Real fixed points and dynamics of one parameter family of function (bx-1)/x

AbstractIn this paper, the real fixed points and dynamics of one parameter family of functions fλ(x)=λf(x),λ>0, where f(x)=(bx-1)/x,x≠0 and f(0)=lnb for b>0,b≠1, are investigated. The real fixed points of fλ(x) as well as their nature are explored. For 0<b<1, it is seen that one fixed point of fλ(x) is attracting and one fixed point is repelling for 0<λ<λ∗ and fλ(x) has no real fixed points for λ>λ∗. It is also found that the bifurcation in the real dynamics of fλ(x) occurs at the real parameter value λ∗. For b>1, similar results are shown