Robust chaos in the generalized ceil map

In this study, we present a predominantly numerical investigation of a novel class of one-dimensional discontinuous dynamical systems, referred to as the generalized ceil map, which combines a power-law nonlinearity with a discontinuous ceiling operation. Despite its simplicity, the map exhibits remarkably rich dynamics, including fixed point stability, robust chaos, and analytically tractable invariant density expressions. Through a detailed investigation of the system’s bifurcation structure, we identify clearly defined stability boundaries (analytically and numerically) and demonstrate the onset of robust chaos in both one and two-parameter spaces involving the nonlinearity exponent α, offset parameter c, and vertical parameter A. Notably, we observe the rare phenomenon of a monotonically increasing Lyapunov exponent within the regime of robust chaos. Analytical expressions for the fixed points and their stability thresholds are derived, allowing us to compute critical parameter values that separate stable and chaotic dynamical regimes. The invariant density function ρ(x) is calculated both numerically and analytically, with the analytical expression becoming asymptotically flat as α→∞, aligning well with simulations. We further analyze the role of parameter variations, revealing that increasing the offset c disrupts robust chaos and induces periodicity, while changes in A have a negligible topological impact and robust chaos persists in such case. We also briefly introduce a similar 1D discontinuous mapping based on the rounding function. These alternative systems show significantly smaller regions of robust chaos and greater susceptibility to periodic windows. Finally, we have explored various types of spatiotemporal patterns observed in the ring-star network configuration including synchronized state, cluster synchronization, and cluster chimera state. To understand the transition of various spatiotemporal patterns with simultaneous variations of the ring and star coupling strengths, we computed a two-parameter regime map in the coupling strength plane highlighting transitions of various novel spatiotemporal patterns.