Editing Attractor (section)

===Linear equation or system===

An univariate linear homogeneous difference equation <math>x_t=ax_{t-1}</math> diverges to infinity if <math>|a|>1</math> from all initial points except 0; there is no attractor and therefore no basin of attraction. But if <math>|a|<1</math> 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 [[coordinate vector|vector]] <math>X</math>, of the homogeneous form <math>X_t=AX_{t-1}</math> in terms of [[square matrix]] <math>A</math> will have all elements of the dynamic vector diverge to infinity if the largest [[eigenvalue]]s of <math>A</math> 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 <math>n</math>-dimensional space of potential initial vectors is the basin of attraction.

Similar features apply to linear [[differential equation]]s. The scalar equation <math> dx/dt =ax</math> causes all initial values of <math>x</math> except zero to diverge to infinity if <math>a>0</math> but to converge to an attractor at the value 0 if <math>a<0</math>, making the entire number line the basin of attraction for 0. And the matrix system <math>dX/dt=AX</math> gives divergence from all initial points except the vector of zeroes if any eigenvalue of the matrix <math>A</math> 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.