Invariant Straight Lines Research Articles

A Polynomial System of Degree Four with an Invariant Square Containing At Least Five Limit Cycles

We consider a class of polynomial systems of degree four with four real invariant straight lines that form a square, called this an invariant square, and also that contains in its interior at least five small amplitude limit cycles for a certain choice of the parameters. Moreover, we will obtain the necessary and sufficient conditions for the critical point inside the square to be a center.

Global Analysis of Riccati Quadratic Differential Systems

In this paper, we study the family of quadratic Riccati differential systems. Our goal is to obtain the complete topological classification of this family on the Poincaré disk compactification of the plane. The family was partially studied before but never from a truly global viewpoint. Our approach is global and we use geometry to achieve our goal. The geometric analysis we perform is via the presence of two invariant parallel straight lines in any generic Riccati system. We obtain a total of 119 topologically distinct phase portraits for this family. Furthermore, we give the complete bifurcation diagram in the 12-dimensional space of parameters of this family in terms of invariant polynomials, meaning that it is independent of the normal forms in which the systems may be presented. This bifurcation diagram provides an algorithm to decide for any given quadratic system in any form it may be presented, whether it is a Riccati system or not, and in case it is to provide its phase portrait.

The bifurcation diagram of the configurations of invariant lines of total multiplicity exactly three in quadratic vector fields

We denote by ${\mbox{\boldmath $QSL$}}_3$ the family of quadratic differential systems possessing invariant straight lines, finite and infinite, of total multiplicity exactly three. In a sequence of papers the complete study of quadratic systems with invariant lines of total multiplicity at least four was achieved. In addition three more families of quadratic systems possessing invariant lines of total multiplicity at least three were also studied, among them the Lotka-Volterra family. However there were still systems in ${\mbox{\boldmath ${\mbox{\boldmath $QSL$}}$}}_3$ missing from all these studies. The goals of this article are: to complete the study of the geometric configurations of invariant lines of ${\mbox{\boldmath ${\mbox{\boldmath $QSL$}}$}}_3$ by studying all the remaining cases and to give the full classification of this family modulo their configurations of invariant lines together with their bifurcation diagram. The family ${\mbox{\boldmath ${\mbox{\boldmath $QSL$}}$}}_3$ has a total of 81 distinct configurations of invariant lines. This classification is done in affine invariant terms and we also present the bifurcation diagram of these configurations in the 12-parameter space of coefficients of the systems. This diagram provides an algorithm for deciding for any given system whether it belongs to ${\mbox{\boldmath $QSL$}}_3$ and in case it does, by producing its configuration of invariant straight lines.

Number of Limit Cycles for Planar Systems with Invariant Algebraic Curves

AbstractFor planar polynomials systems the existence of an invariant algebraic curve limits the number of limit cycles not contained in this curve. We present a general approach to prove non-existence of periodic orbits not contained in this given algebraic curve. When the method is applied to parametric families of polynomial systems that have limit cycles for some values of the parameters, our result leads to effective algebraic conditions on the parameters that force non-existence of the periodic orbits. As applications we consider several families of quadratic systems: the ones having some quadratic invariant algebraic curve, the known ones having an algebraic limit cycle, a family having a cubic invariant algebraic curve and other ones. For any quadratic system with two invariant algebraic curves we prove a finiteness result for its number of limit cycles that only depends on the degrees of these curves. We also consider some families of cubic systems having either a quadratic or a cubic invariant algebraic curve and a family of Liénard systems. We also give a new and simple proof of the known fact that quadratic systems with an invariant parabola have at most one limit cycle. In fact, what we show is that this result is a consequence of the similar result for quadratic systems with an invariant straight line.

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.

The family of cubic differential systems with two real and two complex distinct infinite singularities and invariant straight lines of the type ( 3 , 1 , 1 , 1 )

In this article we consider the class CSL 7 2 r 2 c ∞ of non-degenerate real planar cubic vector fields, which possess two real and two complex distinct infinite singularities and invariant straight lines of total multiplicity 7, including the line at infinity. The classification according to the configurations of invariant lines of systems possessing invariant straight lines was given in articles published from 2014 up to 2022. We continue our investigation for the family CSL 7 2 r 2 c ∞ possessing configurations of invariant lines of type ( 3 , 1 , 1 , 1 ) and prove that there are exactly 42 distinct configurations of this type. Moreover we construct all the orbit representatives of the systems in this class with respect to affine group of transformations and a time rescaling.

Center conditions for a cubic system with two homogeneous in variant straight lines and exponential factors

In this paper for a cubic differential system with a singular pointO(0,0) of acenter or a focus type, having two homogeneous invariant straight lines and exponentialfactors, we determine conditions under which the singular point is a center.

Bifurcation of limit cycles in piecewise quadratic differential systems with an invariant straight line

We solve the center-focus problem in a class of piecewise quadratic polynomial differential systems with an invariant straight line. The separation curve is also a straight line which is not invariant. We provide families having at the origin a weak-foci of maximal order. In the continuous class, the cyclicity problem is also solved, being 3 such maximal number. Moreover, for the discontinuous class but without sliding segment, we prove the existence of 7 limit cycles of small amplitude.

Darboux integrability of a cubic differential system with two parallel invariant straight lines

In this paper we prove the Darboux integrability of a cubic differential system with a singular point of a center typer having at least two parallel invariant straight lines.

