Part II: Two-dimensional systems: flows in the plane

This part is dedicated to the study of systems having a two-dimensional phase space. We shall revisit the two-dimensional systems of the opening part with new eyes thanks to the concepts and techniques we are going to introduce. We shall also see new examples coming from diverse fields.

As we have seen, the solutions of a one-dimensional system $\dot{x}=f(x)$ behave in a simple way because they are forced to move monotonically or remain constant. In higher-dimensional systems, solutions can follow trajectories that have much room to maneuver, which implies a wider range of dynamical behavior. We focus here on two-dimensional systems for which this range is rather well understood. Later on, we shall see that adding one or more dimensions brings some new dynamical behaviors that are impossible in dimension two.

The general form of a two-dimensional system is

$$ \begin{cases} \dot{x}=f(x,y)\\ \dot{y}=g(x,y) \end{cases} $$

where $(x,y)\in\mathbb{R}^2$ and $f,g:\mathbb{R}^2\to\mathbb{R}$. In vector notation this can be written as

$$ \dot{\boldsymbol{x}}=\boldsymbol{f}(\boldsymbol{x}) $$

where $\boldsymbol{x}=(x,y)$ and $\boldsymbol{f}(\boldsymbol{x})=(f(\boldsymbol{x}),g(\boldsymbol{x}))$. As we have seen in the opening part, $\boldsymbol{x}$ represents a point in the $xy$-plane and $\dot{\boldsymbol{x}}$ is the velocity vector at that point which is determined by the vector field $\boldsymbol{f}(\boldsymbol{x})$. This vector field can be thought of as giving the velocity of an imaginary fluid at each point. Then, a particle dropped at position $\boldsymbol{x}_0$ at some time $t_0$ in this fluid will follow a certain trajectory that corresponds to the curve traced out by the solution $\boldsymbol{x}(t)$ such that $\boldsymbol{x}(t_0)=\boldsymbol{x}_0$. This is made precise and rigorous by an extension of the fundamental theorem of existence and uniqueness stated for one-dimensional systems.