Limit cycles and chaos in equations of the pendulum type (Q1175812)

The author proves that for sufficiently small \(\varepsilon\) the equation \[ \ddot x + \sin x = \varepsilon \dot x\cos nx,\quad n\in\mathbb{N}, \tag{*} \] has exactly \(n-1\) coarse limit cycles. Equations considered are part of a bigger family of equations: \(\ddot x+A(x)=\varepsilon f(x,\dot x,\nu t,\varepsilon)\), where \(A\) is a \(2\pi\)-periodic function and \(f\) is a periodic function and \(\varphi=\nu t\) with the same period. Equations of this kind govern the motions of various pendulums. Equation (*) is considered, together with the Van der Pol equation, one of the most fundamental in the nonlinear oscillation theory. In the paper a full qualitative investigation of (*) is presented.