Computation of the anharmonic orbits in two piecewise monotonic maps with a single discontinuity

In this paper, the bifurcation values for two typical piecewise monotonic maps with a single discontinuity are computed. The variation of the parameter of those maps leads to a sequence of border-collision and period-doubling bifurcations, generating a sequence of anharmonic orbits on the boundary of chaos. The border-collision and period-doubling bifurcation values are computed by the word-lifting technique and the Maple fsolve function or the Newton–Raphson method, respectively. The scaling factors which measure the convergent rates of the bifurcation values and the width of the stable periodic windows, respectively, are investigated. We found that these scaling factors depend on the parameters of the maps, implying that they are not universal. Moreover, if one side of the maps is linear, our numerical results suggest that those quantities converge increasingly. In particular, for the linear-quadratic case, they converge to one of the Feigenbaum constants δ F = 4.66920160 ⋯ .