This study investigates the use of the Symmetric Projection Attractor Reconstruction (SPAR) method as a novel approach to visualising and analysing musical sounds and the dynamic changes in their signals. Rooted in the principles of deterministic chaos, the SPAR method reconstructs attractors in N-dimensional phase space to reveal unique geometrical patterns that distinguish different instruments, unveiling complex non-linear structures. SPAR is relatively robust to noise as it constructs the underlying attractor structure. Using an unsupervised machine learning method (hierarchical clustering), we identify distinct attractor shapes for string and wind instruments. String instruments exhibit more symmetric attractors, whereas wind instruments show greater variability and asymmetry. The clustering accuracy for distinguishing between string and wind instruments was 93%, and 71% for differentiating between string, woodwind, and brass instruments, demonstrating the effectiveness of this method for automatic timbre classification. The study further highlights significant variability in attractor shapes in one-second-long windows of the sound of an instrument playing a given pitch. The observed rapid transitions between stable and chaotic states underscore the complexity and dynamic nature of sound signals. Additionally, a comparison of the attractors between sounds generated using the VST (Virtual Studio Technology) libraries (BBC Orchestra, Apple Studio Strings) and recorded samples for violin was performed to show differences associated with the sound source. This research demonstrates the potential of the SPAR method for sound analysis, providing valuable insights for music information retrieval and serving as an input option for machine learning models.