Rational difference equation

A rational difference equation is a nonlinear difference equation of the form^[1]^[2]^[3]^[4]

x_{n+1} = \frac{\alpha+\sum_{i=0}^k \beta_ix_{n-i}}{A+\sum_{i=0}^k B_ix_{n-i}}~,

where the initial conditions $x_{0}, x_{-1},\dots, x_{-k}$ 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

w_{t+1} = \frac{aw_t+b}{cw_t+d}.

When $a,b,c,d$ and the initial condition $w_{0}$ are real numbers, this difference equation is called a Riccati difference equation.^[3]

Such an equation can be solved by writing $w_t$ as a nonlinear transformation of another variable $x_t$ which itself evolves linearly. Then standard methods can be used to solve the linear difference equation in $x_t$ .

Solving a first-order equation

First approach

One approach ^[5] to developing the transformed variable $x_t$ , when $ad-bc \neq 0$ , is to write

y_{t+1}= \alpha - \frac{\beta}{y_t}

where $\alpha = (a+d)/c$ and $\beta = (ad-bc)/c^{2}$ and where $w_t = y_t -d/c$ .

Further writing $y_t = x_{t+1}/x_t$ can be shown to yield

x_{t+2} - \alpha x_{t+1} + \beta x_t =0. \,

Second approach

This approach ^[6] gives a first-order difference equation for $x_t$ instead of a second-order one, for the case in which $(d-a)^{2}+4bc$ is non-negative. Write $x_t = 1/(\eta + w_t)$ implying $w_t = (1- \eta x_t)/x_t$ , where $\eta$ is given by $\eta = (d-a+r)/2c$ and where $r=\sqrt{(d-a)^{2}+4bc}$ . Then it can be shown that $x_t$ evolves according to

x_{t+1} =\left( \frac{d-\eta c}{\eta c+a}\right)x_t + \frac{c}{\eta c+a}.

Third approach

The equation

w_{t+1} = \frac{aw_t+b}{cw_t+d}

can also be solved by treating it as a special case of the more general matrix equation

X_{t+1} = -(E+BX_t)(C+AX_t)^{-1},

where all of A, B, C, E, and X are n×n matrices (in this case n=1); the solution of this is^[7]

X_{t}=N_tD_t^{-1}

where

\begin{pmatrix} N_{t} \\ D_{t}\end{pmatrix} = \begin{pmatrix} -B & -E \\ A & C \end{pmatrix}^t\begin{pmatrix} X_0\\ I \end{pmatrix}.

Application

It was shown in ^[8] that a dynamic matrix Riccati equation of the form

H_{t-1} = K +A'H_tA - A'H_tC(C'H_tC)^{-1}C'H_tA, \,

which can arise in some discrete-time optimal control problems, can be solved using the second approach above if the matrix C has only one more row than column.

References

↑ 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
1 2 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.
↑ 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.