Simple non-autonomous differential equations with many limit cycles

Consider the family of differential equations on the cylinder, dx/dt = a(t) + b(t) x , where x, t ∈ ℝ, and a, b are real, 1-periodic and smooth functions. The solutions satisfying x(0) = x(1) are called periodic orbits of the equation. The periodic orbits that are isolated in the set of all the periodic orbits are usually called limit cycles. We give a proof, which is self contained, that there is no upper bound for the number of limit cycles of the above type of equations.