Christmas light leds

I bought a solar christmas light LED with only 2 wires for the whole string but the controller can control 2 groups of led. So I hooked it up to a scope and saw that it uses PWM to control brightness and it reverses the polarity to control which string should light up. Can anyone point me to the right direction on how I can replace the built in controller with arduino.

I have seen some simple tutorials out there that shows how to light up an LED using a mosfet, so what I am planning right now is to use the output of that mosfet and feed that to an h-bridge to switch the polarity.

Is there a simpler way to do this?

H-bridge should be enough, no need for an additional MOSFET.

No, an H-bridge controls both polarity and PWM. You do not use an additional FET with it.