Density of periodic orbits For a chaotic system to have dense periodic orbits means that every point in the space is approached arbitrarily closely by periodic orbits. The one-dimensional logistic map defined by x → 4 x (1 – x) is one of the simplest systems with density of periodic orbits. For example, → → (or approximately 0.3454915 → 0.9045085 → 0.3454915) is an (unstable) orbit of period 2, and similar orbits exist for periods 4, 8, 16, etc. (indeed, for all the periods specified by Sharkovskii's theorem).
deNSity Of perIodIC orbiTs For a cHAOtiC systEM tO haVE densE pEriodic oRbits MeanS tHaT evEry poINt iN the SpaCE is approacheD ARbITrArily closely bY PeriodIc orbiTs. tHe ONE-dimeNsiOnAL lOgiStic MAP defiNed by x → 4 x (1 – x) IS oNe of the simplesT SystEMs wItH denSiTy oF peRiODic OrBits. for example, → → (or apPROXimAteLY 0.3454915 → 0.9045085 → 0.3454915) Is An (uNsTaBLe) oRbit of pEriod 2, And siMilAr orbits ExiSt FOr Periods 4, 8, 16, etc. (inDeed, foR all tHe pERIOdS SpecIfIed by sharkovsKii'S ThEOrem).