Abstract
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.
Highlights
Consider a uniformly charged ring with a conducting wire placed along the axis of the ring
We study the mathematical model of the point charge oscillator which has been derived by A
We prove that the period increases monotone with the amplitude and derive an expression for its growth rate as the amplitude tends to infinity
Summary
Consider a uniformly charged ring with a conducting wire placed along the axis of the ring. Lemma 5 implies that any trajectory of system (2.7) starting in the fourth quadrant of the phase plane has E0 as ω-limit point. The interesting question concerns the existence of a trajectory T0 in the first quadrant having E0 as ω-limit set and separating the regions G0 and G1 such that all trajectories starting in G0 have E0 as ω-limit set, while all trajectories starting in G1 cross the z-axis (see Figure 3) To answer this question we apply the method of blowing up the equilibrium point E0 by means of the transformation H:. System (2.9) is invariant under the transformation v → −v, t → −t, that means its phase portrait possesses the mirror symmetry with respect to the w-axis This system exhibits the peculiarity that the invariant straight line w = 0 consists of equilibria. We are interested in the dependence of the primitive period on the amplitude
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
