"BP", looks like "Bush Putton".

Or "Button, Push" may be more grammatically correct in their original language (French?).
Seems the purpose is to pulse the circuit just to give an initial kick.
Q2 and Q3 appear to be swapped and reversed. I think they are intended to be a complementary emitter follower, to drive the gate; as shown they would just short out the supply, not very fun. R4 isn't really needed and can be removed (shorted out).
Note that the loop between Q4, D3 and C3 must be short; place them close together.
Beware that the output transistor is unlimited in current, and is dumping all the reactive (magnetic) energy from the inductor into a diode, which is shorting out the inductor half the time, which won't do any favors for discharging said magnetic field. It may be better to place a TVS in series with D3, though it will probably get rather hot rather fast. Better still, dump the energy into a capacitor (making a boost circuit), then dissipate the energy more gradually with a power resistor (discharging the capacitor's excess charge into the supply).
The next best thing is switching both the top and bottom of the coil at the same time (called a two-switch flyback converter), but this requires more hardware to pull off, so I won't try to list all the changes that would need..
The concern is that, if you expect real mechanical power out of this system (i.e., a few watts of friction is probably spent bouncing around that magnet), expect poor efficiency, say 10s of watts input. A lot of which will be lost in the magnetic field, which is weakly coupled to the magnet on account of the air-core coil, the large distance between them and so on. So the circuit will be a boost converter, first and foremost, with a quirky side-effect of also making a magnet spin.
Tim