Rational difference equation explained

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

x _{n+1} = (\alpha + \sum ^{k} _{i =0} \beta _{ix} _{n-i})/(A + \sum ^{k} _{i =0} B _{ix} _{n-i}) ~, where the initial conditions x _{0} , x _{-1} , ... , x _{-k} are such that the denominator never vanishes for any .

First-order rational difference equation

A first-order rational difference equation is a nonlinear difference equation of the form

w _{t+1} = (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} .

Equations of this form arise from the infinite resistor ladder problem.^{[5] [6]}

Solving a first-order equation

First approach

One approach^[7] to developing the transformed variable

x _{t} , when ad - bc ≠ 0 , is to writey _{t+1} = \alpha - (\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 yieldx _{t+2} - \alpha x _{t+1} + \beta x _{t} = 0 .

Second approach

This approach^[8] gives a first-order difference equation for

x _{t} instead of a second-order one, for the case in which (d - a) ^{2} +4 bc is non-negative. Write x _{t} = 1 /( η + w _{t)} implying w _{t} = ( 1- η x _{t)/x} _{t} , where η is given by η = (d - a + r)/ 2 c and where r = \sqrt{(d - a) ^{2} +4 bc} . Then it can be shown that x _{t} evolves according tox _{t+1} = \left( (d -η c)/(η c + a) \right)x _{t} + (c)/(η c + a) .

Third approach

The equation

w _{t+1} = (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} ^{-1} ^{t)} ,

where all of A, B, C, E, and X are n × n matrices (in this case n = 1); the solution of this is^[9]

X _{t} = N _{tD} ^{-1} ^{t}

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 ^[10] that a dynamic matrix Riccati equation of the form

H _{t-1} = K + A'H _{tA} - A'H _{tC(C'H} ^{-1} ^{tC)} 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

1. Skellam, J.G. (1951). “Random dispersal in theoretical populations”, Biometrika 38 196−–218, eqns (41,42)
2. Book: Dynamics of Third-Order Rational Difference Equations with Open Problems and Conjectures. Elias. Camouzis. G.. Ladas. November 16, 2007. CRC Press. 9781584887669. Google Books.
3. Book: Dynamics of Second Order Rational Difference Equations: With Open Problems and Conjectures. Mustafa R. S.. Kulenovic. G.. Ladas. July 30, 2001. CRC Press. 9781420035384. Google Books.
4. Newth, Gerald, "World order from chaotic beginnings", Mathematical Gazette 88, March 2004, 39-45 gives a trigonometric approach.
5. Web site: Equivalent resistance in ladder circuit . Stack Exchange . 21 February 2022.
6. Web site: Thinking Recursively: How to Crack the Infinite Resistor Ladder Puzzle! . Youtube . 21 February 2022.
7. Brand, Louis, "A sequence defined by a difference equation," American Mathematical Monthly 62, September 1955, 489 - 492. online
8. Mitchell, Douglas W., "An analytic Riccati solution for two-target discrete-time control," Journal of Economic Dynamics and Control 24, 2000, 615 - 622.
9. 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.
10. 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.

Further reading

• Simons, Stuart, "A non-linear difference equation," Mathematical Gazette 93, November 2009, 500–504.