I am training myself on oscillations. The topic is self-sustained oscillations. The claim is these oscillations are NOT forgetful about their initial condition as opposed to conservative oscillators which are. What I don't understand is what are our prior information when we want to recover the initial condition? In conservative oscillators, we know that the state of the system is a closed curve and in self-sustained ones it is a limit cycle, but according to uniqueness and existence theorem for ODEs, knowing the state of the system at time $t$, i.e. $x(t)$, will let you recover $x(0)$. So what is "it" that is not forgotten in the former and is forgotten in the latter? And what is the prior knowledge that helps the recovery?
Note: If there are people out there who don't have the time to write the answer, I would be happy to be referred to a reference book.