# A Smooth and Discontinuous OscillatorCo-Dimension Two Bifurcation

A Smooth and Discontinuous Oscillator: Co-Dimension Two Bifurcation [This chapter investigates the fundamental properties of the behaviour of the SD oscillatorSD oscillator of the perturbed degenerate case. As we have already seen, there is a degenerate singularityDegenerate singularity due to the change of hyperbolicity whenHyperbolicity parameter α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha$$\end{document} varies crossing α=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha =1$$\end{document}, where the system is transformed from a single well dynamics to that of a double-wellDouble-well dynamics, in other words from a single stability to a bistabilityMultiple stability. This singularity can also be treated as the transition from non snap through buckling (α>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha >1$$\end{document}) to a snap through bucklingSnap-through buckling (α<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha <1$$\end{document}) under a static load. We turn our attention to the interesting perturbed behaviour of the complex codimension two bifurcations of the oscillator at the degenerate equilibrium point (0, 0) near α=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha =1$$\end{document}. It is found that the universal unfolding with two parameters is the perturbed SD oscillator with a nonlinear visco-damping. This universal unfolding reveals the complicated codimension two bifurcation phenomena in the physical parameter space with homoclinic bifurcation, closed orbit bifurcation, Hopf bifurcations and also the pitchfork bifurcations demonstrated at the same time when the geometrical parameter varies.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

# A Smooth and Discontinuous OscillatorCo-Dimension Two Bifurcation

Springer Journals — Sep 28, 2016
12 pages

Publisher
Springer Berlin Heidelberg
ISBN
978-3-662-53092-4
Pages
53 –65
DOI
10.1007/978-3-662-53094-8_5
Publisher site
See Chapter on Publisher Site

### Abstract

[This chapter investigates the fundamental properties of the behaviour of the SD oscillatorSD oscillator of the perturbed degenerate case. As we have already seen, there is a degenerate singularityDegenerate singularity due to the change of hyperbolicity whenHyperbolicity parameter α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha$$\end{document} varies crossing α=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha =1$$\end{document}, where the system is transformed from a single well dynamics to that of a double-wellDouble-well dynamics, in other words from a single stability to a bistabilityMultiple stability. This singularity can also be treated as the transition from non snap through buckling (α>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha >1$$\end{document}) to a snap through bucklingSnap-through buckling (α<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha <1$$\end{document}) under a static load. We turn our attention to the interesting perturbed behaviour of the complex codimension two bifurcations of the oscillator at the degenerate equilibrium point (0, 0) near α=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha =1$$\end{document}. It is found that the universal unfolding with two parameters is the perturbed SD oscillator with a nonlinear visco-damping. This universal unfolding reveals the complicated codimension two bifurcation phenomena in the physical parameter space with homoclinic bifurcation, closed orbit bifurcation, Hopf bifurcations and also the pitchfork bifurcations demonstrated at the same time when the geometrical parameter varies.]

Published: Sep 28, 2016

Keywords: Degenerate Equilibrium Point; Universal Unfolding; Physical Parameter Space; Homoclinic Bifurcation; Hopf Bifurcation