Lyness' Equation

                Lyness' Equation has the form:
 

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

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

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          The invariant of this equation is:
 

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

    The Lyness equation has the equilibrium value:
 

p=(1±(1+ 4a)1/2)/2

 

THEOREM (Discrete version of  the Dirichlet's theorem).  Consider the difference equation

xn+1 = f( xn)

where x_n is in R^k and f: D® D is continuous, where D Ì R^k. Suppose that I:R^k® R is a continuous invariant that is I(f(x)) = I(x) for every x Î D. If I attains an isolated local minimum or maximum value at the equilibrium (fixed) point p of this system, then there exists a Liapunov function equal to ±(I(x) - I(p)) and so the equilibrium p is stable.
 

By using the above theorem, the Liapunov function, V(x,y) for this equation is found to be given by:
 

V(x, y)= I(x, y)-I(p, p)= I(x, y) - (p+1)3 /p V(x, y)=(1+1/x)(1+1/y)(a+x+y)-(p+1)3 /p

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3-d  Plot of the Invariant of Lyness' Equation

View an animated version of the above graph.

For more information about Lyness' Equation:

• Geometric Unfolding of a Difference Equation  a lecture by:

   Sir Christopher Zeeman
       "Technical skill is mastery of complexity while creativity is mastery of simplicity."
                            Catastrophe Theory, 1977.

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Click here to simulate the solutions of this equation for different choices of parameters and initial points