Joan Torregrosa

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We study the center-focus problem as well as the number of limit cycles which bifurcate from a weak focus for several families of planar discontinuous ordinary differential equations. Our computations of the return map near the critical point are performed with a new method based on a suitable decomposition of certain one-forms associated with the(More)
We consider a system of the form ẋ = Pn(x, y) + xRm(x, y), ẏ = Qn(x, y) + yRm(x, y), where Pn(x, y), Qn(x, y) and Rm(x, y) are homogeneous polynomials of degrees n, n and m, respectively, with n ≤ m. We prove that this system has at most one limit cycle and that when it exists it can be explicitly found. Then we study a particular case, with n = 3 and m =(More)
Consider the planar ordinary differential equation ẋ = −y(1 − y)m, ẏ = x(1 − y)m, where m is a positive integer number. We study the maximum number of zeroes of the Abelian integral M that controls the limit cycles that bifurcate from the period annulus of the origin when we perturb it with an arbitrary polynomial vector field. One of the key points of our(More)
In this paper, we provide a lower bound for the maximum number of limit cycles of planar discontinuous piecewise linear differential systems defined in two half-planes separated by a straight line. Here, we only consider nonsliding limit cycles. For those systems, the interior of any limit cycle only contains a unique equilibrium point or a unique sliding(More)
Let / : M —» M be a C1 map on a C1 differentiate manifold. The map f is called transversal if for all m £ N the graph of fm intersects transversally the diagonal of M x M at each point (x, x) such that x is a fixed point of fm . We study the set of periods of / by using the Lefschetz numbers for periodic points. We focus our study on transversal maps(More)
A particular version of the 16th Hilbert’s problem is to estimate the number, M(n), of limit cycles bifurcating from a singularity of center-focus type. This paper is devoted to finding lower bounds for M(n) for some concrete n by studying the cyclicity of different weak-foci. Since a weak-focus with high order is the most current way to produce high(More)
Christopher in 2006 proved that under some assumptions the linear parts of the Lyapunov constants with respect to the parameters give the cyclicity of an elementary center. This paper is devote to establish a new approach, namely parallelization, to compute the linear parts of the Lyapunov constants. More concretely, it is showed that parallelization(More)
This paper concerns the study of small-amplitude limit cycles that appear in the phase portrait near an unfolded fake saddle singularity. This degenerate singularity is also known as a impassable grain. The normal form of the unperturbed vector field is like a degenerate flow box. Near the singularity,the phase portrait consists of parallel fibers, all of(More)