Global dynamics and asymmetric fractal dimension in a nontwist circle map

We consider the standard nontwist map with strong dissipation that leads the system to a 1D circular map with a quadratic sinusoidal oscillation and two control parameters. The 2D Lyapunov and isoperiodic diagrams reveal a complex interplay between domains of periodicity embedded in regions dominated by quasiperiodic and chaotic behaviors. Arnold tongues and shrimp-like, among other sets of periodicities, compose this rich dynamical scenario in the parameter space. Cobwebs and bifurcation diagrams help reveal the behavior of attractors, including multistability, period-doubling, pitchfork bifurcations, as well as boundary, merging, and interior crises that influence the structures of periodicity. Furthermore, we bring to light the global organization of shrimp-like structures by carrying out a new concept of orbits, the extreme orbits, and announce that the fractal dimension, believed to be universal in the parameter space for decades, has its symmetry breaking in the vicinity of shrimp-like cascades.