A Parallel Interval Modeling Framework for Nonlinear Systems: Application to a Modified Duffing Oscillator

The paper presents a mathematical framework for converting nonlinear dynamical systems into parallel forms. This framework replaces the exact system motion equations with interval equations, enabling the representation of nonlinear functions over piecewise linear domains. Such representation enables the description of system motions using linear-like differential equations, which can be analyzed and manipulated using well-known control methods. One such method is eigenvalue analysis, a powerful tool in classical control theory since many techniques rely on the system’s characteristic polynomial and its eigenvalues. We apply this method to define interval system eigenvalues and track their variation during system operation. These eigenvalues serve as the basis for partial fraction decomposition, transforming the series system model into a parallel structure. This transformation incorporates both real and complex-conjugate eigenvalues. Unlike classical approaches, our method accounts for subsystems with complex components, producing complex vector signals. We demonstrate the approach by converting a modified Duffing oscillator into parallel form and analyzing its outputs. The contribution is primarily conceptual and methodological, providing a decomposition framework rather than a performance-optimized numerical scheme.