Todd's Equation

Todd's Equation has the form:

xn+1 = (a + xn + xn-1 )/ xn-2

      where the parameter a > 0, x _n-1 > 0, x₀ > 0, x ₁ > 0

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          The invariant of this equation is:
 

I(xn , xn-1, xn-2) = (1+1/xn )(1+1/xn-1 )(1+1/xn-2 )(a+xn +xn-1+xn-2 )

                The equilibrium value, p, of Todd's equation:
 

p=(1±(1+ a)1/2)

     satisfies p² - 2p - a = 0.

    Using the discrete version of the Dirichlet's theorem (see previous page on Lyness' Equation), the Liapanov function,
        V(x,y,z) for this equation is given by:
 

V(x, y, z) = I(x, y, z) - I(p, p, p) = I(x, y, z) - (p+1)4 /p2 V(x, y, z) = (1 + 1/x)(1 + 1/y)(1 + 1/z)(a + x + y + z) - (p+1)4 /p2

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3-d Implicit Plot of Invariant of Todd's Equation

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