Riccati Difference Equation

THE RICCATI DIFFERENCE EQUATION

The Riccati difference equation is of the form

n = 0,1,... (1) where the parameters a, b, c, d and the initial condition x₀are real numbers.

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_nwill 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 x₀not 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

transforms the Riccati difference equation into the difference equation (which we shall refer to as Equation (1))

n = 0,1,...

The nonzero real number

is known as the Riccati number.

We can observe that when

Equation(1) has the two equilibrium points

and

We see that w₁< 0 if R < 0 and w₁> 0 if 0 < R < .25 When

Equation(1) has exactly one equilibrium point

When

Equation(1) has no equilibrium points.

We now observe that the change of variables

n = 0,1,... with y₀ = 1 and y₁= w₀ reduces Equation(1) to the second order linear difference equation y_n+2- y_n+1+ R y_n= 0 If we assume that

then w₁and w₂are the only equilibrium points of Equation (1).

The sequence of points

n = 1,2,... is the forbidden set F of Equation (1).

When w₀ is not in the forbidden set F, the solution of Equation (1) is given by

n = 1,2,... The equilibrium w₂is a global attractor if w₀ is not in the union of the forbidden set F and w₁ and the equilibrium w₁is a repeller.

If we assume that

then

is the only equilibrium point of the equation.

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

n = 0,1,.... is the forbidden set F of the equation.

When w₀ is not in the forbidden set F, the solution of the equation is given by

n = 0,1,.... Finally, if we assume that

and let f in the set between 0 and p/2 be such that

and

Then the sequence of points

n = 1,2,... is the forbidden set F of Equation (1).

When w₀ is not in the forbidden set F, the solution of Equation (1) is given by

n = 0,1,... The equilibrium w = 1/2 of Equation (1) is a global attractor for all solutions with w₀not in the forbidden set F. However, the equilibrium is locally unstable, it is a source from the left and a sink from the right.

If we assume that

where k and p are positive comprimes and suppose that

and

then every solution of Equation (1) with w₀not equal to

n = 1,2,...,p-1 is periodic with period p.

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 w₀ not in the forbidden set F is dense in R.

CLICK HERE TO EXPERIMENT WITH THE PARAMETERS AND INITIAL VALUES

Links Related to the Riccati Difference Equation

mathworld.wolfram.com
www.sosmath.com

Bio of Jacopo Francesco Riccati

Jacopo Riccati

Click here to go to Home Page for Difference Equations at URI