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Discontinuity, Nonlinearity, and Complexity 14(3) (2025) 589--605 | DOI:10.5890/DNC.2025.09.011

Bernd Binder

Quanics, Rosenbühlstr. 32, 89182 Bernstadt, Germany

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Abstract

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 $X_{i}-X_{i-2} = -X_{i-1}\times F_{i}\left(X_{i-1},X_{i-2},X_{i-3}\right)\in\mathbb{R}^{3}\textrm{ or}\in\mathbb{R}^{7}$, where the symmetry invariant $C=X_{i}\cdot X_{i-1}=X_{i-1}\cdot X_{i-2}=...$ 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 3\textit{d} and 7\textit{d}. 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.

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