Real-time sound synthesis from normal forms of dynamical systems

Traditional sound synthesis methods offer distinct advantages but often face challenges in either computational efficiency or realism. Nonlinear dynamical systems present a compelling middle ground, especially with recent advances in the numerical integration of ordinary differential equations, allowing efficient implementation while capturing the essential dynamics of acoustic instruments. Here, we explore the application of nonlinear dynamical systems, particularly the theory of normal forms, to sound synthesis, offering a novel perspective for creating virtual instruments. The theory of normal forms provides a rigorous mathematical framework for understanding how oscillations can be generated, destroyed, or how they interact in nonlinear systems, making it a powerful tool in sound design. Despite its potential, this approach remains underexplored in the context of sound synthesis. We present an overview of relevant concepts in dynamical systems such as phase space, fixed points, and bifurcations. Three specific dynamical systems are then examined to analyze their suitability for sound synthesis. Finally, we present a package for the Max programming environment for real-time synthesis and show that it is possible to develop virtual instruments based on the examples presented before.