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

Closed-form waveforms of the modified Emden oscillators x¨ + (q + 2)kxqx˙ + k2x2q+1 + ω2x = 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 lin- earizing 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 2π/ω independent of the amplitude, within an explicitly bounded range of the integration constant, while for even q the oscillations decay algebraically, as t−1/q. The dichotomy is confirmed by an elliptic polar represen- tation, 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, ω2x2 + (x˙ + kxq+1)2 is a Darboux polynomial providing a global Jacobi last multiplier. Explicit waveforms and phase portraits are given for q = 1, . . . , 4, with and without damping, and a recent application to magnon dynamics in the magnetic semiconductor CrSBr is discussed.