In the most general abstract sense suppose you have a system which is in some state and there is a deterministic rule which updates it to the next state. For now we can just represent the states and transitions as a bunch of circles and arrows connecting them. Draw it on a whiteboard if you must. Since transitions are deterministic, each state only transitions to a single other state. There might be two states that lead to one, but never one that leads to two. In general then, no matter which state you start in, it must either eventually lead to some closed loop of the same states over and over again or it runs off the page in an infinite succession of states. The first option is what we would call an "attractor" and the latter which could be aptly called a "repulsor". There's a third possibility I didn't mention (which is hard to conceptualize drawing things as a 2d diagram of states and transitions). You can have the state run off in an infinite succession of states, but rather than running off the page to infinity, it goes around and around visiting similar states over and over but never technically visiting the same state twice. Its a strange attractor.
Now to be less general. If your system states are some continuous parameter, instead of circles and arrows it makes sense to represent the nth state as x_n and to represent the transition rule to the next state as a difference equation x_{n+1} = f(x_n). If your system has more than one defining parameter, do the same thing but with vector valued quantities. Lastly, if your transitions themselves are continuous rather than discrete, replace the difference equation with a differential equation dx/dt = f(x). No matter which of these you are using to represent system states and define how that system updates, the same basic patterns as before are still the only possibility. The solution to your equation either converges to some fixed point, diverges to infinity (usually exponential growth), or the curve stays bounded within some region of state space getting arbitrarily close to forming a loop but always just missing itself.