1. Choose a \(2\times2\) system
The system is \(\mathbf{x}'=A\mathbf{x}\), where \(A=\begin{bmatrix}a&b\\c&d\end{bmatrix}\).
Matrix \(A\)
Initial condition \(\mathbf{x}_0\)
2. Vector field and trajectory
Gray arrows show the vector field \(A\mathbf{x}\). The orange curve is the Euler trajectory from \(\mathbf{x}_0\).
3. Forward Euler multiplier
For an eigenvalue \(\lambda\), forward Euler uses the multiplier \(1+h\lambda\). The eigenmode decays when \(|1+h\lambda|<1\).
4. Graph heat equation on \(1-2-3\)
The Laplacian is \(L=\begin{bmatrix}1&-1&0\\-1&2&-1\\0&-1&1\end{bmatrix}\). We simulate \(\mathbf{u}'=-L\mathbf{u}\).
5. Independent-study tasks with answers
Task A. Choose the saddle preset. What are the signs of the eigenvalues?
Answer: One eigenvalue has positive real part and one has negative real part, so one direction grows and one direction decays.
Answer: One eigenvalue has positive real part and one has negative real part, so one direction grows and one direction decays.
Task B. Choose the center preset. Why do trajectories rotate instead of decay?
Answer: The eigenvalues are purely imaginary, so the real parts are zero.
Answer: The eigenvalues are purely imaginary, so the real parts are zero.
Task C. Try a larger Euler step for a stable system. What can go wrong?
Answer: Euler may become numerically unstable when \(|1+h\lambda|\ge 1\) for some eigenvalue.
Answer: Euler may become numerically unstable when \(|1+h\lambda|\ge 1\) for some eigenvalue.
Task D. In the graph heat equation, what is the limiting vector?
Answer: For the connected path graph, the solution converges to the average of the initial values times \((1,1,1)^T\).
Answer: For the connected path graph, the solution converges to the average of the initial values times \((1,1,1)^T\).