This book, for the first time, presents a review on transition from regular dynamics to quantum chaos in a quantum degenerate system -- a harmonic oscillator perturbed by a monochromatic wave. The model, being one of the simplest dynamical systems, exhibits the fundamental properties of nonlinearity, quantum instability and chaos. And, more importantly, it describes practical systems such as an ion in a linear ion trap interacting with laser radiation, an electron in magnetic field interacting with a plasma wave, an acoustic quantum cyclotron resonance in metals, and a two-dimensional electron gas in semiconductor heterostructures. This book explains in details how the transition to quantum chaos occurs and how the theoretical predictions can be tested in experiments. The book is useful to students and scientists who are interested to understand the nature of quantum instabilities and chaos in dynamic systems and their applications in technological problems.