The number of limit cycles due to polynomial perturbations of the harmonic oscillator

Abstract

We consider arbitrary polynomial perturbationsformula hereof the harmonic oscillator. In (1), f and g are polynomials of x, y with coefficients depending analytically on the small parameter ε. Let us denote n = max (deg f, deg g), H = ½(x2 + y2). Using the energy level H = h as a parameter, we can express the first return mapping of (1) in terms of h and ε. For the corresponding displacement function d(h, ε) = [Pscr ](h, ε)−h we obtain the following representation as a power series in ε:formula herewhich is convergent for small ε. The Melnikov functions Mk(h) are defined for h[ges ]0. Each isolated zero h0∈ (0, ∞) of the first non-vanishing coefficient in (2) corresponds to a limit cycle of (1) emerging from the circle x2 + y2 = 2h0 when ε increases from zero. Our main result in this paper is the following.

Limit cycles in planar piecewise linear differential systems with nonregular separation line

“…It is well known that, up to a first order analysis in ε, perturbing the linear center with arbitrary polynomials of degree n, we can only obtain [(n−1)/2] limit cycles for the perturbed system, where [·] denotes the integer part function, see [16]. On the other hand but in the same class of systems, in [20] it is proved that the maximum number of limit cycles is lower or equal than [N (n − 1)/2]. This upper bound, in general, is reached when n is large enough and N = 2.…”

“…, 10. In [20], considering perturbations of a linear center by quadratic polynomials, it is shown that when N = 1, . .…”

On the Bifurcation of Limit Cycles Due to Polynomial Perturbations of Hamiltonian Centers

“…Following the ideas of Iliev, in [2] the authors study the number of zeros of the Melnikov function at any order for system (2) in the case in which G = (x 2 + y 2 ) m−1 . Our system (2) generalizes the systems studied in [8] and [2] because the unperturbed part is taken to be more general and we extend their results to this more general situation.…”

“…The first one goes back to Iliev [8] were he was the first one in doing so. Due to the fact that this is an extremely hard problem involving very difficult computations, he started with the linear center, i.e., system (2) with G = 1.…”

Limit cycles of discontinuous piecewise polynomial vector fields

“…Iliev [19] in 1999 considered the polynomial vector fields X (x, y) = − y + εP (x, y, ε), x + εQ(x, y, ε) , of degree n > 1 (i.e. the maximum of the degrees of polynomials P and Q is n), which depend analytically on the small parameter ε, and he studied how many limit cycles can bifurcate from the periodic orbits of the linear centerẋ = −y,ẏ = x when ε > 0 is sufficiently small.…”