Yudy Bolaños - Academia.edu (original) (raw)
Papers by Yudy Bolaños
We study the polynomial vector fields X = n+1
Rocky Mountain Journal of Mathematics, 2015
We provide explicit expressions for the Liouvillian first integrals of the quadratic polynomial d... more We provide explicit expressions for the Liouvillian first integrals of the quadratic polynomial differential systems having an integrable saddle.
International Journal of Bifurcation and Chaos, 2012
Using sophisticated techniques of Algebraic Geometry, Jouanolou in 1979 showed that if the number... more Using sophisticated techniques of Algebraic Geometry, Jouanolou in 1979 showed that if the number of invariant algebraic hypersurfaces of a polynomial vector field in ℝn of degree m is at least [Formula: see text], then the vector field has a rational first integral. Llibre and Zhang used only Linear Algebra to provide a shorter and easier proof of the result given by Jouanolou. We use ideas of Llibre and Zhang to extend the Jouanolou result to polynomial vector fields defined on algebraic regular hypersurfaces of ℝn+1, this extended result completes the standard results of the Darboux theory of integrability for polynomial vector fields on regular algebraic hypersurfaces of ℝn+1.
Communications in Contemporary Mathematics, 2013
We characterize the global phase portraits in the Poincaré disc of all the planar Lotka–Volterra ... more We characterize the global phase portraits in the Poincaré disc of all the planar Lotka–Volterra quadratic polynomial differential systems having a Darboux invariant.
Bulletin des Sciences Mathématiques, 2013
The quadrics here considered are the nine real quadrics: parabolic cylinder, elliptic cylinder, h... more The quadrics here considered are the nine real quadrics: parabolic cylinder, elliptic cylinder, hyperbolic cylinder, cone, hyperboloid of one sheet, hyperbolic paraboloid, elliptic paraboloid, ellipsoid and hyperboloid of two sheets. Let Q be one of these quadrics. We consider a polynomial vector field X = (P, Q, R) in R 3 whose flow leaves Q invariant. If m 1 = degree P , m 2 = degree Q and m 3 = degree R, we say that m = (m 1 , m 2 , m 3) is the degree of X. In function of these degrees we find a bound for the maximum number of invariant conics of X that result from the intersection of invariant planes of X with Q. The conics obtained can be degenerate or not. Since the first six quadrics mentioned are ruled surfaces, the degenerate conics obtained are formed by a point, a double straight line, two parallel straight lines, or two intersecting straight lines; thus for the vector fields defined on these quadrics we get a bound for the maximum number of invariant straight lines contained in invariant planes of X. In the same way, if the conic is non degenerate, it can be a parabola, an ellipse or a hyperbola and we provide a bound for the maximum number of invariant non degenerate conics of the vector field X depending on each quadric Q and of the degrees m 1 , m 2 and m 3 of X .
We study the polynomial vector fields X = n+1
Rocky Mountain Journal of Mathematics, 2015
We provide explicit expressions for the Liouvillian first integrals of the quadratic polynomial d... more We provide explicit expressions for the Liouvillian first integrals of the quadratic polynomial differential systems having an integrable saddle.
International Journal of Bifurcation and Chaos, 2012
Using sophisticated techniques of Algebraic Geometry, Jouanolou in 1979 showed that if the number... more Using sophisticated techniques of Algebraic Geometry, Jouanolou in 1979 showed that if the number of invariant algebraic hypersurfaces of a polynomial vector field in ℝn of degree m is at least [Formula: see text], then the vector field has a rational first integral. Llibre and Zhang used only Linear Algebra to provide a shorter and easier proof of the result given by Jouanolou. We use ideas of Llibre and Zhang to extend the Jouanolou result to polynomial vector fields defined on algebraic regular hypersurfaces of ℝn+1, this extended result completes the standard results of the Darboux theory of integrability for polynomial vector fields on regular algebraic hypersurfaces of ℝn+1.
Communications in Contemporary Mathematics, 2013
We characterize the global phase portraits in the Poincaré disc of all the planar Lotka–Volterra ... more We characterize the global phase portraits in the Poincaré disc of all the planar Lotka–Volterra quadratic polynomial differential systems having a Darboux invariant.
Bulletin des Sciences Mathématiques, 2013
The quadrics here considered are the nine real quadrics: parabolic cylinder, elliptic cylinder, h... more The quadrics here considered are the nine real quadrics: parabolic cylinder, elliptic cylinder, hyperbolic cylinder, cone, hyperboloid of one sheet, hyperbolic paraboloid, elliptic paraboloid, ellipsoid and hyperboloid of two sheets. Let Q be one of these quadrics. We consider a polynomial vector field X = (P, Q, R) in R 3 whose flow leaves Q invariant. If m 1 = degree P , m 2 = degree Q and m 3 = degree R, we say that m = (m 1 , m 2 , m 3) is the degree of X. In function of these degrees we find a bound for the maximum number of invariant conics of X that result from the intersection of invariant planes of X with Q. The conics obtained can be degenerate or not. Since the first six quadrics mentioned are ruled surfaces, the degenerate conics obtained are formed by a point, a double straight line, two parallel straight lines, or two intersecting straight lines; thus for the vector fields defined on these quadrics we get a bound for the maximum number of invariant straight lines contained in invariant planes of X. In the same way, if the conic is non degenerate, it can be a parabola, an ellipse or a hyperbola and we provide a bound for the maximum number of invariant non degenerate conics of the vector field X depending on each quadric Q and of the degrees m 1 , m 2 and m 3 of X .