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Yayın Sönümlemeli sistemlerin eşlenmesi üzerine(Afyon Kocatepe Üniversitesi, 2021) Esen, Oğul; Özcan, Gökhan; Sütlü, SerkanBu makalede karşılıklı etki tepki içindeki iki sönümlemeli sistemin beraber (kollektif- eşlenmiş) hareketinin analizi üzerine bir yöntem öneriyoruz. Aşikardır ki; eşlenmiş hareketi kontrol eden diferansiyel denklemler iki sistemin denklemlerini bir arada yazmak dışında karşılıklı etki tepkinin doğurduğu fazladan terimler içerecektir. Karşılıklı etkiyi belirleyen ilave terimler, Lie cebirlerinin karşılıklı etkisi ile üretilecektir ve bu şekilde pür geometrik/cebirsel bir inşa gerçekleştirilecektir. Sonrasında elde ettiğimiz sonuçları 3 ve 4 boyutlu örneklerde göstereceğiz.Yayın On extensions, Lie-Poisson systems, and dissipation(Heldermann Verlag, 2022-07-06) Esen, Oğul; Özcan, Gökhan; Sütlü, SerkanLie-Poisson systems on the dual spaces of unified products are studied. Having been equipped with a twisted 2-cocycle term, the extending structure framework allows not only to study the dynamics on 2-cocycle extensions, but also to (de)couple mutually interacting Lie-Poisson systems. On the other hand, symmetric brackets; such as the double bracket, the Cartan-Killing bracket, the Casimir dissipation bracket, and the Hamilton dissipation bracket are worked out in detail. Accordingly, the collective motion of two mutually interacting irreversible dynamics, as well as the mutually interacting metriplectic flows, are obtained. The theoretical results are illustrated in three examples. As an infinite-dimensional physical model, decompositions of the BBGKY hierarchy are presented. As for the finite-dimensional examples, the coupling of two Heisenberg algebras, and the coupling of two copies of 3D dynamics are studied.Yayın On extensions, Lie-Poisson systems, and dissipation(Cornell Univ, 2021-01-08) Esen, Oğul; Özcan, Gökhan; Sütlü, SerkanOn the dual space of extended structure, equations governing the collective motion of two mutually interacting Lie-Poisson systems are derived. By including a twisted 2-cocycle term, this novel construction is providing the most general realization of (de)coupling of Lie-Poisson systems. A double cross sum (matched pair) of 2-cocycle extensions are constructed. The conditions are determined for this double cross sum to be a 2-cocycle extension by itself. On the dual spaces, Lie-Poisson equations are computed. We complement the discussion by presenting a double cross sum of some symmetric brackets, such as double bracket, Cartan-Killing bracket, Casimir dissipation bracket, and Hamilton dissipation bracket. Accordingly, the collective motion of two mutually interacting irreversible dynamics, as well as mutually interacting metriplectic flows, are obtained. The theoretical results are illustrated in three examples. As an infinite-dimensional physical model, decompositions of the BBGKY hierarchy are presented. As finite-dimensional examples, the coupling of two Heisenberg algebras and coupling of two copies of 3D dynamics are studied.
