Difference equation ($z_{n+1}=F(z_n)$)

A difference equation relates values of a sequence at different indices, or its finite differences. A first-order autonomous system has the displayed form, with a scalar or vector state. Writing $z_{n+1}-z_n=F(z_n)-z_n$ gives its equivalent finite-difference form. Its fixed point stability is determined by the Jacobian matrix of the update and the unit circle, whereas a continuous-time differential equation uses the real parts of its linearized eigenvalues.