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*.

Department of Mathematics