Hopf Bifurcation on an Invariant Algebraic Surface of a Generalized Nosé–Hoover Oscillator

In this work, we present a qualitative study of a generalization of the system characterizing the Nosé–Hoover oscillator, within the framework of Darboux integrability theory. We establish algebraic conditions for the existence of invariant algebraic surfaces of the form z + f (x, y) = 0. Furthermore, we show that under certain conditions it is always possible to obtain a Hopf bifurcation on the invariant algebraic surface, generating nonlinear oscillations in the generalized system.