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Equations bearing some similarity to the autonomous version of (1) have been considered in the literature; see, e.g. [4, Sec.2.5]. These studies have concentrated on the global stability of equilibrium. Equation (1) also generalizes equations introduced in some of the heuristic business cycle models in macroeconomics. For example, special cases of (1) includes Hicks’ model (its 2nd order case) and Goodwin’s model (its discrete version) as well as Puu’s discrete second order equation. A detailed comparative analysis of the mathematics behind these classical models appears in [7]. In this note we take a close look at Equation (1) and obtain criteria for the occurrence of attracting periodic solutions as well as conditions that imply the occurrence of chaotic behavior. We use the fact that under conditions (2), Equation (1) decomposes into a weakly coupled system of first order difference equations. Such decompositions are instances of semiconjugacy; see [7] for some background on this subject.

Periodic and chaotic behavior in a class of second order difference equations