Help me! Aggressive cat invading my home!!!

Hi I'm new to the forum & I desperately need some helpful advice! I have a 3 year old neutered male cat who's black & white & looks like the Felix cat. His name is Billie & he's the most placid cat I've every met. He has a magnetic cat flap, that I thought only he could use but I've discovered that a VERY aggressive Chocolate coloured unneutered Oriental type cat comes into my home, nearly every day & not only attacks my cat, if he's home, but sprays EVERY where!!! I discovered him in my home on a couple of occasions & have sprayed him with water, scared him by shouting at him, scared him with the vacuum cleaner, but he still comes back. His VERY cleaver & if he hears the slightest noise or see the slightest shadow, he runs away. He comes through the cat flap with brute force. He always attack my cat & although I check Billie out all the time, to make sure not scratches get infected, it's not always, possible to see them as he has black fur. This alien cat ( I call him that as that's what he reminds me of) has cost me a fortune in vet bills, as quite often the scratches get infected & I have to take Billie to the vet for antibiotics. I don't know who owns this cat but I'm at my wits end & feel that this situation has gone on for nearly 3 years. The only solution that I've found is to close every door in my flat, so neither cats can go anywhere but the hall way. This causes distress to Billie, as he must think I'm doing it to him but I don't want my flat to smell of cat wee! I clean the wee with a special solutions I got from the pet shop. He's also quite depressed & often doesn't want to stay home or sits in front of the cat flap waiting for the alien cat to try to come in. I've also got Felliway, to try to calm Billie, but it doesn't help that much. I don't know who owns alien nor where he lives. Someone said to write a note & put it on his collar & hope the owners might do some thing but I need to catch him first! Some one out there must have had a similar problem, I can't be the only one. PLEASE help my poor Billie, he's such a nice cat!!