# Part III: Beyond flows in the plane: quasi-periodicity & chaos

In this part, we touch upon systems having a phase space with three or more dimensions. There is a huge gap in complexity with respect to two-dimensional systems. Indeed, we saw that the fate of solutions in dimension two is well classified. In particular, the two main types of attractors are fixed points and limit cycles.

In higher dimension, an amazing phenomenon is possible: *deterministic chaos*, which means unpredictability. It manifests itself with the presence of a complicated object called a *strange attractor*, typically of fractal nature. We had a glimpse of this phenomenon in the opening part with a population model.

We shall also see another new phenomenon: *quasi-periodicity*. We shall study an innocent-looking mechanical system, namely a double pendulum. Such a system can be periodic, quasi-periodic, and even chaotic!