# A Smooth and Discontinuous OscillatorDiscontinuous Dynamics

A Smooth and Discontinuous Oscillator: Discontinuous Dynamics [In this chapter we focus our attention mainly on the limit case as the geometrical parameter α→0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \rightarrow 0$$\end{document}. The content of this chapter presents a new type of discontinuous oscillator with bistability which is a piecewise linear system with the discontinuity at x=0.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x=0.$$\end{document} Due to the limitation of the numerical approximation, an analytical method to formulate the solutions of the discontinuous oscillator is proposed avoiding the barriers encountered in the conventional calculations due to the discontinuity. This scheme made it possible to obtain a chaotic solution theoretically, which gives the chaotic attractor analytically for this discontinuous oscillator. All the results presented here in this chapter and there after in this book for discontinuous dynamics are given by using this analytical procedure.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

# A Smooth and Discontinuous OscillatorDiscontinuous Dynamics

Springer Journals — Sep 28, 2016
18 pages

/lp/springer-journals/a-smooth-and-discontinuous-oscillator-discontinuous-dynamics-jDx260ZMoz
Publisher
Springer Berlin Heidelberg
ISBN
978-3-662-53092-4
Pages
23 –41
DOI
10.1007/978-3-662-53094-8_3
Publisher site
See Chapter on Publisher Site

### Abstract

[In this chapter we focus our attention mainly on the limit case as the geometrical parameter α→0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \rightarrow 0$$\end{document}. The content of this chapter presents a new type of discontinuous oscillator with bistability which is a piecewise linear system with the discontinuity at x=0.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x=0.$$\end{document} Due to the limitation of the numerical approximation, an analytical method to formulate the solutions of the discontinuous oscillator is proposed avoiding the barriers encountered in the conventional calculations due to the discontinuity. This scheme made it possible to obtain a chaotic solution theoretically, which gives the chaotic attractor analytically for this discontinuous oscillator. All the results presented here in this chapter and there after in this book for discontinuous dynamics are given by using this analytical procedure.]

Published: Sep 28, 2016

Keywords: Periodic Solution; Bifurcation Diagram; Chaotic Attractor; Chaotic Orbit; Poincar Section