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Subject: Jacobi elliptic function ×

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    The point charge oscillator: Qualitative and analytical investigations
    (Vilnius : Vilnius Gediminas Technical University, 2019) Schneider, Klaus R.
    We study the mathematical model of the point charge oscillator which has been derived by A. Belendez et al. [2]. First we determine the global phase portrait of this model in the Poincare 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.
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    The point charge oscillator: Qualitative and analytical investigations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Schneider, Klaus R.
    We determine the global phase portrait of a mathematical model describing the point charge oscillator. It shows that the family of closed orbits describing the point charge oscillations has two envelopes: an equilibrium point and a homoclinic orbit to an equilibrium point at infinity. We derive an expression for the growth rate of the primitive period Ta of the oscillation with the amplitude a as a tends to infinity. Finally, we determine an exact relation between period and amplitude by means of the Jacobi elliptic function cn.