Bartomeu Coll

Some theorems on the existence, uniqueness, and nonexistence of limit cycles for quadratic systems

Abstract Given a quadratic system (QS) with a focus or a center at the origin we write it in the form ẋ = y + P 2 (x, y) , ẏ = − x + dy + Q 2 (x, y) where P 2 and Q 2 are homogeneous polynomials of degree 2. If we define F ( x , y ) = ( x − dy ) P 2 ( x , y ) + yQ 2 ( x , y ) and g ( x , y ) = xQ 2 ( x , y ) − yP 2 ( x , y ) we give results of existence, nonexistence, and uniqueness of limit cycles if F ( x , y ) g ( x , y ) does not change of sign. Then, by using these results plus the properties on the evolution of the limit cycles of the semicomplete families of rotated vector fields we can study some particular families of QS, i.e., the QS with a unique finite singularity and the bounded QS with either one or two finite singularities.

THE CENTER PROBLEM FOR DISCONTINUOUS LIÉNARD DIFFERENTIAL EQUATION

We prove that the Lyapunov constants for differential equations given by a vector field with a line of discontinuities are quasi-homogeneous polynomials. This property is strongly used to derive the general expression of the Lyapunov constants for two families of discontinuous Lienard differential equations, modulus some unknown coefficients. In one of the families these coefficients are determined and the center problem is solved. In the other family the center problem is just solved by assuming that the coefficients which appear in these expressions are nonzero. This assumption on the coefficients is supported by their computation (analytical and numerical) for several cases.

PHASE PORTRAITS OF THE QUADRATIC SYSTEMS WITH A POLYNOMIAL INVERSE INTEGRATING FACTOR

We classify the phase portraits of all planar quadratic polynomial differential systems having a polynomial inverse integrating factor.

Quadratic systems with a unique finite rest point.

We study phase portraits of quadratic systems with a unique finite singularity. We prove that there are 111 different phase portraits without limit cycles and that 13 of them are realizable with exactly one limit cycle. In order to finish completely our study two problems remain open: the realization of one topologically possible phase portrait, and to determine the exact number of limit cycles for a subclass of these systems.