Abstract

We present some new results concerning the dynamics of the real third order Lyness equation $x$_n+3=(a+x_n+2+x_n+1)/x_n, where a is a real parameter and the initial conditions are in $R3$. Among others, we prove the existence of continua of initial conditions giving rise to periodic orbits of period $2,4,5,6,7$ and $4p$ for $p>2$, as well as continua of initial conditions giving rise to dense orbits in the real line. The main results are obtained by using the well known first integral (invariant) of the discrete dynamical system associated with the difference equation: $V(x,y,z)=(x+1)(y+1)(z+1)(a+x+y+z)/(xyz)$. The dynamics at one of the level sets of $V$ can be studied using a new system for which it can be found a new first integral. The key point is that on each leaf of the foliation induced by this new first integral the dynamics is given by a real Möbius transformation.

This is a joint work done in collaboration with Anna Cima and Armengol Gasull.