Phase Plane

For a 2-dimensional ODE system, we can plot solutions as curves in the plane of the dependable variables.

The phase plane allows us to see solutions of both linear and nonlinear differential equations. Let us start with a 2-dimensional linear homogeneous system with constant coefficients.

dx/dt = a x + b y
dy/dt = c x + d y

Of course, it can be written in the matrix form

dX/dt = A X

where X is the column vector (x, y)'and the matrix A consists of a, b, c, and d.

The geometric properties of the phase portrait are closely related to the algebraic characteristics of eigenvalues and eigenvectors of the matrix A.

If there is no zero eigenvalue, the origin is the only equilibrium. If there is at least one zero eigenvalue, there are infinitely many equilibria, and they form a straight line or the whole plane.

When the real parts of all eigenvalues are negative, the origin is a sink, also known as an asymptotically stable equilibrium.

• Sink (two distinct negative eigenvalues);
• Sink (two equal negative eigenvalues with only one linearly independent eigenvector);
• Sink (two equal negative eigenvalues with two linearly independent eigenvectors);
• Spiral Sink (two imaginary eigenvalues with negative real parts)

When the real parts of all eigenvalues are positve, the origin is a source, which is unstable.

• Source (two distinct positive eigenvalues)
• Source (two equal positive eigenvalues with only one linearly independent eigenvector);
• Source (two equal positive eigenvalues with two linearly independent eigenvectors);
• Spiral Source (two imaginary eigenvalues with positive real parts)

When one eigenvalue is positive, and the other is negative the origin is a saddle point, which is also unstable.

• Saddle point

Other situations

• Center (two imaginary eigenvalues with zero real parts)
• A line of stable equilibria (a negative eigenvalue and a zero eigenvalue)
• A line of unstable equilibria (a positive eigenvalue and a zero eigenvalue)
• A line of unstable equilibria and other solutions are parallel to this line (two zero eigenvalues with only one linearly indepedent eigenvector)
• A plane of equilibria (two zero eigenvalues with two linearly indepedent eigenvector)