Periodic Solutions of a Logistic Difference Equation

Abstract

Periodic solutions of the difference equation xn+ mxn(1-x) are studied for values of m, 0 _-<m -< 4. It is shown that asm increases from zero, solutions having successively higher periods branch from old ones until the valuem 3.57 is reached, after which there is an infinity of periodic solutions. The solution set is said to be chaotic if there is an infinity of periodic solutions. This investigation focuses on solution behavior in the chaotic regime. It is shown how as m increasesfromm, solutions having various other periods are added to the solution set until atm 3.83, solutions of period three, and hence all periods are present. Finally, density functions are calculated numerically to describe the dynamics of solutions in portions of the chaotic regime.