<p>Delayed feedback control presents significant challenges, particularly with the emergence of limit cycles (LCs) and entrainment phenomena in oscillators exposed to free and forced vibrations. Within the entrainment region, vibration amplitudes can increase to dangerously high levels, potentially damaging the structure and actuator, or even exciting the system into higher-order modes. The effects of control gain, delay, and damping parameters on these dynamics have not been thoroughly examined in the existing literature. This study investigates the LC oscillations and entrainment phenomena of a cubic-quintic Duffing oscillator under delayed velocity feedback, exploring a wide range of gain and delay parameters. In the first part of this work, we establish the criteria for the existence of LCs and compute their frequencies and amplitudes using the method of describing function. It has been observed that an undamped system can have an infinite number of LCs, regardless of the control gain and delay values, while a damped system exhibits a finite number that progressively increases with delay. The corresponding forced oscillator is then analyzed in the frequency domain using the method of slowly varying parameters. The frequency, amplitude, and stability of LCs identified through the free vibration analysis are supported through the observations of entrainment phenomena under forced vibrations. In the entrainment region of a LC, the response amplitudes can suddenly jump to large values, leading to a state known as vibration anti-control. Therefore, control system designers must be vigilant regarding high-amplitude entrainment regions to prevent unbounded system responses. The high vibration amplitudes observed in the entrainment region suggest that the concepts discussed in this article have the potential to generate controlled vibrations for various macro- and micro-electromechanical applications. However, extensive research is needed to fully explore and understand this potential.</p>