As we’ve seen, in one-dimensional phase spaces the flow is extremely confined— all trajectories are forced to move monotonically or remain constant. In higher-dimensional phase spaces, trajectories have much more room to maneuver, and so a wider range of dynamical behavior becomes possible. Rather than attack all this complexity at once, we begin with the simplest class of higher-dimensional systems, namely linear systems in two dimensions. These systems are interesting in their own right, and, as we’ll see later, they also play an important role in the classification of fixed points of nonlinear systems. We begin with some definitions and examples.