Duffing-type equations: singular points of amplitude profiles and bifurcations

TL;DR

This paper investigates the singular points and bifurcations of amplitude profiles in Duffing-type equations with polynomial nonlinearities, providing a global structure and connecting to jump phenomena in nonlinear dynamics.

Contribution

It determines the global structure of singular points in amplitude profiles and applies asymptotic methods to predict qualitative dynamical changes in Duffing equations.

Findings

Computed bifurcation sets containing all singular points.

Connected singular points to jump phenomena in amplitude profiles.

Applied techniques to solutions from various asymptotic approaches.

Abstract

We study the Duffing equation and its generalizations with polynomial nonlinearities. Recently, we have demonstrated that metamorphoses of the amplitude response curves, computed by asymptotic methods in implicit form as $F\left( \Omega ,\ A\right) =0$, permit prediction of qualitative changes of dynamics occurring at singular points of the implicit curve $F\left(\Omega ,\ A\right) =0$. In the present work we determine a global structure of singular points of the amplitude profiles computing bifurcation sets, i.e. sets containing all points in the parameter space for which the amplitude profile has a singular point. We connect our work with independent research on tangential points on amplitude profiles, associated with jump phenomena, characteristic for the Duffing equation. We also show that our techniques can be applied to solutions of form $\Omega _{\pm }=f_{\pm }\left( A\right) $, obtained within other asymptotic approaches.