Linear Algebra Dynamical System Help

Question

I was just wondering, for a dynamic system does the origin always have to be an attractor, saddle point, or repellor?

Also if a matrix isn't diagonalizable then does that mean the origin cannot be a repellor the matrix?

Answer 1

For a linear dynamical system, the origin could also be a center if the eigenvalues are purely imaginary. For example, the system \begin{align} \begin{bmatrix} x_1^\prime \\x_2^\prime \end{bmatrix} = \begin{bmatrix} 0 & 1\\ -1 & 0 \end{bmatrix} \begin{bmatrix} x_1 \\x _2 \end{bmatrix} \end{align} has eigenvalues $\pm i$. The solution trajectories circle around the origin.

When the eigenvalues are complex with positive real components, then the trajectories spiral away from the origin. When the eigenvalues are complex with negative real components, then the trajectories spiral toward the origin.

A matrix that is not diagonalizable can be a repeller, although not all initial conditions will be permissible. For example, let \begin{align} \begin{bmatrix} x_1^\prime \\x_2^\prime \end{bmatrix} = \begin{bmatrix} 1 & 0\\ 1 & 1 \end{bmatrix} \begin{bmatrix} x_1 \\x _2 \end{bmatrix} \end{align} There is only one linearly independent eigenvector $\mathbf{v} = [0, 1]^T$, with eigenvalue $1$. Thus if the initial condition $\mathbf{x}_0$ is a scalar multiple of $\mathbf{v}$, then the solution trajectories will be repelled from the origin. If the initial condition is not a scalar multiple of $\mathbf{v}$, then no solution exists.

Note that for nonlinear dynamical systems, the origin is not in general the location of a source, sink, center, or saddle point.