Why does my homemade liquid engine run rich then surge at full pressure?

Seen almost this exact thing on my nitrous test rig. Check the tank side first. When you blow down a bottle of nitrous fast, the liquid in the tank boils and the whole bottle chills toward its saturation temperture at that pressure. The pressure sags during the burn, and if your chamber pressure wiggles around that sagging feed pressure you get exactly the lazy-then-surging cycle you are describing. Weigh the bottle before and after a run. If it lost almost nothing but the pressure drop is big, you are pulling vapor off the top instead of liquid, and vapor flow is what gives you that sputtering pulsing burn. Keeping some liquid sloshing in there and running off the liquid side with a dip tube is the usual fix. Line layout can cause the same symptom though, so do not skip it. Long runs in warm shop air, or a line that passes near the hot engine mounts, will boil off little vapor pockets that collapse and slam open again. Feel the line after a run and see where it got cold. A cold spot right after a fitting is your boiling point. Simplest test: run it once with a fresh bottle topped up and still, once with the same bottle warm and nearly empty, and once with the feed line packed in ice. If the pulsing moves with the bottle condition it is the tank. If it follows the line temperature it is the plumbing.

And log the tank pressure during a burn if you can. Watching the gauge sag in step with each lazy phase is very convincing. Four seconds is right about where the fun starts, the first second or two everything is cold and happy and then the thermal side of things wakes up.

One thing I would add after chasing a similar cycle on my own rig is that a marginal pressure drop across the injector can turn a small wobble into a self sustaining oscillation, the classic chugging behavior. If your feed pressure and chamber pressure are sitting too close togehter the whole loop just amplifies every little disturbance instead of damping it out. Try raising the tank pressure a notch and see if the pulse period shortens or disappears. If the character of the oscillation changes with tank pressure rather than with temperature, that points you at the injector instead of the plumbing.

The replies above cover the nitrous side well, so I'll add the fuel half. On my little IPA burner I chased a similar pulse for weeks before I realized the fuel was flashing right at the injector face once the chamber walls soaked through. First couple seconds the head is cold and acts like a heat sink, then around the four second mark the whole injector comes up to temperature and everything downstream of it changes. Mine settled out once I added a small orifice upstream of the injector to drop pressure ahead of the hot zone. Also worth doing: film the plume and count pulses per second. If it's a steady rhythm rather than random sputter, that tells you it's a coupling problem, not just a supply hiccup.

Update: weighed everything against the replies and the timing lines up with the heat soak idea. It pulses right around when the walls should be getting hot, not in the first second. So if the fuel is flashing at the face once the head warms up, would extra cooling around the injector help, or does the feed pressure still need more margin first?