Dark Type Dynamical Systems: The Integrability Algorithm and Applications

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Authors

PRYKARPATSKY Yarema A URBANIAK Ilona KYCIA Radoslaw Antoni PRYKARPATSKI Anatolij K

Year of publication 2022
Type Article in Periodical
Magazine / Source ACM Transactions on Algorithms
MU Faculty or unit

Faculty of Science

Citation
Web https://doi.org/10.3390/a15080266
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
Description 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.
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