Phase Portraits

phase portraits

A Phase Portrait is one type of qualitative tool for studying first order systems of ODEs. For the first order system,

x' = f(x,y)
y' = g(x,y),

one can sketch its phase portrait on what is called the phase plane. The phase plane is simply an xy-plane (for the current example). That is, the phase plane is the plane whose coordinate axes are determined by the dependent variables of the given first order system. The phase portrait is the xy-plane which contains information about the equilibrium solutions as well as other solution trajectories of the given system. Choose an initial point (x(0),y(0)) in the xy-plane. By varying the independent variable t and plotting the curve of points given by(x(t),y(t)) originating from that our initial point, we get a solution curve or trajectory in the xy-plane. The phase portrait is simply the xy-plane containing several solution curves which describe the behavior of the solution trajectories of the first order system. Constructing the phase portrait amounts to sketching several of these solution curves based on the equilibrium point analysis and on information about the long-term behavior of other solution curves.

Here is an alternate way of thinking about it. We think of the independent variable t as the time variable and the vector (x(t),y(t)) as representing the position of the solution curve in the xy-plane at time t. As t varies, the point (x(t),y(t)) ``moves'' along the solution trajectory in the xy-plane. The velocity at which the point moves is given by the vector of derivatives (x'(t),y'(t)) (we sometimes use the notation (dx/dt,dy/dt) in class). First, we know that the given first order system holds. If we choose a point (x,y) in the xy-plane, then the velocity vector for the solution curve passing through that point in the plane is given by (x',y')=(f(x,y),g(x,y)). That is, we can pick a point in the xy-plane, evaluate the functions on the right hand side of the system at that point, and these values determine the velocity vector of the solution curve at that point. When we plot the phase portrait, we simply compute and diagram a variety of velocity vectors in the plane and sketch solution curves by following along the velocity vectors.

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Updated on:08/22/07.