Isochronous and underdamped waveforms of modified Emden oscillators

Abstract

Bernoulli-type waveforms for modified Emden nonlinear oscillators of arbitrary natural power $q$ are obtained through a generalized commutative factorization approach. These oscillators display a well-defined odd-even dynamical dichotomy, which is discussed in detail: the odd-$q$ cases entail isochronous oscillators whose period $T = 2π/ω$ is independent of amplitude and initial conditions, while the even-$q$ cases display underdamped behavior. The Lagrangian formulation is presented in the Lurie's dissipative description. The isochronous regime and the period of the solutions in the odd case are also confirmed through a generalized polar-coordinate analysis in the spirit of Sabatini's work. The absence of periodic orbits for even $q$ is shown to be a consequence of the Bendixson-Dulac criterion applied to the radial velocity function. Explicit waveforms and their phase portraits are presented for $q = 1, 2, 3, 4$, along with the non exponential envelope formulas for the damped cases and singular-region bounds for the isochronous ones. A few possible applications are also mentioned.