The* Riccati difference
equation* is of the form

In order to avoid unwanted cases, we must assume that both d and ad-bc do not equal zero.

The forbidden set F
of the *Riccati difference equation* is the set of initial conditions
which the denominator c+dx_{n }will become zero for some non negative
value of n.

Assume that the previous
conditions hold and that the *Riccati difference equation *does possess
a periodic solution of prime period two.

When b+c is equal to
zero, we observe the every solution of the *Riccati difference equation
*with
x_{0 }not equal to b/c is periodic of period 2.

In addition to the previous conditions, we will assume that b+c does not zero.

The change of variable for n non negative

The nonzero real number

We can observe that when

We now observe that the change of variables

The sequence of points

When w_{0} is
not in the forbidden set F, the solution of Equation (1) is given by

If we assume that

The sequence of points which converges to the equilibrium from the left

When w_{0} is
not in the forbidden set F,
the solution of the equation is given by

When w_{0} is
not in the forbidden set F,
the solution of Equation (1) is given by

If we assume that

If we assume that the
number f is
not a rational multiple of p,then
the following statements are true:

(i) No solution of
the equation is periodic.

(ii) The set of limit
points of a solution of Equation 1 with w_{0} not in the forbidden
set F
is dense in R.

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Links Related to the Riccati Difference Equation

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Bio of Jacopo Francesco Riccati

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