Binder, Bernd (2024) (Split)Quaternion and (Split)Octonion Dynamics in DiscreteTime Recurrent Frenet Frames. [Preprint]
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Abstract
We consider and apply a multidimensional discretetime delay autonomous third order nonlinear vector difference equation system, where the orthogonal change after two reflections is given by a vector cross product leading to spinor rotations $X_{i}X_{i2} = X_{i1}\times F_{i}\left(X_{i1},X_{i2},X_{i3}\right)\in\mathbb{R}^{3}\textrm{ or}\in\mathbb{R}^{7}$, where the symmetry invariant $C=X_{i}\cdot X_{i1}=X_{i1}\cdot X_{i2}=...$ allows at every step for a memory sign flip including expansion/contraction by a scalar factor. Using the Frenet frame approach defining orthogonal comoving 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 subcycles, where the resulting smooth orbital spinor dynamics shows discrete atomictype orbital eigenstates or local waves with characteristic numbers, nonlocal reflection, instability, hysteresis, interaction, helical emissions, and transition to chaos.
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Item Type:  Preprint  

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Keywords:  Discrete Frenet Octonion Quaternion Split Reflection Cycles Waves Modular Symmetry  
Subjects:  Specific Sciences > Mathematics > Methodology Specific Sciences > Mathematics > Practice Specific Sciences > Complex Systems 

Depositing User:  Dr. Bernd Binder  
Date Deposited:  28 Aug 2024 13:42  
Last Modified:  28 Aug 2024 13:42  
Item ID:  23842  
DOI or Unique Handle:  DNCD2400044_R11  
Subjects:  Specific Sciences > Mathematics > Methodology Specific Sciences > Mathematics > Practice Specific Sciences > Complex Systems 

Date:  26 July 2024  
URI:  https://philsciarchive.pitt.edu/id/eprint/23842 
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