Geometric Stability of Dynamical Systems with Self-Generated State-Space Memory
Abstract
We develop a geometric stability framework for dynamical systems coupled to memory fields generated by their own state trajectories. Temporal age and state-space displacement enter a nonlinear hereditary force admitting an exact, trajectory-dependent velocity representation. A common velocity-history phase space contains both resting and uniformly translating solutions. We establish global velocity arrest under a uniform geometric memory-gain bound and obtain exact translating-speed balance laws from the full radial imprint. The response slope yields a positive longitudinal exponent on rising-speed branches, while a complementary contraction criterion gives local exponential attraction to selected descending branches in the scalar velocity dynamics, in fading-memory topology. Our principal result is a constructive geometric non-identifiability theorem: two systems with identical nonlinear force operators on an open neighborhood of rest can exhibit sharply different complete-motion behavior at the same subcritical gain. In the first, every admissible trajectory with bounded velocity prehistory comes to rest exponentially; in the second, locally attracting rest coexists with a locally attracting translating branch and an additional spectrally unstable translating branch. The infinite-memory case exhibits exponentially flat force differences, with all finite-order equilibrium derivatives agreeing despite the possibility of distinct global translating families. The framework also yields classical Volterra kernels and distributed-derivative transfer functions by linearization.
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BibTeX
@misc{vrabel2026geometric,
title = {{Geometric Stability of Dynamical Systems with Self-Generated State-Space Memory}},
author = {Robert Vrabel},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24152},
eprint = {2610.24152},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24152},
}
RIS
TY - UNPB
TI - Geometric Stability of Dynamical Systems with Self-Generated State-Space Memory
AU - Robert Vrabel
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.24152
UR - https://arxiv.org/abs/2610.24152
AB - We develop a geometric stability framework for dynamical systems coupled to memory fields generated by their own state trajectories. Temporal age and state-space displacement enter a nonlinear hereditary force admitting an exact, trajectory-dependent velocity representation. A common velocity-history phase space contains both resting and uniformly translating solutions. We establish global velocity arrest under a uniform geometric memory-gain bound and obtain exact translating-speed balance laws from the full radial imprint. The response slope yields a positive longitudinal exponent on rising-speed branches, while a complementary contraction criterion gives local exponential attraction to selected descending branches in the scalar velocity dynamics, in fading-memory topology. Our principal result is a constructive geometric non-identifiability theorem: two systems with identical nonlinear force operators on an open neighborhood of rest can exhibit sharply different complete-motion behavior at the same subcritical gain. In the first, every admissible trajectory with bounded velocity prehistory comes to rest exponentially; in the second, locally attracting rest coexists with a locally attracting translating branch and an additional spectrally unstable translating branch. The infinite-memory case exhibits exponentially flat force differences, with all finite-order equilibrium derivatives agreeing despite the possibility of distinct global translating families. The framework also yields classical Volterra kernels and distributed-derivative transfer functions by linearization.
ER -