Cubic differential systems with invariant straight lines of total multiplicity seven and four real distinct infinite singularities

In this article we consider the class \(\mathrm{CSL}_7^{4s\infty}\) of non-degenerate real planar cubic vector fields possessing four distinctreal infinite singularities and invariant straight lines, including the line at infinity of total multiplicity 7. We prove that there are exactly 93 distinct configurations of invariant straight lines for this class, and present corresponding examples for the realization of each one of the detected configurations. For more information see https://ejde.math.txstate.edu/Volumes/2021/83/abstr.html

On the dynamic of the isotropic star

In this paper, we investigate the dynamical behavior of the planar polynomial system which describes the evolution of the isotropic star. It is shown that the system has no global [Formula: see text] and no global analytic first integrals. It has an invariant algebraic curve with algebraic multiplicity [Formula: see text] and an exponential factor that comes from the multiplicity of the infinite invariant straight line. It is proved that the system can be changed into the Liénard system. By using a dominant balance analysis, we prove the system has a general solution which eventuates to a finite-time singularity. Finally, we prove the trajectories of the vector field associated with the planar system of the isotropic star do not create a trajectory manifold which means there is no pseudo-Riemannian metric [Formula: see text] in the sense of trajectory metric such that the trajectories of the isotropic star system be geodesic.

Phase Portraits of Random Planar Homogeneous Vector Fields

We study the phase portraits with positive probability of random planar homogeneous vector fields of degree n. In particular, for n=1,2,3, we give a complete solution of the problem and, moreover, either we give the exact value of each probability or we estimate it by using the Monte Carlo method. It is remarkable that all but two of these phase portraits are characterized by their index at the origin and by their number of invariant straight lines through it.

CENTER PROBLEM FOR CUBIC DIFFERENTIAL SYSTEMS WITH THE LINE AT INFINITY AND AN AFFINE REAL INVARIANT STRAIGHT LINE OF TOTAL MULTIPLICITY FOUR

In this article, we show that a non-degenerate monodromic critical point of differential systems with the line at infinity and an affine real invariant straight line of total multiplicity four is a center type if and only if the first four Lyapunov quantities vanish.

On the centers of cubic polynomial differential systems with four invariant straight lines

Assume that a cubic polynomial differential system in the plane has four invariant straight lines in generic position, i.e., they are not parallel and no more than two straight lines intersect in a point. Then such a differential system only can have $0$, $1$ or $3$ centers.

Cyclicity of a Family of Generic Reversible Quadratic Systems with One Center

This paper is concerned with the quadratic perturbations from one parameter family of generic reversible quadratic vector fields having a simple center and an invariant straight line. It is shown that the system can generate at least two limit cycles. As the parameter is rational, we propose a procedure for finding the upper bound to cyclicity of period annulus based on the Chebyshev criterion for Abelian integrals together with one rationalizing transformation. To illustrate our approach, we determine the cyclicity of three representative reversible systems. Our results may be viewed as a contribution to proving the conjecture on cyclicity proposed by Iliev [1998].

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.

On some attractors of a two-dimensional quadratic map

In this paper, we study the appearance, evolution and neighbourhood of two attractors of a dynamical system defined by a quadratic polynomial map T: R2 → R2. The first is a Cantor-type attractor located on an invariant straight line. Thus, it suffices to study the restriction of the map T to this invariant line. The second is a closed curves cycle of period 2. We show, by a numerical approach, that when a parameter of the system varies, the evolution of the orbits in the region close to this second attractor is dependent on the evolution of the stable and unstable sets (homoclinic tangency) of a saddle cycle of period 2 located in this region.

Axial crystals – macroscopic symmetry and tensor properties

ABSTRACTAxial crystals have axial symmetry which keeps invariant straight line with a fixed point. Axial symmetry groups include 27 non-cubic crystallographic point groups and 5 limit groups describing symmetry of textures and liquid crystals. We show that, except for four cases, each axial symmetry belongs to one of five axial types: polar, chiral, pseudopolar (three basic axial types), directional (possessing none of characteristic properties of basic types) or rotational (exhibiting characteristic properties of all basic types). Each basic type can appear in two structurally different variants with the same symmetry. These variants can coexist and form a mesoscopic structure (antiparallel ferroelectric or chiral domain structure, mixture of enantiomers). We examine macroscopic properties of axial types and variants, and experimental accessivity of their characteristic features.

First integrals and phase portraits of planar polynomial differential cubic systems with invariant straight lines of total multiplicity eight

First integrals and phase portraits of planar polynomial differential cubic systems with invariant straight lines of total multiplicity eight

First Integrals and Phase Portraits of Planar Polynomial Differential Cubic Systems with the Maximum Number of Invariant Straight Lines

In the article Llibre and Vulpe (Rocky Mt J Math 38:1301–1373, 2006) the family of cubic polynomial differential systems possessing invariant straight lines of total multiplicity 9 was considered and 23 such classes of systems were detected. We recall that 9 invariant straight lines taking into account their multiplicities is the maximum number of straight lines that a cubic polynomial differential systems can have if this number is finite. Here we complete the classification given in Llibre and Vulpe (Rocky Mt J Math 38:1301–1373, 2006) by adding a new class of such cubic systems and for each one of these 24 such classes we perform the corresponding first integral as well as its phase portrait. Moreover we present necessary and sufficient affine invariant conditions for the realization of each one of the detected classes of cubic systems with maximum number of invariant straight lines when this number is finite.