Limit cycles of piecewise differential equations on the cylinder (Q2037518)

The authors consider a class of scalar non-autonomous differential equations with a discontinuity in the independent variable \[ \frac{{dx}}{{dt}} = \begin{cases} f(t,x)& \text{for} \quad 0 \le t \le \pi,\\ g(t,x) & \text{for} \quad \pi \le t \le 2\pi \end{cases} \tag{1} \] with \[ f(t,x) = \sum_{i=0}^n a_i(t)x^i, \qquad g(t,x) = \sum_{j=0}^m b_j(t)x^j, \] where \(a_i\) and \(b_j\) are \(2\pi\)-periodic functions. Thus, (1) can be understood as a differential equation on a cylinder. The maximum number of solutions \(x(t)\) of (1) satisfying \(x(0)=x(2\pi)\) is denoted by \(H(n,m)\). The authors prove \[ H(1,1) = 1, \quad H(1,2) = 2, \quad H(1,3) = \infty. \] Reviewer's remark: The authors forgot to assume some smoothness for \(a_i\) and \(b_j\).