Small limit cycles bifurcating in pendulum systems under trigonometric perturbations.

In this paper, we consider the bifurcation of small-amplitude limit cycles near the origin in perturbed pendulum systems of the form x ˙ = y , y ˙ = − sin ⁡ (x) + ε Q (x , y) , where Q (x , y) is a smooth or piecewise smooth polynomial in the triple (sin ⁡ (x) , cos ⁡ (x) , y) with free coefficients. We obtain sharp upper bounds on the number of positive zeros of its associated first order Melnikov function near h = 0 for Q (x , y) being smooth and piecewise smooth with the discontinuity at y = 0 or x = 0 , respectively. [ABSTRACT FROM AUTHOR]