<p>We study a memristor-based chaotic circuit composed of an inductor and a negative resistor in series with a parallel combination of a passive memristor and a capacitor. The system is modeled by a three-dimensional differential system with seven parameters. A comprehensive bifurcation analysis is carried out by varying the parameters, revealing the existence of first integrals and periodic oscillations with distinct amplitudes, organized into centers located on invariant surfaces in the phase space, defined by these first integrals. We also prove the occurrence of a zero-Hopf bifurcation at the origin, which gives rise to a stable periodic orbit. As one of the system parameters is further varied, this periodic orbit undergoes a cascade of period-doubling bifurcations, eventually leading to a chaotic attractor. This provides a theoretical explanation for one of the mechanisms by which chaos emerges in the circuit, thereby complementing and extending previous results in the literature.</p>