Directional Relational Manifolds with Relativistic-Driven Nonlinear Dynamics

We introduce a framework coupling Directional Relational Manifolds (DRM) — configuration spaces with point-dependent dimensionality—to nonlinear discretedynamics driven by a relativistic scaling parameter. The central object is an iterated map xn+1 = F (xn; A(ω, r), D(xn), gxn ) whose control parameter A(ω, r) = (4/π) ωr γ(ωr), with γ the Lorentz factor, diverges as the tangential velocity v = ωr approaches the speed of light c. We show that, under standard topological conjugacy assumptions, this divergence induces a cascade of period-doubling bifurcations in the dynamics, establish conditions under which bounded orbits converge to toroidal attractors, and show that the effective dimensionality of the underlying manifold can change along trajectories. The framework generalises fixed-dimensional Riemannian geometry and provides a setting in which emergent dimensional structure, relativistic kinematics, and nonlinear dynamics interact within a single, mathematically consistent formalism. The bifurcation results hold for map families satisfying explicit conjugacy hypotheses; connections to Feigenbaum universality are established as limiting cases, not general claims. No claim of physical novelty beyond the mathematical framework is made; interpretational aspects are confined to a clearly delimited section.