Nonlinear Normal Modes (NNMs) provides a useful analytical framework by extending the principles of Linear Normal Modes (LNMs) to nonlinear dynamical systems. The goal is to decouple the dynamical equations and use linear superposition on the solutions of individual ODEs. Two primary approaches for quantifying NNMs are the Kauderer-Rosenberg approach, which emphasizes the frequency-energy characteristics of the underlying Hamiltonian dynamics, and the Shaw-Piere invariant manifold approach, which focuses on the geometrical aspects of NNMs without excluding dissipation terms from the ODEs. While Shaw’s technique is applicable to a broader range of nonlinear systems, its parameterization accuracy diminishes away from the equilibrium point of the system, and the invariant manifold lacks uniqueness. Recognizing these limitations, different techniques have emerged for parameterizing the Shaw-Piere invariant manifold. One such method involves utilizing the spectral properties of the Koopman operator, an operator theoretic approach for identifying NNMs and representing the dynamics in terms of Koopman eigen functions.

This study aims to advance the framework by demonstrating the linear independence of Koopman eigen functions, ensuring the uniqueness of parameterization. The applicability of the Koopman operator framework is typically confined to systems with hyperbolic equilibrium points, and the uniqueness of invariant manifolds is guaranteed only for non-resonating eigenvalues. This work investigates on the impact of nonlinear internal resonance and dissipation on the accuracy of this framework, utilizing a 2-degree-of-freedom system consisting of two Duffing oscillators, where the primary oscillator is grounded, and the secondary oscillator is coupled to the primary oscillator. Additionally, the research focuses into the examination of the Koopman operator framework’s existence and behavior when one of the system’s eigenvalues approaches zero arbitrarily closely. The investigation aims to stretch the applicability beyond these limitations through numerical simulations, exploring scenarios beyond the established constraints.