From Quaternions to Pentarnions II: quadratic differential equations
Abstract
We investigate differential equations defined through elements of a nonassociative five-dimensional real algebra 𝒜γ, and show that every system of quadratic differential equations ṗ=F(p) on ℝ5 can be written as a differential equation over such a non-flexible quadratic algebra. We then study quadratic differential equations and their solutions over the whole family {𝒜γ}0≤γ≤1 of algebras while varying γ across the parameter interval [0,1], exploiting the fact that by definition, 𝒜0 contains Hamilton's quaternion algebra and that 𝒜1=ℙ is the real Pentarnion algebra. We construct explicit Jacobi elliptic solutions that deform a quaternion-valued solution at γ=0 into a Pentarnion-valued one at γ=1, and an algebra-valued Riccati equation whose equilibrium p=0 loses stability at a critical value of γ. Numerical experiments indicate period doubling, coexisting stable periodic orbits and irregular dynamics.
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BibTeX
@misc{poshtkohi2026from,
title = {{From Quaternions to Pentarnions II: quadratic differential equations}},
author = {Alireza Poshtkohi and Susanne Pumpluen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34201},
eprint = {2610.34201},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34201},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - From Quaternions to Pentarnions II: quadratic differential equations
AU - Alireza Poshtkohi
AU - Susanne Pumpluen
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34201
UR - https://arxiv.org/abs/2610.34201
AB - We investigate differential equations defined through elements of a nonassociative five-dimensional real algebra 𝒜_γ, and show that every system of quadratic differential equations ṗ=F(p) on ℝ^5 can be written as a differential equation over such a non-flexible quadratic algebra. We then study quadratic differential equations and their solutions over the whole family {𝒜_γ}_0≤γ≤1 of algebras while varying γ across the parameter interval [0,1], exploiting the fact that by definition, 𝒜_0 contains Hamilton's quaternion algebra and that 𝒜_1=ℙ is the real Pentarnion algebra. We construct explicit Jacobi elliptic solutions that deform a quaternion-valued solution at γ=0 into a Pentarnion-valued one at γ=1, and an algebra-valued Riccati equation whose equilibrium p=0 loses stability at a critical value of γ. Numerical experiments indicate period doubling, coexisting stable periodic orbits and irregular dynamics.
ER -