On the number of limit cycles for perturbed pendulum equations (Q285654)

The authors study the number of limit cycles that bifurcate from the closed ovals of the unperturbed pendulum equation NEWLINE\[NEWLINE\ddot{x}+\sin(x)= 0.NEWLINE\]NEWLINE This equation can be written as the Hamiltonian system NEWLINE\[NEWLINE \dot{x}=y, \quad \dot{y}=-\sin(x), NEWLINE\]NEWLINE which is defined on the cylinder \([-\pi,\pi] \times \mathbb{R}\). The total energy (i.e. Hamiltonian) is NEWLINE\[NEWLINE H(x,y)=\frac{y^2}{2}+1 - \cos(x). NEWLINE\]NEWLINE Note that the orbits of this system form three period annuli in the separated regions: \(\mathcal{R}^0\) denotes the period annulus covered by periodic orbits surrounding the origin of coordinates and with energy levels \(h \in (0,2)\); \(\mathcal{R}^{+}\) (resp. \(\mathcal{R}^{-}\)) denotes the period annulus covered by periodic orbits around the cylinder in which the system is defined with energy levels \(h \in (2,\infty)\) and \(y>0\) (resp. \(y<0\)). The region \(\mathcal{R}^0\) is called oscillatory region and the regions \(\mathcal{R}^{\pm}\) form the rotary region.NEWLINENEWLINEThe authors consider the following perturbation of this equation NEWLINE\[NEWLINE \ddot{x}+\sin(x)=\varepsilon\sum_{s=0}^{m} Q_{n,s}(x)\dot{x}^{s}, NEWLINE\]NEWLINE where for each \(s\) the functions \(Q_{n,s}(x)\) are trigonometric polynomials of degree at most \(n\) and \(\varepsilon>0\) is a small parameter. The authors are interested in quantifying in terms of \(m\) and \(n\) the number of limit cycles that bifurcate from the closed ovals of the unperturbed pendulum equation. To do so the notion of first Melnikov function is used. A first result provides the structure of this first Melnikov function in each of the regions \(\mathcal{R}^0\) and \(\mathcal{R}^{\pm}\) and an upper bound for its number of isolated zeros is provided. A second and third result gives a sharp upper bound by adding some hypothesis on the exponents \(s\), first for the region \(\mathcal{R}^0\) and then for the regions \(\mathcal{R}^{\pm}\).NEWLINENEWLINEThe tools used to prove these results make use of some general concepts such as Chebyshev system and can be used in other frameworks. As a last part, the simultaneous bifurcation of limit cycles in the oscillatory and rotary regions is studied for some particular cases.