Invariant Algebraic Curves Research Articles

Normal forms of polynomial differential systems in $${\mathbb{R}}^3$$ having at least three invariant algebraic surfaces

In this paper, we find the normal forms of polynomial differential systems in $${\mathbb{R}}^3$$ which have at least three invariant algebraic surfaces. Also, we deduce the normal forms of polynomial differential systems in $${\mathbb{R}}^3$$ having a parabolic cylinder with the equation $${\mathcal{P}} : y^2-z$$ , or having a hyperbolic parabolic with the equation $${\mathcal{H}} : x^2-y^2-z$$ as invariant objects. The conditions to find a lower bound for the number of invariant algebraic curves for the deduced systems are obtained.

Classification of Invariant Algebraic Curves and Nonexistence of Algebraic Limit Cycles in Quadratic Systems from Family (I) of the Chinese Classification

We give the complete classification of irreducible invariant algebraic curves in quadratic systems from family [Formula: see text] of the Chinese classification, that is, of differential system [Formula: see text] with [Formula: see text]. In addition, we provide a complete and correct proof of the nonexistence of algebraic limit cycles for these equations.

On the Poincaré problem and Liouvillian integrability of quadratic Liénard differential equations

AbstractWe present the complete classification of irreducible invariant algebraic curves of quadratic Liénard differential equations. We prove that these equations have irreducible invariant algebraic curves of unbounded degrees, in contrast with what is wrongly claimed in the literature. In addition, we classify all the quadratic Liénard differential equations that admit a Liouvillian first integral.

不变代数曲线法求解积分因子

不变代数曲线法求解积分因子

An improvement of the non-existence region for limit cycles of the Bogdanov-Takens system

In this paper, an improvement of the global region for the non-existence of limit cycles of the Bogdanov-Takens system, which is well-known in the Bifurcation Theory, is given by two ideas. The first is to apply the existence of the algebraic invariant curve of the system to the Bendixson-Dulac criterion, and the second is to consider a necessary condition in order that a closed orbit of the system includes two equilibrium points. In virtue of these methods, it shall be shown that our previous result and the result of Gasull et al. are improved partially.

Algebraic Traveling Wave Solutions to Nonlinear Evolution Equations

In this paper, we employ planar dynamical systems and invariant algebraic curves to characterize all algebraic traveling wave solutions to nonlinear evolution equations. In order to demonstrate the applicability and efficiency of the method, we apply the approach to four (2+1)-dimensional integrable extensions of the Kadomtsev–Petviashvili equation. The numerical simulations are also plotted for better understanding the physical phenomena.

Foliations on $$\mathbb {CP}^2$$ with a unique singular point without invariant algebraic curves

We prove that a foliation on $$\mathbb {CP}^2$$ of degree d with a singular point of type saddle-node with Milnor number $$d^2+d+1$$ does not have invariant algebraic curves. We give a family of this kind of foliations. We also present a family of foliations of degree d with a unique nilpotent singularity without invariant algebraic curves for d odd greater than 1. Finally we prove that the space of foliations on $$\mathbb {CP}^2$$ of degree $$d \ge 2$$ with a unique singular point has dimension at least $$3d+2$$ .

On the global dynamics and integrability of the Chemostat system

We study a Chemostat system of the form ẋ=−qx+R̃K+yxy,ẏ=(c̃−y)q−R̃ã(K+y)xy,where q>0, R̃>0, K>0, c̃>0 and ã≠0. This system appears in competition modelling in biology. We describe its global dynamics on the Poincaré disc and study its Liouvillian integrability. For the first topic we use the well-known Poincaré compactification theory and for the second one we make use of the Puiseux series to derive the structure of all the irreducible invariant algebraic curves.

Birational Geometry of Foliations Associated to Simple Derivations

We propose a study of the foliations of the projective plane induced by simple derivations of the polynomial ring in two indeterminates over the complex field. These correspond to foliations which have no invariant algebraic curve nor singularities in the complement of a line. We establish the position of these foliations in the birational classification of foliations and prove the finiteness of their birational symmetries. Most of the results apply to wider classes of foliations.

Limit Cycles and Invariant Curves in a Class of Switching Systems with Degree Four

In this paper, a class of switching systems which have an invariant conicx2+cy2=1,c∈R, is investigated. Half attracting invariant conicx2+cy2=1,c∈R, is found in switching systems. The coexistence of small-amplitude limit cycles, large amplitude limit cycles, and invariant algebraic curves under perturbations of the coefficients of the systems is proved.

