Bifurcation analysis of quasi-periodic orbits of mechanical systems with 1:2 internal resonance via spectral submanifolds

• Published in: Nonlinear dynamics Vol. 113; no. 11; pp. 12609 - 12640
• Main Authors: Liang, Hongming, Jain, Shobhit, Li, Mingwu
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Nature B.V 01.06.2025
• Subjects: Bifurcations; Finite element method; Fourier transforms; Harmonic excitation; Manifolds (mathematics); Mechanical systems; Orbits; Random vibration; Reduced order models; Resonance
• ISSN: 0924-090X; 1573-269X
• DOI: 10.1007/s11071-024-10794-6

A 1:2 internally resonant mechanical system can undergo secondary Hopf (Neimark-Sacker) bifurcations, resulting in a quasi-periodic response when the system is subject to harmonic excitation. While these quasi-periodic orbits have been observed in practice, their bifurcations are not well studied, especially in high-dimensional mechanical systems. This is mainly because of the challenges associated with the computation and bifurcation detection of these quasi-periodic motions. Here we present a computational framework to address these challenges via reductions on spectral submanifolds, which transforms quasi-periodic orbits of high-dimensional systems as limit cycles of four-dimensional reduced-order models. We apply the proposed framework to analyze bifurcations of quasi-periodic orbits in several mechanical systems exhibiting 1:2 internal resonance, including a finite element model of a shallow-curved shell. We uncover local bifurcations such as period-doubling and saddle-node, as well as global bifurcations such as homoclinic connections, isolas, and simple bifurcations of quasi-periodic orbits. We also observe cascades of period-doubling bifurcations of quasi-periodic orbits that eventually result in chaotic motions, as well as the coexistence of chaotic and quasi-periodic attractors. These findings elucidate the complex bifurcation mechanism of quasi-periodic orbits in 1:2 internally resonant systems.