k + 2-th order Lyness's Equation

k+2-th order Lyness's Equation has the form:

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

      where the parameter a > 0, x_k > 0, ... , x ₁ > 0

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          The invariant of this equation is:
 

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

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

p=(k+1±{(k+1)2 + 4a}1/2)/2

satisfies p² - (k+1)p - a = 0.

    Using the discrete version of the Dirichlet's theorem, the Liapanov function,
        V(x,y,z) for this equation is found to be given by:
 

V(xn , xn-1, .., xn-k-1) = I(xn , xn-1, .., xn-k-1) - I(p, p, p) = I(xn , xn-1, .., xn-k-1) - (p+1)k+3 /pk+1

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Bifurication Diagram of the k+2th order Lyness' Equation

 

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