Intrinsic localized modes behavior in a nonlinear oscillators system with nonreciprocal coupling spring

Intrinsic localized modes (ILMs) arise from nonlinear vibration localization and have been observed in photonic crystals, coupled cantilevers, and nonlinear oscillators. Although prior research has explored ILMs in arrays of nonlinear oscillators linked by different coupling methods, such as weak and magnetic springs, they have not yet examined the study of the system with nonreciprocal springs. However, this study focuses on ILMs within two Duffing oscillator arrays connected by a nonreciprocal spring. Nonreciprocal springs, known for their asymmetric response to applied forces, exhibit varying stiffness depending on force direction. Through numerical analysis, the system with a nonreciprocal spring is studied and compared to its reciprocal counterpart. The system response was initially explored for various nonreciprocal spring configurations. Subsequently, the influence of nonreciprocal spring stiffness on the ILMs was studied. Then, perturbation analysis and energy transfer between oscillators were investigated. The results indicate that nonreciprocal springs shift ILMs’ starting frequencies and adjust the amplitude difference between ILMs and stable branches for equivalent frequencies and spring stiffness as the reciprocal setup. Also, increasing the stiffness of the nonreciprocal spring can lead to a single low–high (L-H) mode within the ILMs. In addition to that, perturbation analysis shows that a system with a nonreciprocal spring has a higher propensity to shift from High–High (H-H) mode to Low–Low (L-L) mode compared to a reciprocal system at the same excitation frequencies when perturbed from the 1st oscillator. When the oscillators are set in the ILMs, the nonreciprocal system demonstrates notable potential for energy transfer/suppression when the system is set at ILMs compared to its reciprocal counterpart. Integrating nonreciprocal springs into nonlinear systems holds promise for various engineering applications, particularly in vibration absorption and energy harvesting.