# A Smooth and Discontinuous OscillatorElliptic and Hyperbolic Functions

A Smooth and Discontinuous Oscillator: Elliptic and Hyperbolic Functions [In this chapter, we go on into the methods for obtaining the analytical solutions of the SD oscillator. A series of irrational elliptic functionsIrrational elliptic function and hyperbolic functionsIrrational hyperbolic function is proposed for the unperturbed oscillator to provide the analytical solutionsAnalytical solution for both the smooth and discontinuous cases with periodic solutions and the homoclinic ones which could not be expressed using classical tools, the traditional methodologies being applicable only for rational or polynomial systems. It is found that the solutions of the discontinuous case can also be given by letting 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}. With the help of the defined elliptic functions and the hyperbolic functions for the periodic and homoclinic orbits, the chaotic behaviours of the perturbed system can be detected analytically.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

# A Smooth and Discontinuous OscillatorElliptic and Hyperbolic Functions

Springer Journals — Sep 28, 2016
17 pages

/lp/springer-journals/a-smooth-and-discontinuous-oscillator-elliptic-and-hyperbolic-Sv4C3NgS9D
Publisher
Springer Berlin Heidelberg
ISBN
978-3-662-53092-4
Pages
121 –138
DOI
10.1007/978-3-662-53094-8_9
Publisher site
See Chapter on Publisher Site

### Abstract

[In this chapter, we go on into the methods for obtaining the analytical solutions of the SD oscillator. A series of irrational elliptic functionsIrrational elliptic function and hyperbolic functionsIrrational hyperbolic function is proposed for the unperturbed oscillator to provide the analytical solutionsAnalytical solution for both the smooth and discontinuous cases with periodic solutions and the homoclinic ones which could not be expressed using classical tools, the traditional methodologies being applicable only for rational or polynomial systems. It is found that the solutions of the discontinuous case can also be given by letting 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}. With the help of the defined elliptic functions and the hyperbolic functions for the periodic and homoclinic orbits, the chaotic behaviours of the perturbed system can be detected analytically.]

Published: Sep 28, 2016

Keywords: Periodic Solution; Periodic Orbit; Phase Portrait; Elliptic Function; Homoclinic Orbit