THE POINT CHARGE OSCILLATOR: QUALITATIVE AND ANALYTICAL INVESTIGATIONS

We study the mathematical model of the point charge oscillator which has been derived by A. Beléndez et al. [2]. First we determine the global phase portrait of this model in the Poincaré disk. It consists of a family of closed orbits surrounding the unique finite equilibrium point and of a continuu...

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Bibliographic Details
Published inMathematical modelling and analysis Vol. 24; no. 3; pp. 372 - 384
Main Author Schneider, Klaus R.
Format Journal Article
LanguageEnglish
Published Vilnius Vilnius Gediminas Technical University 01.04.2019
Subjects
amplitude-period relation
Amplitudes
closed orbits
Elliptic functions
global phase portrait
Infinity
Jacobi elliptic function
Mathematical models
Orbits
Oscillators
Point charge
point charge oscillator
Qualitative analysis
Qualitative research
Germany
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Summary:We study the mathematical model of the point charge oscillator which has been derived by A. Beléndez et al. [2]. First we determine the global phase portrait of this model in the Poincaré disk. It consists of a family of closed orbits surrounding the unique finite equilibrium point and of a continuum of homoclinic orbits to the unique equilibrium point at infinity. Next we derive analytic expressions for the relationship between period (frequency) and amplitude. Further, we prove that the period increases monotone with the amplitude and derive an expression for its growth rate as the amplitude tends to infinity. Finally, we determine a relation between period and amplitude by means of the complete elliptic integral of the first kind K(k) and of the Jacobi elliptic function cn.
ISSN:1392-6292
1648-3510
DOI:10.3846/mma.2019.023