# Dynamics of observables in a $q$-deformed harmonic oscillator

## TL;DR

This paper investigates the dynamical behavior of a $q$-deformed harmonic oscillator, revealing how its expectation values can exhibit periodic, quasi-periodic, or chaotic behavior depending on system parameters, with analysis supported by various chaos indicators.

## Contribution

It extends the analysis of quantum chaos by studying expectation values in a $q$-deformed harmonic oscillator, providing explicit classification based on deformation parameters.

## Key findings

- System exhibits periodic, quasi-periodic, or chaotic dynamics.
- Chaos characterized using recurrence plots, power spectra, and Lyapunov exponents.
- Behavior depends on deformation parameter $q$ and amplitude $\alpha_q$.

## Abstract

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.