Polynomial Differential Systems Research Articles

Quadratic Differential Systems with a Finite Saddle-Node and an Infinite Saddle-Node (1, 1)SN - (B)

This paper presents a global study of the class [Formula: see text] of all real quadratic polynomial differential systems which have a finite semi-elemental saddle-node and an infinite saddle-node formed by the coalescence of a finite singularity and an infinite singularity. This class can be divided into two different families, namely, [Formula: see text] phase portraits possessing a finite saddle-node as the only finite singularity and [Formula: see text] phase portraits possessing a finite saddle-node and also a simple finite elemental singularity. Each one of these two families is given by a specific normal form. The study of family [Formula: see text] was reported in [Artés et al., 2020b] where the authors obtained [Formula: see text] topologically distinct phase portraits for systems in the closure [Formula: see text]. In this paper, we provide the complete study of the geometry of family [Formula: see text]. This family which modulo the action of the affine group and time homotheties is three-dimensional and we give the bifurcation diagram of its closure with respect to a specific normal form, in the three-dimensional real projective space. The respective bifurcation diagram yields 631 subsets with 226 topologically distinct phase portraits for systems in the closure [Formula: see text] within the representatives of [Formula: see text] given by a specific normal form. Some of these phase portraits are proven to have at least three limit cycles.

Four-dimensional Zero-Hopf Bifurcation of Quadratic Polynomial Differential System, via Averaging Theory of Third Order

Four-dimensional Zero-Hopf Bifurcation of Quadratic Polynomial Differential System, via Averaging Theory of Third Order

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 $4$.

Polynomial differential systems with even degree have no global centers

Let x˙=P(x,y), y˙=Q(x,y) be a differential system with P and Q real polynomials, and let d=max⁡{degP,degQ}. A singular point p of this differential system is a global center if R2∖{p} is filled with periodic orbits. We prove that if d is even then the polynomial differential systems have no global centers.

Bifurcations of the Riccati Quadratic Polynomial Differential Systems

In this paper, we characterize the global phase portrait of the Riccati quadratic polynomial differential system [Formula: see text] with [Formula: see text], [Formula: see text] nonzero (otherwise the system is a Bernoulli differential system), [Formula: see text] (otherwise the system is a Liénard differential system), [Formula: see text] a polynomial of degree at most [Formula: see text], [Formula: see text] and [Formula: see text] polynomials of degree at most 2, and the maximum of the degrees of [Formula: see text] and [Formula: see text] is 2. We give the complete description of the phase portraits in the Poincaré disk (i.e. in the compactification of [Formula: see text] adding the circle [Formula: see text] of the infinity) modulo topological equivalence.

Generalized Analytic Integrability of a Class of Polynomial Differential Systems in $\mathbb{C}^{2}$

This paper study the type of integrability of differential systems with separable variables \(\dot{x}=h\left (x\right )f\left (y\right )\), \(\dot{y}= g\left (y\right )\), where \(h\), \(f\) and \(g\) are polynomials. We provide a criterion for the existence of generalized analytic first integrals of such differential systems. Moreover we characterize the polynomial integrability of all such systems.In the particular case \(h\left (x\right )=\left (ax+b\right )^{m}\) we provide necessary and sufficient conditions in order that this subclass of systems has a generalized analytic first integral. These results extend known results from Giné et al. (Discrete Contin. Dyn. Syst. 33:4531–4547, 2013) and Llibre and Valls (Discrete Contin. Dyn. Syst., Ser. B 20:2657–2661, 2015). Such differential systems of separable variables are important due to the fact that after a blow-up change of variables any planar quasi-homogeneous polynomial differential system can be transformed into a special differential system of separable variables \(\dot{x}=xf\left (y\right )\), \(\dot{y}=g\left (y\right )\), with \(f\) and \(g\) polynomials.

Algebraic and topological classification of homogeneous quartic vector fields in the plane

We provide canonical forms for the homogeneous polynomials of degree five. Then we characterize all the phase portraits in the Poincare disk for all quartic homogeneous polynomial differential systems. More precisely, there are exactly 23 different topological phase portraits for the quartic homogeneous polynomial differential systems.

The Number of Limit Cycles Bifurcating from a Degenerate Center of Piecewise Smooth Differential Systems

For two families of planar piecewise smooth polynomial differential systems, whose unperturbed system has a degenerate center at the origin, we study the biggest lower bound for the maximum number of limit cycles bifurcating from the periodic orbits of the center. These results are extensions of the known ones on unperturbed nondegenerate [Formula: see text]-center, derived from a nonsmooth harmonic oscillator model, to degenerate [Formula: see text]-center. Our study involves some new computational treatments. The main tools are the generalized polar coordinate change and the generalized Lyapunov polar coordinate change together with an averaging theory for one-dimensional piecewise smooth differential equations. Finally, we present two Maple programs for computing the averaging functions and consequently the biggest lower bound on the maximum number of limit cycles of degenerate [Formula: see text]-center under general polynomial perturbations of degree [Formula: see text].

Limit Cycles of a Class of Perturbed Differential Systems via the First-Order Averaging Method

By means of the averaging method of the first order, we introduce the maximum number of limit cycles which can be bifurcated from the periodic orbits of a Hamiltonian system. Besides, the perturbation has been used for a particular class of the polynomial differential systems.

