(Split-)Quaternion and (Split-)Octonion Dynamics in Discrete-Time Recurrent Frenet Frames - Binder, Bernd (Discontinuity, Nonlinearity, and Complexity, 2026)
We consider and apply a multidimensional discrete-time delay autonomous third order non-linear vector difference equation system, where the orthogonal change after two reflections is given by a vector cross product leading to spinor rotations X_i-X_i-2 = -X_i-1× F_i(X_i-1,X_i-2,X_i-3)R^3 orR^7, where the symmetry invariant C=X_i· X_i-1=X_i-1· X_i-2=... allows at every step for a memory sign flip including expansion/contraction by a scalar factor. Using the Frenet frame approach defining orthogonal co-moving components with torsion and curvature parameter, both, the orthogonal frame and the change of the frame are represented by the three position memory terms recurrently. The necessary cross and dot vector product (split-) algebra is encoded in variable multiplication tables in 3d and 7d. We discuss two special F_i types showing stable point densities, which are drifting limit cycles with sub-cycles, where the resulting smooth orbital spinor dynamics shows discrete atomic-type orbital eigenstates or local waves with characteristic numbers, non-local reflection, instability, hysteresis, interaction, helical emissions, and transition to chaos.