New family of cubic Hamiltonian centers

We characterize the 11 non topological equivalent classes of phase portraits in the Poincaré disc of the new family of cubic polynomial Hamiltonian differential systems with a center at the origin and Hamiltonian H=1/2((x+ax2+bxy+cy2)2+y2), with a2+b2+c2≠0

Configuration of zeros of isochronous vector fields of degree 5

In this paper, we give the algebraic conditions that a configuration of 5 points in the plane must satisfy in order to be the configuration of zeros of a polynomial isochronous vector field. We use the obtained results to analyze configurations having some of its zeros satisfying some particular geometric conditions

Polynomials in Control Theory Parametrized by Their Roots

The aim of this paper is to introduce the space of roots to study the topological properties of the spaces of polynomials. Instead of identifying a monic complex polynomial with the vector of its coefficients, we identify it with the set of its roots. Viète's map gives a homeomorphism between the space of roots and the space of coefficients and it gives an explicit formula to relate both spaces. Using this viewpoint we establish that the space of monic (Schur or Hurwitz) aperiodic polynomials is contractible. Additionally we obtain a Boundary Theorem

On Solving Systems of Autonomous Ordinary Differential Equations by Reduction to a Variable of an Algebra

A new technique for solving a certain class of systems of autonomous ordinary differential equations over n is introduced ( being the real or complex field). The technique is based on two observations: (1), if n has the structure of certain normed, associative, commutative, and with a unit, algebras over , then there is a scheme for reducing the system of differential equations to an autonomous ordinary differential equation on one variable of the algebra; (2) a technique, previously introduced for solving differential equations over ℂ, is shown to work on the class mentioned in the previous paragraph. In particular it is shown that the algebras in question include algebras linearly equivalent to the tensor product of matrix algebras with certain normal forms

New family of cubic Hamiltonian centers

We characterize the 11 non topological equivalent classes of phase portraits in the Poincaré disc of the new family of cubic polynomial Hamiltonian differential systems with a center at the origin and Hamiltonian H=1/2((x+ax2+bxy+cy2)2+y2), with a2+b2+c2≠0

On Solving Systems of Autonomous Ordinary Differential Equations by Reduction to a Variable of an Algebra

A new technique for solving a certain class of systems of autonomous ordinary differential equations over K n is introduced K being the real or complex field . The technique is based on two observations: 1 , if K n has the structure of certain normed, associative, commutative, and with a unit, algebras A over K, then there is a scheme for reducing the system of differential equations to an autonomous ordinary differential equation on one variable of the algebra; 2 a technique, previously introduced for solving differential equations over C, is shown to work on the class mentioned in the previous paragraph. In particular it is shown that the algebras in question include algebras linearly equivalent to the tensor product of matrix algebras with certain normal forms

Algebrization of Nonautonomous Differential Equations

Given a planar system of nonautonomous ordinary differential equations, dw/dt=F(t,w), conditions are given for the existence of an associative commutative unital algebra A with unit e and a function H:Ω⊂R2×R2→R2 on an open set Ω such that F(t,w)=H(te,w) and the maps H1(τ)=H(τ,ξ) and H2(ξ)=H(τ,ξ) are Lorch differentiable with respect to A for all (τ,ξ)∈Ω, where τ and ξ represent variables in A. Under these conditions the solutions ξ(τ) of the differential equation dξ/dτ=H(τ,ξ) over A define solutions (x(t),y(t))=ξ(te) of the planar system