Hilbert Number for a Family of Piecewise Nonautonomous Equations

For family x'=(a₀+a₁cos t+a₂ sin t)|x|+b₀+b₁ cos t+b₂ sin t x ′ = ( a 0 + a 1 cos t + a 2 sin t ) | x | + b 0 + b 1 cos t + b 2 sin t , we solve three basic problems related with its dynamics. First, we characterize when it has a center (Poincaré center focus problem). Second, we show that each equation has a finite number of limit cycles (finiteness problem), and finally we give a uniform upper bound for the number of limit cycles (Hilbert problem 16).