Hector Giacomini

On the nonexistence, existence and uniqueness of limit cycles

We present two new criteria for studying the nonexistence, existence and uniqueness of limit cycles of planar vector fields. We apply these criteria to some families of quadratic and cubic polynomial vector fields, and to compute an explicit formula for the number of limit cycles which bifurcate out of the linear centre , when we deal with the system . Moreover, by using the second criterion we present a method to derive the shape of the bifurcated limit cycles from a centre.

Darboux integrability and the inverse integrating factor

Abstract We mainly study polynomial differential systems of the form dx / dt = P ( x , y ), dy / dt = Q ( x , y ), where P and Q are complex polynomials in the dependent complex variables x and y , and the independent variable t is either real or complex. We assume that the polynomials P and Q are relatively prime and that the differential system has a Darboux first integral of the form H=f 1 λ 1 ⋯f p λ p exp h 1 g 1 n 1 μ 1 ⋯ exp h q g q n q μ q , where the polynomials f i and g j are irreducible, the polynomials g j and h j are coprime, and the λ i and μ j are complex numbers, when i =1,…, p and j =1,…, q . Prelle and Singer proved that these systems have a rational integrating factor. We improve this result as follows. Assume that H is a rational function which is not polynomial. Following to Poincare we define the critical remarkable values of H . Then, we prove that the system has a polynomial inverse integrating factor if and only if H has at most two critical remarkable values. Under some assumptions over the Darboux first integral H we show, first that the system has a polynomial inverse integrating factor; and secondly that if the degree of the system is m , the homogeneous part of highest degree of H is a multi-valued function, and the functions exp( h j / g j ) are exponential factors for j =1,…, q , then the system has a polynomial inverse integrating factor of degree m +1. We also present versions of these results for real polynomial differential systems. Finally, we apply these results to real polynomial differential systems having a Darboux first integral and limit cycles or foci.

The problem of distinguishing between a center and a focus for nilpotent and degenerate analytic systems

Abstract In this work we study the centers of planar analytic vector fields which are limit of linear type centers. It is proved that all the nilpotent centers are limit of linear type centers and consequently the Poincare–Liapunov method to find linear type centers can be also used to find the nilpotent centers. Moreover, we show that the degenerate centers which are limit of linear type centers are also detectable with the Poincare–Liapunov method.

Integrals of motion and the shape of the attractor for the Lorenz model

Abstract We consider three-dimensional dynamical systems, as for example the Lorenz model. For these systems, we introduce a method for obtaining families of two-dimensional surfaces such that trajectories cross each surface of the family in the same direction. To obtain these surfaces, we are guided by the integrals of motion that exist for particular values of the parameters of the system. Nonetheless families of surfaces are obtained for arbitrary values of these parameters. Only a bounded region of the phase space is not filled by these surfaces. The global attractor of the system must be contained in this region. In this way, we obtain information on the shape and location of the global attractor. These results are more restrictive than similar bounds that have been recently found by the method of Lyapunov functions.

A unique extremal metric for the least eigenvalue of the Laplacian on the Klein bottle

We prove the following conjecture recently formulated by Jakobson, Nadirashvili and Polterovich \cite{JNP}: on the Klein bottle

Liouvillian first integrals for the planar Lotka-Volterra system

\mathbb{K}

Inverse Jacobi multipliers

, the metric of revolution

Analytical approximation schemes for solving exact renormalization group equations. II Conformal mappings

g_0= {9+ (1+8\cos ^2v)^2\over 1+8\cos ^2v} \left(du^2 + {dv^2\over 1+8\cos ^2v}\right),