The point charge oscillator: qualitative and analytical investigations. (English) Zbl 1470.34087
Summary: We study the mathematical model of the point charge oscillator which has been derived by A. Beléndez et al. [Phys. Lett., A 373, No. 7, 735–740 (2009; Zbl 1227.78007)]. 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 \(cn\).
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 \(cn\).
MSC:
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
| 34C15 | Nonlinear oscillations and coupled oscillators for ordinary differential equations |
| 34C37 | Homoclinic and heteroclinic solutions to ordinary differential equations |
| 78A35 | Motion of charged particles |
Keywords:
point charge oscillator; global phase portrait; closed orbits; amplitude-period relation; Jacobi elliptic functionCitations:
Zbl 1227.78007References:
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