Energy exchange in isochronous Liénard oscillators applied to active mechanical systems
de la Cruz Díaz, Josué Domingo^{1,2}; Rosu, Haret-Codratian^{1}; Salgado Blanco, Daniel Igancio^{1,3,4}
$^1$División de Materiales Avanzados, IPICYT, Camino a la Presa San José 2055, Col. Lomas 4a Sección, San Luis Potosí, S. L. P., 78216, México. $^2$ Universidad Nacional Rosario Castellanos, Av. Universidad Tecnológica S/N, Col. Rancho Nuevo, Soledad de Graciano Sánchez, S.L.P., 78430, México. $^3$Grupo de Ciencia e Ingeniería Computacionales, Centro Nacional de Supercómputo. IPICYT, Camino a la Presa San José 2055, Col. Lomas 4a Sección, San Luis Potosí, S. L. P., 78216, México. $^4$Investigadores por México SECIHTI - Grupo de Ciencia e Ingeniería Computacionales, Centro Nacional de Supercómputo. IPICYT, Camino a la Presa San José 2055, 78216, San Luis Potosí, México
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Abstract
We investigate energy exchange in isochronous Liénard oscillators through a Rayleigh formulation, obtaining the mechanical energy rate $\dot H=-f(x)\dot{x}^{2}$. The sign of the state-dependent function $f(x)$ determines whether energy is extracted or supplied, while periodic motion requires exact balance between these contributions over a complete cycle. This framework is applied to the modified Emden oscillator and a generalized odd-power family, preserving an analytically prescribed period while allowing the nonlinear gain--loss mechanism to vary. We further interpret the Liénard functions as reduced representations of active mechanical systems, with $g(x)$ describing the restoring response and $f(x)$ an effective state-dependent combination of dissipation and active energy injection. The resulting balance provides a compact criterion for sustained self-oscillation and a measure of mechanical energy throughput, offering a reduced-order framework for active resonators.
Keywords: Liénard equation; Active materials; Rayleigh dissipation.