The phase plane

Systems of differential equations: Non-linear differential equations

The phase plane

For linear systems of first-order differential equations we have introduced concepts like phase plane, phase portrait, direction field, isoclines, and equilibrium. These can also be used for non-linear systems of first-order differential equations, for example, for systems of differential equations of the form $\left\{\begin{aligned} \dfrac{\dd x}{\dd t} &= f(x,y)\\[0.25cm] \dfrac{\dd y}{\dd t} &= g(x,y)\end{aligned}\right.$ in the plane, where $f$ and $g$ are "neat" functions (here, "neat" means that derivatives exist and are continuous functions).

With a given choice of $(x_0,y_0)\in \mathbb{R}^2$ , there is exactly one curve $t\mapsto \bigl(x(t),y(t)\bigr)$ that satifies to this system, and for which $\bigl(x(0),y(0)\bigr)=(x_0,y_0)$ .

If $f(x_0,y_0)=g(x_0,y_0)=0$ then $(x_0,y_0)$ is called a singular point, and also a equilibrium, and then this curve is constant: $\bigl(x(t),y(t)\bigr)=(x_0,y_0)$ for all $t$ . When $(x_0,y_0)$ is not a singular point, then the function $t\mapsto \bigl(x(t),y(t)\bigr)$ is not constant (that is, really a curve instead of a point) and this curve is called the solution curve through $(x_0,y_0)$ .

Often, one draws a number of solution curves in one diagram, and something like this is then called a phase portrait. Usually one also draws then the vector field associated with the 2-dimensional system of differential equations. At each point $(x,y)$ in the phase portrait we can compute the vector $\cv{x'\\y'}$ as $\cv{f(x,y)\\g(x,y)}$ and draw it in the plane. Often one pays attention to the set of points in the plane where the vector $\cv{x'\\y'}$ is constant, the so-called isoclines. Especially one considers $\boldsymbol{x}$ -nullclines and $\boldsymbol{y}$ -nullclines, that is, points for which $\frac{\dd x}{\dd t}=0$ and $\frac{\dd y}{\dd t}=0$ , respectively. Singular points are intersections of $x$ -nullclines and $y$ -nullclines.

Lotka-Volterra equation

We consider a Lotka-Volterra system, used for example in predator-prey models: $\left\{\begin{aligned} \dfrac{\dd x}{\dd t} & =a x-b x y\\[0.25cm] \dfrac{\dd y}{\dd t} & =-c y+d x y\end{aligned}\right.$ As example we take $a=c=1$ and $b=d=2$ . The phase portrait below illustrates that there are various solution curves: periodic trajectories in the first quadrant catch the eye. The periodic orbits go around the equilibrium $(\tfrac{1}{2},\tfrac{1}{2})$ . If you are in such a periodic orbit and there occurs a disturbance, then you go to a periodic orbit with another 'amplitude'. The equilibrium $(0,0)$ is a saddle point. When $x$ and $y$ are variables which only have nonnegative values, then this saddle point is not so interesting. The red lines represent the $x$ -nullclines, that is, the points where the direction of solution curves is vertical, and the green lines form the $y$ -nullclines, that is, the points where the direction of solution curves is horizontal.

Van der Pol equation

We consider the Van der Pol equation $\left\{\begin{aligned} \dfrac{\dd x}{\dd t} & =\frac{1}{\varepsilon}\left(y+x-\frac{x^3}{3}\right)\\[0.25cm] \dfrac{\dd y}{\dd t} & =-\varepsilon x\end{aligned}\right.$ with $0<\varepsilon\ll 1$ . As example we choose a fairly large value $\varepsilon=\tfrac{1}{2}$ in order to show more clearly what happens. The phase portrait below illustrates that there now exists another type of solution curve, namely the limit cycle. This is a periodic trajectory, here drawn in the colour magenta, to which solutions curves converge. We have drawn in the figure below two types of solution curves: three curves converge from the inside to the limit cycle (black curves), and two curves converge from the outside to the limit cycle (blue curves). A small perturbation of the system does not really matter because the solution curve obviously converges then back to the limit cycle. The red lines represent the $x$ -nullclines, that is, the points where the direction of solution curves is vertical, and the green lines form the $y$ -nullclines, that is, the points where the direction of solution curves is horizontal.

You can also play with the following interactive computerversion of a phase portrait of a non-linear system of differential equation with a parameter $a$ .

In this section we will look again at the behaviour of solutions near an equilibrium.