From Nonlinear to Linear Dynamics: A Structural Approach via Wedderburn–Artin Decomposition

Nonlinear dynamical systems resist global analysis when approached through classical linearization techniques which rely on differential equations and local approximations. In seeking approaches to structurally reduce nonlinear dynamics to linear components, we propose interpreting the evolution of a nonlinear system as a sequence of non-commuting operations within a finite-dimensional associative algebra, where interaction rules are captured abstractly through algebraic composition rather than defined analytically. By embedding the nonlinear system into a semisimple algebra, we employ the Wedderburn–Artin decomposition to represent its dynamics as a direct sum of matrix algebras over division rings. Each matrix block defines a linear action on an irreducible subspace, corresponding to a dynamically invariant mode grounded in the system’s internal symmetries. This block structure reveals a modular architecture, demonstrating how nonlinear interactions can give rise to intrinsically linear behaviours governed by underlying algebraic principles. We apply our method to three distinct systems—symbolic rewriting systems, operator-driven vector dynamics and partially associative bitwise systems—selected to represent symbolic, quantitative and hybrid forms of nonlinearity. This range ensures that our decomposition framework effectively captures both regular and irregular compositional structures across diverse classes of nonlinear behaviour. We demonstrate that our method is able to isolate invariant subsystems and uncover underlying structure, by revealing the latent linear organization embedded within complex nonlinear behaviour. Overall, our framework extends matrix-based analysis into domains that are traditionally nonlinear, bridging symbolic computation, algebraic structure and dynamical behavior and providing an alternative approach to tackle nonlinear systems through their decomposable linear representations.