Consider the self-maps $$ F_{\mu_1,\mu_2}(t) : \mathbb{R}/\mathbb{Z}\to\mathbb{R}/\mathbb{Z};\quad t\mapsto t + \mu_1 + \mu_2\sin(2\pi t) $$ of the circle. (Here $\mu_1\in\mathbb{R}$ and $\mu_2>0$ are parameters.) This family, known as the standard family, was introduced by Arnold in 1961 to model periodically forced nonlinear oscillators, and has since served as one of the simplest models of...