Rational difference equation
where the initial conditions are such that the denominator never vanishes for any n.
First-order rational difference equation
A first-order rational difference equation is a nonlinear difference equation of the form
When and the initial condition are real numbers, this difference equation is called a Riccati difference equation.
Such an equation can be solved by writing as a nonlinear transformation of another variable which itself evolves linearly. Then standard methods can be used to solve the linear difference equation in .
Solving a first-order equation
One approach  to developing the transformed variable , when , is to write
where and and where .
Further writing can be shown to yield
This approach  gives a first-order difference equation for instead of a second-order one, for the case in which is non-negative. Write implying , where is given by and where . Then it can be shown that evolves according to
can also be solved by treating it as a special case of the more general matrix equation
where all of A, B, C, E, and X are n×n matrices (in this case n=1); the solution of this is
- Skellam, J.G. (1951). “Random dispersal in theoretical populations”, Biometrika 38 196−218, eqns (41,42)
- Dynamics of third-order rational difference equations with open problems and Conjectures
- Dynamics of Second-order rational difference equations with open problems and Conjectures
- Newth, Gerald, "World order from chaotic beginnings", Mathematical Gazette 88, March 2004, 39-45 gives a trigonometric approach.
- Brand, Louis, "A sequence defined by a difference equation," American Mathematical Monthly 62, September 1955, 489–492. online
- Mitchell, Douglas W., "An analytic Riccati solution for two-target discrete-time control," Journal of Economic Dynamics and Control 24, 2000, 615–622.
- Martin, C. F., and Ammar, G., "The geometry of the matrix Riccati equation and associated eigenvalue method," in Bittani, Laub, and Willems (eds.), The Riccati Equation, Springer-Verlag, 1991.
- Balvers, Ronald J., and Mitchell, Douglas W., "Reducing the dimensionality of linear quadratic control problems," Journal of Economic Dynamics and Control 31, 2007, 141–159.
- Simons, Stuart, "A non-linear difference equation," Mathematical Gazette 93, November 2009, 500-504.