Related papers: Planar quadratic vector fields with invariant line…

In this article we consider the action of affine group and time rescaling on planar quadratic differential systems. We construct a system of representatives of the orbits of systems with four invariant lines, including the line at infinity…

In this article we consider the behavior in the vicinity of infinity of the class of all planar quadratic differential systems. This family depends on twelve parameters but due to action of the affine group and re-scaling of time the family…

We classify all cubic systems possessing the maximum number of invariant straight lines (real or complex) taking into account their multiplicities. We prove that there are exactly 23 topological different classes of such systems. For every…

We construct a polynomial planar vector field of degree two with one invariant algebraic curves of large degree. We exhibit an explicit quadratic vector fields which invariant curves of degree nine, twelve, fifteen and eighteen degree.

We study the existence of invariant quadrics for a class of systems of difference equations in ${\mathbb R}^n$ defined by linear fractionals sharing denominator. Such systems can be described in terms of some square matrix $A$ and we prove…

The new necessary and sufficient affine invariant conditions for the existence and for determining the number of centers for general quadratic system are pointed out. These conditions correspond to the partition of 12-dimensional…

We consider an integrable system in five unknowns having three quartics invariants. We show that the complex affine variety defined by putting these invariants equal to generic constants, completes into an abelian surface; the jacobian of a…

For 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…

Planar quadratic differential systems occur in many areas of applied mathematics. Although more than one thousand papers have been written on these systems, a complete understanding of this family is still missing. Classical problems, and…

In this paper we show how we can transform quadratic systems into new quadratic systems after some kind of birational transformations, the quadratic plane Cremona maps. We afterwards apply these transformations to the families of quadratic…

A genus one curve of degree 5 is defined by the 4 x 4 Pfaffians of a 5 x 5 alternating matrix of linear forms on P^4. We describe a general method for investigating the invariant theory of such models. We use it to explain how we found our…

In this paper, we develop a representation-theoretic formulation of discrete-time linear systems. We show that such systems are naturally viewed as representations of time groups acting on vector spaces, thereby endowing the state space…

We give an algorithm for deciding whether a planar polynomial differential system has a first integral which factorizes as a product of defining polynomials of curves with only one place at infinity. In the affirmative case, our algorithm…

Among the various critical systems that worth to be formally analyzed, a wide set consists of controllers for dynamical systems. Those programs typically execute an infinite loop in which simple com putations update internal states and…

The classification, up to a center-affinity, of the homogeneous quadratic differential systems defined on $\mathbb{R}^{3}$ that have at least a semisimple nonsingular derivation, is achieved. It is proved that there exist four…

This article reveals an analysis of the quadratic systems that hold multiparametric families therefore, in the first instance the quadratic systems are identified and classified in order to facilitate their study and then the stability of…

We consider indecomposable representations of the Klein four group over a field of characteristic $2$ and of a cyclic group of order $pm$ with $p,m$ coprime over a field of characteristic $p$. For each representation we explicitly describe…

Geometric Invariant Theory gives a method for constructing quotients for group actions on algebraic varieties which in many cases appear as moduli spaces parametrizing isomorphism classes of geometric objects (vector bundles, polarized…

The note focuses on the differential geometric approach to the study of nonlinear systems that are affine in control. We first develop normal forms for nonlinear system affine in control. Based on these normal forms, we then address the…

In this paper we study differential forms and vector fields on the orbit space of a proper action of a Lie group on a smooth manifold, defining them as multilinear maps on the generators of infinitesimal diffeomorphisms, respectively. This…