Dynamical behaviour of parametrically driven Duffing and externally driven Helmholtz–Duffing oscillators under nonlinear dissipation

In this paper, we mainly focus our attention on the global dynamical behaviour of some ubiquitous nonlinear oscillators under the presence of nonlinear dissipation. We particularly consider the parametrically driven Duffing oscillator and externally driven Helmholtz–Duffing oscillators with nonlinear dissipation term proportional to the power of velocity (vvp-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(v\left| v \right| ^{p-1})$$\end{document}. We obtain the threshold condition for the occurrence of chaos analytically as well as numerically for all the cases p=\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=$$\end{document}1, 2, 3 and 4. We also identify the regions of 2D parameter space (consisting of external forcing amplitude and damping coefficient) corresponding to various asymptotic dynamics and analyse the effect of nonlinear damping on the overall dynamical behaviour of these nonlinear oscillators.