On the diversity of Poincaré mappings for cubic equations with variable coefficients (Q1566489)

Consider the scalar differential equation \[ dx/dt = a_3 (t)x^3 + a_2 (t) x^2 + a_1 (t)x + a_0 (t) \tag \(*\) \] for \( t \in [0,1]\) with \(a_i \in C^1\) for \(i=1,2,3\). Let \(\varphi (t, x)\) be the solution to \((*)\) satisfing \(\varphi (0,x)=x\), by \(\pi (x): = \varphi (1,x)\) we denote the Poincaré map of \((*)\). A solution \(\varphi\) to \((*)\) satisfying \(\varphi (1,x)=x\) is called a cyclic solution. By studying the set of Poincaré maps, the author proves that \((*)\) can have arbitrarily many (nondegenerate) cyclic solutions.