The global dynamics of a quartic difference equation
In general, polynomial difference equations and polynomial maps in the plane have been studied in both the real and complex domains (see [8, 9]). First results on quadratic polynomial difference equation have been obtained in [1, 2] but these results gave us only a part of the basins of attraction of equilibrium points and period-two solutions. In [4], the general second order difference equation is completely investigated and described the regions of initial conditions in the first quadrant for which all solutions tend to equilibrium points, period-two solutions, or the point at infinity, except for the case of infinitely many period-two solutions. In [3], case of infinitely many period-two solutions is completely investigated. Our results are based on the theorems which hold for monotone difference equations. Our principal tool is the theory of monotone maps, and in particular cooperative maps, which guarantee the existence and uniqueness of the stable and unstable invariant manifolds for the fixed points and periodic points (see [5]). Consider the difference equation