The arrows correspond to trajectories starting outside the attractor, but ending up in a continuing cycle along the attractor. When the non-linearity of the system becomes more important, more complicated attractors may appear. The final trajectory in which the system settles may have a very irregular shape, without any apparent periodicity. In general, an attractor is a region in the space of possible states which the system can enter but not leave. Attractors with non-integer, fractal dimensions are called strange attractors.
Trajectories moving inside a strange attractor are totally chaotic.
Another characteristic of highly non-linear systems is that they have in general several attractors. A simple thermodynamical system, on the other hand, has only one attractor, the maximum entropy state. When there is more than one attractor, the main question is in which of those attractors the system will end up. Imagine that each attractor corresponds to a lake or sea, and that the trajectories leading into an attractor correspond to the rivers and streams flowing into those lakes. Depending on where it falls, rainwater will follow either one river or another, ending up in either one lake or another.