A rational difference equation is a nonlinear difference equation of the formSkellam, J.G. (1951). “Random dispersal in theoretical populations”, Biometrika 38 196−218, eqns (41,42)Newth, Gerald, "World order from chaotic beginnings", Mathematical Gazette 88, March 2004, 39-45 gives a trigonometric approach.
where the initial conditions 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
When and the initial condition are real numbers, this difference equation is called a Riccati difference equation. Their theory is closely related to the theory of Mobius transforms
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 .
In natural resource sciences and theoretical ecology, the special case of is known as the Beverton-Holt model of population growth , and is important in fisheries management.
Equations of this form arise from the infinite resistor ladder problem.
Solving a first-order equation
First approach
WLoG, the determinant-like quantity , and this can be easily seen by noting that, with a division on both the numerator and denominator in Equation , you can always set , in which case, if , then the numerator and denominator will cancel away, leaving no difference equation left, being instead so completely reduced as to become exactly the constant . Thus, one approachBrand, Louis, "A sequence defined by a difference equation," American Mathematical Monthly 62, September 1955, 489–492. online to developing the transformed variable , is to write
where and and where .
Further writing can be shown to yield
Second approach
The above approach is already of general applicability. This following approachMitchell, Douglas W., "An analytic Riccati solution for two-target discrete-time control," Journal of Economic Dynamics and Control 24, 2000, 615–622. gives a first-order difference equation for instead of a second-order one. Let . For the case in which , every term will be real-valued and this method may be convenient to use. Otherwise, the method still works, but complex numbers will appear, and it might be more convenient to attempt a trigonometric ansatz instead. Substituting , which implies , into Equation , we find that it is always possible to make evolve according to the simple inhomogeneous first-order linear difference equation
by choosing such that , and it is clear that this can always be done in either of the two choices , even when
Third approach
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 isMartin, 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.
where
Application
It was shown in 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. 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.
References
Further reading
Simons, Stuart, "A non-linear difference equation," Mathematical Gazette 93, November 2009, 500–504.