Rational difference equation

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

where the initial conditions are such that the denominator never vanishes for any n.

First-order rational difference equationEdit

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.[3]

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 equationEdit

First approachEdit

One approach [5] to developing the transformed variable  , when  , is to write

 

where   and   and where  .

Further writing   can be shown to yield

 

Second approachEdit

This approach [6] 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

 

Third approachEdit

The equation

 

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[7]

 

where

 

ApplicationEdit

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

 

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.

ReferencesEdit

  1. ^ Skellam, J.G. (1951). “Random dispersal in theoretical populations”, Biometrika 38 196−–218, eqns (41,42)
  2. ^ Dynamics of third-order rational difference equations with open problems and Conjectures
  3. ^ a b Dynamics of Second-order rational difference equations with open problems and Conjectures
  4. ^ Newth, Gerald, "World order from chaotic beginnings", Mathematical Gazette 88, March 2004, 39-45 gives a trigonometric approach.
  5. ^ Brand, Louis, "A sequence defined by a difference equation," American Mathematical Monthly 62, September 1955, 489–492. online
  6. ^ Mitchell, Douglas W., "An analytic Riccati solution for two-target discrete-time control," Journal of Economic Dynamics and Control 24, 2000, 615–622.
  7. ^ 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.
  8. ^ 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 readingEdit

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