5.6

Eigenvector decomposition of x0

determines what happens to the sequence xk

The origin is called a saddle point if

it attracts solutions from some directions and repels them in other directions. This occurs when one eigenvalue is greater than 1 in magnitude and the other is less than 1 in magnitude. The direction of greatest attraction is determined by an eigenvector for the eigenvalue of smaller magnitude. The direction of greatest repulsion is determined by an eigenvector for the eigenvalue of greater magnitude.

In a linear system, the origin is

the only possible attractor or repeller. This is not the case in nonlinear dynamical systems

If the eigenvalues of A are larger than 1 in magnitude

the origin is a repeller of the dynamical system. All solutions of xk+1 = Axk except the constant (zero) solution are unbounded and tend away from the origin. The direction of the greatest repulsion is the line through 0 and the eigenvector for the eigenvalue of larger magnitude.

If all trajectories tend towards 0

the origin is an attractor of the dynamical system; this occurs when both eigenvalues are less than 1 in magnitude. Direction of greatest attraction is along the line through 0 and the eigenvector v2 for the eigenvalue of smallest magnitude.

Graph of x0, x1 ...

trajectory of dynamical system