Critical fractal boundary between chaos and periodicity: Exploring the Devil’s staircase

We investigate the dynamical behavior of the sine-circle map in a two-parameter space, with particular emphasis on extreme curves that traverse the centers of periodic windows. To analyze rotational dynamics in regimes where the classical definition may become unreliable, we introduce a return number, ρ∗, defined from return events along trajectories, which acts as a rotation-like quantity. Along the extreme curves, we compute ρ∗ and compare the results with those obtained from the classical rotation number (ρ). We find that the return number remains well-defined in parameter regions where the classical approach fails to converge, and its results are consistent with the qualitative behavior observed in cobweb diagrams. The extreme curves reveal hierarchical structures associated with period-adding cascades, in which stable periodic orbits increase their period by fixed increments. Along one such curve, we identify a structure that we term a discontinuous Devil’s Staircase, characterized by sequences of rational values separated by chaotic intervals, in contrast to the classical continuous staircase. Furthermore, we compute the uncertainty exponent α along two distinct extreme curves: one located in a simple period-adding region and the other in a more intricate dynamical regime. Despite their structural differences, both curves yield nearly identical values of α≈0.233(3), suggesting that this fractal measure is robust with respect to the underlying dynamical complexity. These results contribute to the understanding of how extreme curves organize parameter space in nonlinear systems and highlight the uncertainty exponent as a useful diagnostic tool for complex dynamics.