Transport barriers and directed transport in the rational standard nontwist map

We explore the dynamics and transport properties of the rational standard nontwist map (RSNM), which works as an extension of the standard nontwist map (SNM). In addition to the usual parameters of the SNM that govern the twist function profile and the intensity of the nonlinear perturbation, we introduce a new perturbation parameter μ in the RSNM, which makes it possible to break the symmetry of the system. The symmetry breaking leads to directed transport, known as the ratchet effect, where chaotic orbits exhibit a preferential direction of motion. We analyze the impact of μ on both the phase space and the parameter space structure, focusing on the destruction of transport barriers, which acts as separators between chaotic regions. Through numerical simulations and analysis of the fixed points stability, we demonstrate that an increase in μ enhances the chaotic volume in the lower half of the phase space, resulting in the destruction of invariant spanning curves, while simultaneously regularizing the upper half. Additionally, we explore the conditions under which partial transport barriers persist and their role in moderating transport across the phase space. We show that even small variations in a control parameter causes crossings of invariant manifolds from different regions of the phase space, enhancing transport with the mechanism of turnstiles and intercrossing. Our analysis of directed transport reveals that the breaking of symmetry by μ results in either positive or negative net transport in the phase space, depending on the control parameters. We also note that RSNM creates new regions within the parameter space, referred to as holes, due to the emergence of transport within previously null transport regions.