Geometry-dependent chaos: A unified framework for anisotropic dynamical systems via Trinition deformation

Abstract

This work develops a Trinition-based framework for geometry-dependent chaotic dynamics. The Trinition construction is represented on a three-component primitive space, while its multiplication law generates both an anisotropic quadratic geometry and an additional interaction-accessible bivector sector, providing a precise mechanism for dimensional mutation. The associated deformation tensor depends on a parameter [Formula: see text], with [Formula: see text] recovering the Euclidean configuration and the endpoints [Formula: see text] and [Formula: see text] corresponding to singular geometric limits. When the inverse Trinition metric acts on a nonlinear vector field, equilibrium locations are preserved, whereas spectral properties, stability, symmetry, and bifurcation boundaries may vary with [Formula: see text]. Applied to the Lorenz system, the deformation breaks the classical symmetry between the two nontrivial equilibria and produces branch-dependent Hopf stability boundaries. The Rössler system similarly exhibits deformation-dependent modifications of its local stability and bifurcation structure. Numerical simulations further reveal pronounced [Formula: see text]-dependent changes in attractor morphology and long-time dynamics. The resulting framework provides a mathematically controlled mechanism through which algebraically induced anisotropic geometry can participate directly in nonlinear and chaotic evolution.