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.

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.

Local polar invariants and the Poincaré problem in the dicritical case

We develop a study on local polar invariants of planar complex analytic foliations at $(\mathbb{C}^{2},0)$, which leads to the characterization of second type foliations and of generalized curve foliations, as well as to a description of the $GSV$-index. We apply it to the Poincare problem for foliations on the complex projective plane $\mathbb{P}^{2}_{\mathbb{C}}$, establishing, in the dicritical case, conditions for the existence of a bound for the degree of an invariant algebraic curve $S$ in terms of the degree of the foliation $\mathcal{F}$. We characterize the existence of a solution for the Poincare problem in terms of the structure of the set of local separatrices of $\mathcal{F}$ over the curve $S$. Our method, in particular, recovers the known solution for the non-dicritical case, $\deg(S) \leq \deg (\mathcal{F}) + 2$.

Invariant algebraic curves for Liénard dynamical systems revisited

A novel algebraic method for finding invariant algebraic curves for a polynomial vector field in $\mathbb{C}^2$ is introduced. The structure of irreducible invariant algebraic curves for Li\'{e}nard dynamical systems $x_t=y$, $y_t=-g(x)y-f(x)$ with $\text{deg} f=\text{deg} g+1$ is obtained. It is shown that there exist Li\'{e}nard systems that possess more complicated invariant algebraic curves than it was supposed before. As an example, all irreducible invariant algebraic curves for the Li\'{e}nard differential system with $\text{deg} f=3$, $\text{deg} g=2$ are obtained. All these results seem to be new.

Novel algebraic aspects of Liouvillian integrability for two-dimensional polynomial dynamical systems

The general structure of irreducible invariant algebraic curves for a polynomial dynamical system in C2 is found. Necessary conditions for existence of exponential factors related to an invariant algebraic curve are derived. As a consequence, all the cases when the classical force-free Duffing and Duffing–van der Pol oscillators possess Liouvillian first integrals are obtained. New exact solutions for the force-free Duffing–van der Pol system are constructed.

Algebraic Limit Cycles on Quadratic Polynomial Differential Systems

AbstractAlgebraic limit cycles in quadratic polynomial differential systems started to be studied in 1958, and a few years later the following conjecture appeared: quadratic polynomial differential systems have at most one algebraic limit cycle. We prove that a quadratic polynomial differential system having an invariant algebraic curve with at most one pair of diametrically opposite singular points at infinity has at most one algebraic limit cycle. Our result provides a partial positive answer to this conjecture.

Global Phase Portraits of Quadratic Systems with a Complex Ellipse as Invariant Algebraic Curve

In this paper, we study a new class of quadratic systems and classify all its phase portraits. More precisely, we characterize the class of all quadratic polynomial differential systems in the plane having a complex ellipse x2 + y2 + 1 = 0 as invariant algebraic curve. We provide all the different topological phase portraits that this class exhibits in the Poincare disc.

Classical Planar Algebraic Curves Realizable by Quadratic Polynomial Differential Systems

In this paper, we show planar quadratic polynomial differential systems exhibiting as solutions some famous planar invariant algebraic curves. Also we place particular attention to the Darboux integrability of these differential systems.

Unique ergodicity for foliations in $$\mathbb {P}^2$$ P 2 with an invariant curve

Consider a foliation in the projective plane admitting a projective line as the unique invariant algebraic curve. Assume that the foliation is generic in the sense that its singular points are hyperbolic. We show that there is a unique positive ddc-closed (1,1)-current of mass 1 which is directed by the foliation and this is the current of integration on the invariant line. A unique ergodicity theorem for the distribution of leaves follows: for any leaf L, appropriate averages of L converge to the current of integration on the invariant line. This property is surprising because for most of such foliations the leaves (except the invariant line) are dense in the projective plane. So one could expect that they spend a significant amount of hyperbolic time in every open set and that there should be a fat ddc-closed non-closed current with support equal to the projective plane. The proof uses an extension of our theory of densities for currents. Foliations on compact Kaehler surfaces with one or several invariant curves are also considered.

A Geometric Investigation of the Invariant Algebraic Curves in Two Dimensional Lotka–Volterra Systems

The aim of this work is to investigate the geometric behaviour of the invariant algebraic curves in two dimensional Lotka–Volterra systems, first classified by Moulin-Ollagnier.