Phase portrait

Let $ A$ be a $ 2 \times 2$ matrix. By choosing an initial vector $ \mathbf{x}_0 = \begin{bmatrix}x_0 \\
y_0 \end{bmatrix}$, we can obtain the trajectory $ \mathbf{x}_1,\ldots,\mathbf{x}_t$ of the difference equation recursively for $ k = 1,\ldots,t$. Then the trajectory are plotted on the $ xy$-plane to see graphically how the system evolves, called a phase portrait. And it can be repeated with different choices of initial vector  $ \mathbf{x}_0$ 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.

\includegraphics{idemo12a.ps}

EXAMPLE 2. A dynamical system will exhibit a repellor at the origin of phase portrait when both eigenvalues are larger than one in magnitude.

\includegraphics{idemo12b.ps}

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.

\includegraphics{idemo12c.ps}

EXAMPLE 4. Observe that the eigenvalues for the matrix $ A$ 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 $ A$.

\includegraphics{idemo12d.ps}

EXAMPLE 5. A rotation is introduced in the phase portrait when the eigenvalues of $ A$ are complex numbers.

\includegraphics{idemo12e.ps}



Department of Mathematics
Last modified: 2005-10-21