Dependence of isochronous bifurcations on the driving-mode phase shift in two-harmonic standard maps

Some dynamical properties of nonlinear coupled systems can be described by the two-harmonic standard map, a two-dimensional area-preserving system with two parameters, where two distinct arbitrary resonant modes compete. Usually, the initial phase of the resonant modes is considered to be null. In this paper, we consider a non-null phase shift between the two competing isochronous modes that form the system. We observe that a nonzero phase shift alters the phase space, changing the stability and positions of the fixed points. Furthermore, the phase shift can change the dominant mode and create intermediate modes between the main ones. Lastly, we analyze the effect of the phase shift on the onset of secondary shearless curves in the phase space. Thus, different phase shifts result in various scenarios in which secondary shearless curves emerge in the phase space.