Jordi Villadelprat

The period function for second-order quadratic ODEs is monotone

AbstractVery little is known about the period function for large families of centers. In one of the pioneering works on this problem, Chicone [3] conjectured that all the centers encountered in the family of second-order differential equationsn

A NOTE ON THE CRITICAL PERIODS OF POTENTIAL SYSTEMS

ddot x = V(x,dot x)

A Poincaré-Hopf theorem for noncompact manifolds☆

n, beingV a quadratic polynomial, should have a monotone period function. Chicone solved some of the cases but some others remain still unsolved. In this paper we fill up these gaps by using a new technique based on the existence of Lie symmetries and presented in [8]. This technique can be used as well to reprove all the cases that were already solved, providing in this way a compact proof for all the quadratic second-order differential equations. We also prove that this property on the period function is no longer true whenV is a polynomial which nonlinear part is homogeneous of degreen>2.

On the period function of centers in planar polynomial hamiltonian systems of degree four

Given an analytic family of vector fields in 2 having a saddle point, we study the asymptotic development of the time function along the union of the two separatrices. We obtain a result (depending uniformly on the parameters) which we apply to investigate the bifurcation of critical periods of quadratic centres.

An effective algorithm to compute Mandelbrot sets in parameter planes

In this paper, we consider the planar differential system associated with the potential Hamiltonian H(x,y) = (1/2)y2+V(x) where V(x) = (1/2)x2+(a/4)x4+(b/6)x6 with b ≠ 0. This family of differentia...

ISOCHRONICITY FOR SEVERAL CLASSES OF HAMILTONIAN SYSTEMS

Abstract In this paper we show that in dimension greater or equal than 3 the index of a stable critical point can be any integer. More concretely, given any k∈ Z and n⩾3 we construct a C ∞ vector field on R n with a unique critical point which is stable (in positive and negative time) and has index equal to k. This result extends previous ones on the index of stable critical points.

The period function of reversible quadratic centers

Abstract We provide the natural extension, from the dynamical point of view, of the Poincare-Hopf theorem to noncompact manifolds. On the other hand, given a compact set K being an attractor for a flow generated by a C1 tangent vector field X on an n-manifold, we prove that the Euler characteristic of its region of attraction A , χ( A ), is defined and satisfies Ind A (X) = (−1)nχ( A ). Finally we prove that χ( A ) = χ(K) when K is an euclidean neighbourhood retract being asymptotically stable and invariant.

Algebraic and analytical tools for the study of the period function

In this paper we study non-degenerate centers of planar polynomial Hamiltonian systems. We prove that if the differential system has degree four then the period function of the center tends to infinity as we approach to the boundary of its period annulus. The proof takes advantage of the geometric properties of the period annulus in the Poincaré disc and it requires the study of the so called cubic-like Hamiltonian systems, namely the differential systems associated to a Hamiltonian function of the formH(x, y)=A(x)+B(x)y+C(x)y2+D(x)y3. Concerning the centers of this family of differential systems, we obtain an analytic expression of its period function. From our point of view this expression constitutes the first step in order to find the isochronicity conditions in the family.