Ana Cristina Mereu

Limit cycles of the generalized polynomial Liénard differential equations

We apply the averaging theory of first, second and third order to the class of generalized polynomial Lienard differential equations. Our main result shows that for any n , m ≥ 1 there are differential equations of the form ẍ + f ( x ) ẋ + g ( x ) = 0, with f and g polynomials of degree n and m respectively, having at least [( n + m − 1)/2] limit cycles, where [·] denotes the integer part function.

Limit cycles of resonant four-dimensional polynomial systems

We study the bifurcation of limit cycles from four-dimensional centres inside a class of polynomial differential systems. Our results establish an upper bound for the number of limit cycles which can be prolonged in function of the degree of the polynomial perturbation considered, up to first-order expansion of the displacement function with respect to small parameter. The main tool for proving such results is the averaging theory.

INVARIANT TORI FULFILLED BY PERIODIC ORBITS FOR FOUR-DIMENSIONAL

We provide an algorithm for studying invariant tori fulfilled by periodic orbits of a perturbed system which emerge from the set of periodic orbits of an unperturbed linear system in p : q resonance. We illustrate the algorithm with an application.