubic polynomial perturbations of the linear center have more limit cycles than the ones found for continuous and discontinuous linear perturbations. Moreover we provide sufficient and necessary conditions for the existence of a center or a focus at infinity if the discontinuous piecewise perturbations of the linear center are general quadratic polynomials or cubic quasi-homogenous polynomials.

Isochronicity of a [formula omitted]-equivariant quintic system

The main purpose of this paper is to study the problem of simultaneous existence of two isochronous centers in some families of planar polynomial differential systems with symmetry. More precisely, we present conditions for a family of Z2-equivariant systems of degree 5 to have an isochronous bi-center. We also study their global phase portraits in the Poincaré disk and present considerations about the number of simultaneous centers in Z2-equivariant systems of degree 3 and 5. This study provides examples of quintic systems possessing three and five isochronous centers simultaneously.

Limit cycles by perturbing quadratic isochronous centers inside piecewise polynomial differential systems

In this paper, we consider the quadratic isochronous centers perturbed inside piecewise polynomial differential systems of arbitrary degree n with the straight line of discontinuity x=0. The main concerns are the number of zeros of the first order Melnikov functions and the estimate of the number of limit cycles bifurcating from the period annuli. For quadratic isochronous centers S1, S2 and S3, we will provide a sharp upper bound for the number of zeros of the first order Melnikov functions. For quadratic isochronous center S4, we give a rough estimate. However, when the problem is reduced to perturbations inside polynomial differential systems, our result for S4 will improve that in Li et al. (2000) [12] significantly. Moreover, we will reveal out some equivalence between the first order Melnikov function method and the first order averaging method for investigating the number of limit cycles of piecewise polynomial differential systems.

The center problem for [formula omitted]-symmetric nilpotent vector fields

We say that a polynomial differential system x˙=P(x,y), y˙=Q(x,y) having the origin as a singular point is Z2-symmetric if P(−x,−y)=−P(x,y) and Q(−x,−y)=−Q(x,y). It is known that there are nilpotent centers having a local analytic first integral, and others which only have a C∞ first integral. However these two kinds of nilpotent centers are not characterized for different families of differential systems. Here we prove that the origin of any Z2-symmetric system is a nilpotent center if, and only if, there is a local analytic first integral of the form H(x,y)=y2+⋯, where the dots denote terms of degree higher than two.

Non-existence and uniqueness of limit cycles for planar polynomial differential systems with homogeneous nonlinearities

In this paper we study the limit cycles of the planar polynomial differential systemsx˙=ax−y+Pn(x,y),y˙=x+ay+Qn(x,y), where Pn and Qn are homogeneous polynomials of degree n≥2, and a∈R. Consider the functionsφ(θ)=Pn(cos⁡θ,sin⁡θ)cos⁡θ+Qn(cos⁡θ,sin⁡θ)sin⁡θ,ψ(θ)=Qn(cos⁡θ,sin⁡θ)cos⁡θ−Pn(cos⁡θ,sin⁡θ)sin⁡θ,ω1(θ)=aψ(θ)−φ(θ),ω2(θ)=(n−1)(2aψ(θ)−φ(θ))+ψ′(θ). First we prove that these differential systems have at most 1 limit cycle if there exists a linear combination of ω1 and ω2 with definite sign. This result improves previous known results. Furthermore, if ω1(ν1aψ−ν2φ)≤0 for some ν1,ν2≥0, we provide necessary and sufficient conditions for the non-existence, and the existence and uniqueness of the limit cycles of these differential systems. When one of these mentioned limit cycles exists it is hyperbolic and surrounds the origin.