Understanding how energy moves through nonlinear systems is essential for progress in many areas of science and engineering, including fluid dynamics, neuroscience, and the design of advanced materials. This project studies a mathematical model known as a nonlinear oscillator chain, where interactions between neighboring components can create complex, cascading flows of energy between different scales. Such systems serve as simplified yet powerful representations of more complicated physical processes, such as ocean turbulence or signal propagation in the brain. This project supports fundamental research in probability and applied dynamical systems, as well as the development of new computational tools for analyzing high-dimensional stochastic systems that also inform coupled neuronal oscillators and machine learning algorithms. Through student training activities, this work will help build a capable STEM workforce, contributing to national priorities in scientific advancement and education. Recent breakthroughs have drawn new connections between nonlinear dispersive equations and wave kinetic equations (WKE), with particular interest in understanding how energy cascades through scales in weakly nonlinear regimes. A central object in this theory is the Kolmogorov–Zakharov (KZ) spectrum, a formal steady-state solution of the WKE that reflects how energy transfers across modes. This project investigates a class of nonlinear oscillator chains—called energy cascade systems—tha