Let be a matrix. By choosing an initial vector , we can obtain the trajectory of the difference equation recursively for . Then the trajectory are plotted on the -plane to see graphically how the system evolves, called a phase portrait. And it can be repeated with different choices of initial vector to get a geometric description of the system.
EXAMPLE 1. The phase portrait below demonstrates that a dynamical system has an attractor at the origin when both eigenvalues are less than one in magnitude.
EXAMPLE 2. A dynamical system will exhibit a repellor at the origin of phase portrait when both eigenvalues are larger than one in magnitude.
EXAMPLE 3. A dynamical system will exhibit a saddle point at the origin when one eigenvalue is less than one and the other is larger than one in magnitude.
EXAMPLE 4. Observe that the eigenvalues for the matrix in Examples 4 and 5 are exactly the same, and that the dynamical system in Example 5 has a saddle point at the origin. This indicates that the behavior of dynamical systems (attractor, repellor, or saddle point) is determined by the eigenvalues of .
EXAMPLE 5. A rotation is introduced in the phase portrait when the eigenvalues of are complex numbers.