# Math Insight

### Applet: A linear system with phase plane and versus time

Control panel (Show)
Eigenvalues and eigenvectors (Show)
Equilibrium classification (Show)
Solution (Show)

Illustration of the solution to a system of two linear ordinary differential equations. The system is of the form $\diff{\vc{x}}{t} = A\vc{x}$ with prescribed initial conditions $\vc{x}(0)=\vc{x}_0$, where $\vc{x}(t)=(x(t),y(t))$. The solution trajectory $(x(t),y(t))$ is plotted as a cyan curve on the phase plane in the left panel. In the right panel, the components of the solution $x(t)$ (top axes, solid cyan curve) and $y(t)$ (bottom axes, dashed cyan curve) are plotted versus time.

To visualize how the solution changes as a function of time in the phase plane, one can change the time $t$ with the slider in the right panel or press the play button (triangle) in the lower left of one of the panels to start the animation of $t$ increasing. The red points in both panel move with $t$ to correspond to the solution $(x(t),y(t))$.

Values of the matrix $A$ can be changed in the top control panel. The initial condition $\vc{x}(0)= (x_0,y_0)$ can be changed by dragging the cyan points in either panel or by entering numbers in the control panel.

If the eigenvalues of $A$ are real, then one can check the “show eigenvectors” box to show the directions of the eigenvectors of $A$ in the left phase plane. If the corresponding eigenvalue is not zero, arrows along the eigenvector indicates the direction the solution moves along the eigenvector direction. Checking the “show vector” box displays a vector from the origin to $(x(t),y(t))$, allowing one to track the direction of the solution even when the point $(x(t),y(t))$ moves out of view. Checking the “show decompositions” box, shows the decomposition of $(x(t),y(t))$ as a sum of components along the eigenvectors.

If you check the box “show eigenvalues”, then the phase plane plot shows an overlay of the eigenvalues, where the axes are reused to represent the real and imaginary axes of the complex plane. The eigenvalues appear as two points on this complex plane, and will be along the x-axis (the real axis) if the eigenvalues are real. If both eigenvalues are in the left half of the plane (which becomes shaded when the box is checked), then the equilibrium at the origin is stable.

The solution, eigenvalues, eigenvectors, and characterization of the equilibrium at the origin are shown in the sections at the bottom of the applet. These calculations depend on values of $A$ and initial condition chosen.

Applet file: linear_system_phase_plane_versus_time.ggb