information:classification, structural:summarization
Linear equation or system

An univariate linear homogeneous difference equation x t = a x t − 1 {\displaystyle x_{t}=ax_{t-1}} diverges to infinity if | a | > 1 {\displaystyle |a|>1} from all initial points except 0; there is no attractor and therefore no basin of attraction. But if | a | < 1 {\displaystyle |a|<1} all points on the number line map asymptotically (or directly in the case of 0) to 0; 0 is the attractor, and the entire number line is the basin of attraction.

Likewise, a linear matrix difference equation in a dynamic vector X {\displaystyle X}, of the homogeneous form X t = A X t − 1 {\displaystyle X_{t}=AX_{t-1}} in terms of square matrix A {\displaystyle A} will have all elements of the dynamic vector diverge to infinity if the largest eigenvalues of A {\displaystyle A} is greater than 1 in absolute value; there is no attractor and no basin of attraction. But if the largest eigenvalue is less than 1 in magnitude, all initial vectors will asymptotically converge to the zero vector, which is the attractor; the entire n {\displaystyle n} -dimensional space of potential initial vectors is the basin of attraction.

Similar features apply to linear differential equations. The scalar equation d x / d t = a x {\displaystyle dx/dt=ax} causes all initial values of x {\displaystyle x} except zero to diverge to infinity if a > 0 {\displaystyle a>0} but to converge to an attractor at the value 0 if a < 0 {\displaystyle a<0}, making the entire number line the basin of attraction for 0. And the matrix system d X / d t = A X {\displaystyle dX/dt=AX} gives divergence from all initial points except the vector of zeroes if any eigenvalue of the matrix A {\displaystyle A} is positive; but if all the eigenvalues are negative the vector of zeroes is an attractor whose basin of attraction is the entire phase space.