Phase portraits of separable quadratic systems and a bibliographical survey on quadratic systems

Although planar quadratic differential systems and their applications have been studied in more than one thousand papers, we still have no complete understanding of these systems. In this paper we have two objectives.First we provide a brief bibliographical survey on the main results about quadratic systems. Here we do not consider the applications of these systems to many areas as in Physics, Chemist, Economics, Biology, …Second we characterize the new class of planar separable quadratic polynomial differential systems. For such class of systems we provide the normal forms which contain one parameter, and using the Poincaré compactification and the blow up technique, we prove that there exist 10 non-equivalent topological phase portraits in the Poincaré disc for the separable quadratic polynomial differential systems.

The cyclicity of the period annulus of a reversible quadratic system

We prove that perturbing the periodic annulus of the reversible quadratic polynomial differential system $\dot x=y+ax^2$, $\dot y=-x$ with a ≠ 0 inside the class of all quadratic polynomial differential systems we can obtain at most two limit cycles, including their multiplicities. Since the first integral of the unperturbed system contains an exponential function, the traditional methods cannot be applied, except in Figuerasa, Tucker and Villadelprat (2013, J. Diff. Equ., 254, 3647–3663) a computer-assisted method was used. In this paper, we provide a method for studying the problem. This is also the first purely mathematical proof of the conjecture formulated by Dumortier and Roussarie (2009, Discrete Contin. Dyn. Syst., 2, 723–781) for q ⩽ 2. The method may be used in other problems.

Limit cycles in piecewise polynomial systems allowing a non-regular switching boundary

Continuing the investigation for the piecewise polynomial perturbations of the linear center ẋ=−y,ẏ=x from Buzzi et al. (2018) for the case where the switching boundary is a straight line, in this paper we allow that the switching boundary is non-regular, i.e. we consider a switching boundary which separates the plane into two angular sectors with angles α∈(0,π] and 2π−α. Moreover, unlike the aforementioned work, we allow that the polynomial differential systems in the two sectors have different degrees. Depending on α and for arbitrary given degrees we provide an upper bound for the maximum number of limit cycles that bifurcate from the periodic orbits of the linear center using the averaging method up to any order. This upper bound is reached for the first two orders. On the other hand, we pay attention to the perturbation of the linear center inside this class of piecewise polynomial Liénard systems and give some better upper bounds in comparison with the one obtained in the general piecewise polynomial perturbations. Again our results imply that the non-regular switching boundary (i.e. when α≠π) of the piecewise polynomial perturbations usually leads to more limit cycles than the regular case (i.e. when α=π) where the switching boundary is a straight line.

Quadratic Differential Systems with a Finite Saddle-Node and an Infinite Saddle-Node (1,1)SN - (A)

Our goal is to make a global study of the class [Formula: see text] of all real quadratic polynomial differential systems which have a finite semi-elemental saddle-node and an infinite saddle-node formed by the coalescence of a finite and infinite singularities. This class can be divided into two different families, being (A) possessing the finite saddle-node as the only finite singularity and (B) possessing the finite saddle-node and also a finite simple elemental singularity. In this paper we provide the complete study of the geometry of family (A). The family (A) modulo the action of the affine group and time homotheties are four-dimensional and we give the bifurcation diagram of its closure with respect to a specific normal form, in the four-dimensional real projective space [Formula: see text]. As far as we know, this is the first time that a complete family is studied in the four-dimensional real projective space. The respective bifurcation diagram yields 36 topologically distinct phase portraits for systems in the closure [Formula: see text] within the representatives of [Formula: see text] given by a specific normal form.

Formal Weierstrass integrability for a Liénard differential system

A novel algebraic method to verify if a polynomial differential system in C2 is formal Weierstrass integrable was introduced in [18,19]. Here we apply these results to a Liénard differential system finding non-Liouvillian integrable systems.

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.

Quadratic slow-fast systems on the plane

In this paper singularly perturbed quadratic polynomial differential systems εẋ=Pε(x,y)=P(x,y,ε),ẏ=Qε(x,y)=Q(x,y,ε)with x,y∈R,ε⩾0 and (Pε,Qε)=1 for ε>0, are considered. We prove that there are 10 classes of equivalence for these systems. We describe the dynamics of these 10 classes on the Poincaré disc when ε=0. For ε>0, we present the possible local behavior of the solutions near of a finite and infinite equilibrium point under suitable conditions. More specifically, if p0 is a finite equilibrium point then we obtain the local behavior for ε>0 using Fenichel theory. Assuming that p0 is an infinite equilibrium point, there exists K⊂M0 normally hyperbolic and p0∈M0′∩K using the Poincaré compactification and algebraic invariant we describe globally the dynamics for ε>0 small of some classes of equivalence.

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.

Rational first integrals of the Liénard equations: The solution to the Poincaré problem for the Liénard equations.

Poincaré in 1891 asked about the necessary and sufficient conditions in order to characterize when a polynomial differential system in the plane has a rational first integral. Here we solve this question for the class of Liénard differential equations x ¨ + f ( x ) x ˙ + x = 0 , being f ( x ) a polynomial of arbitrary degree. As far as we know it is the first time that all rational first integrals of a relevant class of polynomial differential equations of arbitrary degree has been classified.

The algebraic curves of planar polynomial differential systems with homogeneous nonlinearities

The algebraic curves of planar polynomial differential systems with homogeneous nonlinearities

A number of limit cycle of sextic polynomial differential systems via the averaging theory

The main purpose of this paper is to study the number of limit cycles of sextic polynomial differential systems (SPDS) via the averaging theory which is an extension to the study of cubic polynomial vector fields in (Nonlinear Analysis 66 (2007), 1707--1721), where we provide an accurate upper bound of the maximum number of limit cycles that SPDS can have bifurcating from the period annulus surrounding the origin of a class of cubic system.