Odd-even dichotomy in the waveforms of modified Emden oscillators of arbitrary integer power

Closed-form waveforms of the modified Emden oscillators $\ddot{x}+(q+2)k{x}^q\dot{x}+k^2{x}^{2q+1}+{\omega }^{2}x=0$ of arbitrary natural power $q$, and of their damped $c$-deformed versions, are obtained through a generalized commutative factorization that reduces them to Bernoulli equations. We show that commutative factorizations of Li'enard equations are gauge-conjugated forms of the nonlocal linearizing transformation of Chandrasekar, Senthilvelan, Kundu and Lakshmanan, the factorization constant being half the difference of the roots of the associated linear oscillator. These oscillators display an odd–even dichotomy: for odd $q$ they are isochronous, with period $2\pi \mathrm{/}\omega$ independent of the amplitude, within an explicitly bounded range of the integration constant, while for even $q$ the oscillations decay algebraically, as $t^{-1\mathrm{/}q}$. The dichotomy is confirmed by an elliptic polar representation, in which the phase rotates uniformly for every $q$, and, for even $q$, by the Bendixson–Dulac criterion applied to the divergence of the vector field. We present a dissipative Lagrangian of Lurie–Rayleigh type and, for $q=1$, an autonomous Sundman linearization that yields a rational first integral, a Jacobi last multiplier and a non-dissipative Lagrangian; for every $q$, ${\omega }^2{x}^2+(\dot{x}+k{x}^{q+1}{)}^2$ is a Darboux polynomial providing a global Jacobi last multiplier. Explicit waveforms and phase portraits are given for $q=1,\dots ,4$, with and without damping, and a recent application to magnon dynamics in the magnetic semiconductor CrSBr is discussed.