Coexistence and Basins of Attraction in Nonreciprocal Duffing Chains

Determine the coexistence of multiple attractors and characterize their basins of attraction in the one-dimensional chain of unidirectionally coupled double-well Duffing oscillators arranged into ring-structured units, with forward coupling C, backward compensation coupling C′, and damping γ, as defined in the model equations. Ascertain the parameter regimes and initial-condition dependence under which different attractors (e.g., fixed points, limit cycles, tori, chaotic or hyperchaotic states) simultaneously exist, and delineate the boundaries of their attraction basins within this nonreciprocal, dissipative system.

Background

The paper introduces a nonlinear, dissipative model of a one-dimensional chain composed of ring-structured units of double-well Duffing oscillators with unidirectional (forward) couplings and long-range backward compensation couplings. The paper explores Non-Reciprocal Skin Effect (NRSE) in the nonlinear regime and demonstrates a range of attractors obtained via bifurcations, including limit cycles, tori, chaos, and hyperchaos, depending on coupling and nonlinearity parameters.

Because the system is autonomous and dissipative, the presence of backward compensation C′ is essential to sustain nontrivial attractors. The authors show how adding units synthesizes higher-dimensional attractors due to unidirectional coupling, and how saturation leads to an anti-NRSE pattern in the chaotic regime. While they provide numerical bifurcation diagrams and Lyapunov spectra indicating rich dynamical behavior, they explicitly note that the detailed structure of coexisting attractors and their basins of attraction remains unresolved, motivating a precise classification and basin analysis in this nonreciprocal Duffing chain.