Driving a transistor base with PWM
The circuit does not work because you have the bases of the MJ15003's strapped to +5V. Also, the TIP50 can not supply enough current.

Also, you confuse the device's MAXIMUM ratings with the dynamic characteristics. MAXIMUM rates tell you when the device will release the magic smoke.

You do not specify the maximum current you want to draw, I am assuming the maximum for a MJ15003 (about 20A). From the specs, figure 1, I see that the MJ15003 Hfe (current amplification) is about 20 at 20 Amps, meaning the base-emitter current should be about 1A. Figure 3 tells me that the B-E voltage will be about 1.4 Volts.

A TIP50 can supply at most 1A, so it is not powerful enough for this application. It's Hfe is about 30, so to feed 2A as required by the 2 power transistors, it will draw at least 66 mA from the AtMega. This is more than it can source (40mA), so you will need an extra stage.

Have a look at the Darlington configuration for how to make a power amplifier. You will probable need three transistors, not just two, and use one TIP50 for each MJ15003.