  Chaos in classical systems has been studied in plenty over many years.
Although the search for chaos in quantum systems has been an area of prominent
research over the last few decades, the detailed analysis of many inherently
chaotic quantum systems based on expectation values of dynamical variables has
not been reported in the literature. In this paper, we extend the study of
dynamical behaviour using expectation values of variables to a $q$-deformed
harmonic oscillator. The system is found to be periodic, quasi-periodic or
chaotic depending on the values of the deformation parameter $q$ and the
deformed coherent amplitude $\alpha_{q}$, thus enabling us to explicitly
classify the chaotic nature of the system on the basis of these parameters. The
chaotic properties of the system are clearly illustrated through recurrence
plots, power spectra, first-return-time distributions and Lyapunov exponents of
the time series obtained for the expectation values of the dynamic variables.
