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@article{2227818,
author = {Prykarpatsky, Yarema A and Urbaniak, Ilona and Kycia, Radoslaw Antoni and Prykarpatski, Anatolij K},
article_location = {USA},
article_number = {8},
doi = {http://dx.doi.org/10.3390/a15080266},
keywords = {dark type dynamical systems; evolution flows; conservation laws; Lax-Noether condition; asymptotic solutions; linearization; complete integrability},
language = {eng},
issn = {1549-6325},
journal = {ACM Transactions on Algorithms},
title = {Dark Type Dynamical Systems: The Integrability Algorithm and Applications},
url = {https://doi.org/10.3390/a15080266},
volume = {15},
year = {2022}
}
TY - JOUR
ID - 2227818
AU - Prykarpatsky, Yarema A - Urbaniak, Ilona - Kycia, Radoslaw Antoni - Prykarpatski, Anatolij K
PY - 2022
TI - Dark Type Dynamical Systems: The Integrability Algorithm and Applications
JF - ACM Transactions on Algorithms
VL - 15
IS - 8
PB - ACM
SN - 15496325
KW - dark type dynamical systems
KW - evolution flows
KW - conservation laws
KW - Lax-Noether condition
KW - asymptotic solutions
KW - linearization
KW - complete integrability
UR - https://doi.org/10.3390/a15080266
N2 - Based on a devised gradient-holonomic integrability testing algorithm, we analyze a class of dark type nonlinear dynamical systems on spatially one-dimensional functional manifolds possessing hidden symmetry properties and allowing their linearization on the associated cotangent spaces. We described main spectral properties of nonlinear Lax type integrable dynamical systems on periodic functional manifolds particular within the classical Floquet theory, as well as we presented the determining functional relationships between the conserved quantities and related geometric Poisson and recursion structures on functional manifolds. For evolution flows on functional manifolds, parametrically depending on additional functional variables, naturally related with the classical Bellman-Pontriagin optimal control problem theory, we studied a wide class of nonlinear dynamical systems of dark type on spatially one-dimensional functional manifolds, which are both of diffusion and dispersion classes and can have interesting applications in modern physics, optics, mechanics, hydrodynamics and biology sciences. We prove that all of these dynamical systems possess rich hidden symmetry properties, are Lax type linearizable and possess finite or infinite hierarchies of suitably ordered conserved quantities.
ER -
PRYKARPATSKY, Yarema A, Ilona URBANIAK, Radoslaw Antoni KYCIA and Anatolij K PRYKARPATSKI. Dark Type Dynamical Systems: The Integrability Algorithm and Applications. \textit{ACM Transactions on Algorithms}. USA: ACM, 2022, vol.~15, No~8. ISSN~1549-6325. Available from: https://dx.doi.org/10.3390/a15080266.
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