A rational difference equation is a nonlinear difference equation of the form[ 1] [ 2] [ 3] [ 4]
x
n
+
1
=
α
+
∑
i
=
0
k
β
i
x
n
−
i
A
+
∑
i
=
0
k
B
i
x
n
−
i
,
{\displaystyle x_{n+1}={\frac {\alpha +\sum _{i=0}^{k}\beta _{i}x_{n-i)){A+\sum _{i=0}^{k}B_{i}x_{n-i))}~,}
where the initial conditions
x
0
,
x
−
1
,
…
,
x
−
k
{\displaystyle x_{0},x_{-1},\dots ,x_{-k))
are such that the denominator never vanishes for any n .
First-order rational difference equation [ edit ] A first-order rational difference equation is a nonlinear difference equation of the form
w
t
+
1
=
a
w
t
+
b
c
w
t
+
d
.
{\displaystyle w_{t+1}={\frac {aw_{t}+b}{cw_{t}+d)).}
When
a
,
b
,
c
,
d
{\displaystyle a,b,c,d}
and the initial condition
w
0
{\displaystyle 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
{\displaystyle w_{t))
as a nonlinear transformation of another variable
x
t
{\displaystyle x_{t))
which itself evolves linearly. Then standard methods can be used to solve the linear difference equation in
x
t
{\displaystyle x_{t))
.
Equations of this form arise from the infinite resistor ladder problem.[ 5] [ 6]
Solving a first-order equation [ edit ] One approach[ 7] to developing the transformed variable
x
t
{\displaystyle x_{t))
, when
a
d
−
b
c
≠
0
{\displaystyle ad-bc\neq 0}
, is to write
y
t
+
1
=
α
−
β
y
t
{\displaystyle y_{t+1}=\alpha -{\frac {\beta }{y_{t))))
where
α
=
(
a
+
d
)
/
c
{\displaystyle \alpha =(a+d)/c}
and
β
=
(
a
d
−
b
c
)
/
c
2
{\displaystyle \beta =(ad-bc)/c^{2))
and where
w
t
=
y
t
−
d
/
c
{\displaystyle w_{t}=y_{t}-d/c}
.
Further writing
y
t
=
x
t
+
1
/
x
t
{\displaystyle y_{t}=x_{t+1}/x_{t))
can be shown to yield
x
t
+
2
−
α
x
t
+
1
+
β
x
t
=
0.
{\displaystyle x_{t+2}-\alpha x_{t+1}+\beta x_{t}=0.}
This approach[ 8] gives a first-order difference equation for
x
t
{\displaystyle x_{t))
instead of a second-order one, for the case in which
(
d
−
a
)
2
+
4
b
c
{\displaystyle (d-a)^{2}+4bc}
is non-negative. Write
x
t
=
1
/
(
η
+
w
t
)
{\displaystyle x_{t}=1/(\eta +w_{t})}
implying
w
t
=
(
1
−
η
x
t
)
/
x
t
{\displaystyle w_{t}=(1-\eta x_{t})/x_{t))
, where
η
{\displaystyle \eta }
is given by
η
=
(
d
−
a
+
r
)
/
2
c
{\displaystyle \eta =(d-a+r)/2c}
and where
r
=
(
d
−
a
)
2
+
4
b
c
{\displaystyle r={\sqrt {(d-a)^{2}+4bc))}
. Then it can be shown that
x
t
{\displaystyle x_{t))
evolves according to
x
t
+
1
=
(
d
−
η
c
η
c
+
a
)
x
t
+
c
η
c
+
a
.
{\displaystyle x_{t+1}=\left({\frac {d-\eta c}{\eta c+a))\right)\!x_{t}+{\frac {c}{\eta c+a)).}
The equation
w
t
+
1
=
a
w
t
+
b
c
w
t
+
d
{\displaystyle 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
+
B
X
t
)
(
C
+
A
X
t
)
−
1
,
{\displaystyle 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[ 9]
X
t
=
N
t
D
t
−
1
{\displaystyle X_{t}=N_{t}D_{t}^{-1))
where
(
N
t
D
t
)
=
(
−
B
−
E
A
C
)
t
(
X
0
I
)
.
{\displaystyle {\begin{pmatrix}N_{t}\\D_{t}\end{pmatrix))={\begin{pmatrix}-B&-E\\A&C\end{pmatrix))^{t}{\begin{pmatrix}X_{0}\\I\end{pmatrix)).}
It was shown in [ 10] that a dynamic matrix Riccati equation of the form
H
t
−
1
=
K
+
A
′
H
t
A
−
A
′
H
t
C
(
C
′
H
t
C
)
−
1
C
′
H
t
A
,
{\displaystyle H_{t-1}=K+A'H_{t}A-A'H_{t}C(C'H_{t}C)^{-1}C'H_{t}A,}
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.
^ Skellam, J.G. (1951). “Random dispersal in theoretical populations”, Biometrika 38 196−218, eqns (41,42)
^ Camouzis, Elias; Ladas, G. (November 16, 2007). Dynamics of Third-Order Rational Difference Equations with Open Problems and Conjectures . CRC Press. ISBN 9781584887669 – via Google Books.
^ a b Kulenovic, Mustafa R. S.; Ladas, G. (July 30, 2001). Dynamics of Second Order Rational Difference Equations: With Open Problems and Conjectures . CRC Press. ISBN 9781420035384 – via Google Books.
^ Newth, Gerald, "World order from chaotic beginnings", Mathematical Gazette 88, March 2004, 39-45 gives a trigonometric approach.
^ "Equivalent resistance in ladder circuit" . Stack Exchange . Retrieved 21 February 2022 .
^ "Thinking Recursively: How to Crack the Infinite Resistor Ladder Puzzle!" . Youtube . Retrieved 21 February 2022 .
^ 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.