Nonlinear analysis of forced mechanical systems with internal resonance using spectral submanifolds, Part II: Bifurcation and quasi-periodic response

In Part I of this paper, we have used spectral submanifold (SSM) theory to construct reduced-order models for harmonically excited mechanical systems with internal resonances. In that setting, extracting forced response curves formed by periodic orbits of the full system was reduced to locating the solution branches of equilibria of the corresponding reduced-order model. Here, we use bifurcations of the equilibria of the reduced-order model to predict bifurcations of the periodic response of the full system. Specifically, we identify Hopf bifurcations of equilibria and limit cycles in reduced models on SSMs to predict the existence of two-dimensional and three-dimensional quasi-periodic attractors and repellers in periodically forced mechanical systems of arbitrary dimension. We illustrate the accuracy and efficiency of these computations on finite-element models of beams and plates.

Automated summary

In Part I of this paper, the authors constructed reduced-order models for harmonically excited mechanical systems with internal resonances using spectral submanifold theory. By locating the solution branches of equilibria of the corresponding reduced-order model, they were able to extract forced response curves formed by periodic orbits of the full system. In Part II, the authors use bifurcations of the equilibria of the reduced-order model to predict bifurcations of the periodic response of the full system, specifically predicting the existence of two-dimensional and three-dimensional quasi-periodic attractors and repellers in periodically forced mechanical systems of arbitrary dimension.