Organized structures and multiple attractors in chaotic regimes

Deterministic chaotic systems often exhibit highly organized structures in both phase space and parameter space. In this work, we investigate two-dimensional Hénon-like extensions of the discrete Gaussian and cubic maps incorporating a linear feedback channel. The Gaussian-based model admits a natural realization as a nonlinear electronic circuit consisting of a Gaussian-saturating amplifier coupled to a linear memory feedback loop. A detailed bifurcation analysis reveals that, while the classical one-dimensional Gaussian map is primarily organized by fold and flip bifurcations together with period-bubbling routes to chaos, its two-dimensional extension possesses a substantially richer dynamical structure. In particular, the system exhibits Neimark-Sacker bifurcations, quasiperiodic dynamics, Arnold tongues generated through frequency locking, shrimp-shaped periodic stability regions embedded within chaotic regimes, and extensive multistability. The observed Arnold tongues follow the classical Farey-tree hierarchy of rotation numbers and organize the resonance structure of the parameter space. Furthermore, coexistence of attractors is demonstrated through bistability between period-9 and period-14 oscillations for identical parameter values. The corresponding basins of attraction display strong intermingling, and the computed basin entropy (Sb = 0.6032) indicates fractal basin boundaries and pronounced sensitivity to initial conditions. In addition, although the one-dimensional cubic map already exhibits shrimp-shaped periodic windows, its two-dimensional extension reveals nested shrimp families together with resonance tongue structures, substantially enriching the global bifurcation landscape. These results show that low-dimensional nonlinear maps can generate highly organized parameter-space architectures in which periodic, quasiperiodic, and chaotic dynamics coexist through well-defined bifurcation mechanisms.