Analysis and modeling of music dynamic characteristics based on differential equations

Abstract

Music, as a complex form of human expression, is intrinsically dynamic, with nonlinear characteristics in its structure and behavior over time. Music expresses a variety of emotional qualities and can be represented as a nonlinear dynamical system that frequently exhibits persistence. Traditional analysis approaches are frequently insufficient for capturing these complex dynamics. The research is to utilize Ordinary Differential Equations (ODEs) to investigate the dynamic properties of music, with focus on the nonlinear behavior observed in musical signals. The method involves breaking down music into its core components, examining local characteristics, and generalizing the underlying features. Using ODEs, the system's Lyapunov exponents to measure stability and chaotic behavior, the energy spectrum to analyze oscillatory modes, and the correlation dimension to comprehend the fractality inherent in music signalswere investigated. These strategies use numerical simulations to characterize music's nonlinear features. The findings show that ODE-based approaches can accurately simulate important dynamic characteristics of music, with Lyapunov exponents exposing chaotic behavior the energy spectrum demonstrating oscillatory patterns, and the correlation dimension emphasizing the fractal form of music. Finally, the research verifies the use of differential equations (DE) to explain the dynamic properties of music. The method provides important insights into the nonlinear dynamics of musical signals, paving the way for more detailed models and applications in music analysis.