We present a data-driven nonlinear and chaos theory-based analysis of thermoacoustic instabilities in a simple Rijke tube. Thermoacoustic instability modes in this simple Rijke tube display very rich nonlinear behavior because of the interaction of acoustic modes and unsteady heat-release processes during combustion. This approach of analyzing thermoacoustic instabilities, their evolution, and interactions differs from traditional linear time-series-based approaches, such as the Fourier transform or the autocorrelation function. The objectives of this work are to address the limitations of conventional linear and model-based nonlinear analyses and to describe the potential of data-driven nonlinear and chaos theory-based analyses to gain a deeper understanding of thermoacoustic instability dynamics for reacting flows. The Rijke tube permits investigation of the nonlinear dynamics of thermoacoustic instabilities in a very systematic way by using picosecond time-resolved laser-induced fluorescence (PITLIF) of the OH radical at a rate of 2500 Hz. The 2500-Hz measurement bandwidth is sufficient to capture all thermoacoustic instability modes encountered in the Rijke tube. Through the use of these nontraditional analyses, we find that thermoacoustic instabilities within the Rijke tube contain chaotic behavior. A simple low-order approximate model and a data-adaptive feedback controller are also introduced for the control of thermoacoustic instabilities.