Determining the Stability Criteria for the Suggested Nonlinear Autoregressive Model

       We present in this current study the stability properties of the model, which primarily relies on methods derived from Ozaki's work. The proposed mathematical simplification method makes nonlinear model analysis more achievable through an approximate solution. Stability conditions for the proposed model's behavior throughout time represent the main focus of this research investigation. Many scientific disciplines, including economics and biology, implement nonlinear models broadly because these models offer effective solutions to complex dynamical systems, which regular techniques find difficult to handle, along with oscillatory and chaotic phenomena. The analysis of these systems becomes simpler through localization at the non-zero point using Ozaki's model, which allows the use of standard linear stability analysis methods. Our investigation targets the singular point while developing conditions that describe its stability range and determining the stability rule for newly expected limit cycles. The theoretical findings undergo verification through different numerical examples, which adhere to derivative specifications.