Second-order difference equation ($x_{n+1}=F(x_n,x_{n-1})$)

A second-order difference equation determines the next member of a sequence from its two preceding values, for example $x_{n+1}=a{x}_n+b{x}_{n-1}$. Two initial values are generally needed. Although the state is written with one scalar sequence, the update becomes a first-order discrete dynamical system on pairs:

Trying $x_n=r^n$ gives the characteristic equation $r^2-ar-b=0$, whose roots determine growth, decay and alternating behavior. For $x_{n+1}=2\cos \theta x_n-x_{n-1}$ with $0<\theta <\pi$, the roots are $e^{\pm i\theta }$ and the real solutions oscillate. This is useful when a model has memory of a previous step, or when discretizing a second-order equation of motion.