Global Existence of Bi-Hamiltonian Structures on Orientable Three-Dimensional Manifolds

Abstract

Abstract. In this work, we show that an autonomous dynamical system defined by a nonvanishing vector field on an orientable three-dimensional manifold is globally bi-Hamiltonian if and only if the first Chern class of the normal bundle of the given vector field vanishes. Furthermore, the bi-Hamiltonian structure is globally compatible if and only if the Bott class of the complex codimension one foliation defined by the given vector field vanishes.

Sasakian structure associated with a second order ODE and Hamiltonian dynamical systems

“…In fact this curvature corresponds to the sectional curvature associated with the plane in T q Σ spanned by ξ 1 and ξ 3 at a point q. Recently, it has been shown in [30] that an autonomous dynamical system defined by a nonvanishing vector field on an orientable three-dimensional manifold is globally bi-Hamiltonian if and only if the first Chern class of the normal bundle of the given vector field vanishes. Accordingly, we can say that the dynamical system determined by the Hamiltonian vector field v = ξ 1 (H)ξ 2 on the Sasakian manifold corresponding to an ODE of the form d 2 y/dx 2 = f (x, p) is globally bi-Hamiltonian if and only if f satisfies f x + f f p = ψ(x) for smooth function ψ.…”

Equivalence problem for compatible bi-Hamiltonian structures on three-dimensional orientable manifolds

“…In this section, fundamental ideas of [10] related to our work are summarized. In [10] it was shown that (2) is locally bi-Hamiltonian. For this purpose first we will describe the Poisson structures in three dimensions.…”

“…Therefore, the equivalence problem for nonvanishing vector fields is trivial and we need some further property, or more precisely another geometric structure related to a nonvanishing vector field, to define a nontrivial equivalence problem. In [10] it was shown that any nonvanishing local vector field on an orientable three-dimensional manifold admits a bi-Hamiltonian structure, i.e. if v (x) is a nonvanishing vector field on a three-dimensional manifold M with an arbitrary metric g on it , then there exist two independent functions H 1 (x) and H 2 (x) , and a function ϕ…”

Study on the interior equilibrium point of a special class of 2 × 2 × 2 asymmetric evolutionary games

“…In the light of generalized Hamiltonian system theory, different types of three-dimensional dynamical systems have been analysed, such as Lu ¨ systems, Chen systems and Qi systems known for their chaotic characteristics [44], optical Maxwell-Bloch equations [45], epidemiological Kermack-McKendrick models [46], Lotka-Volterra equations describing species interactions in ecosystems [47,48] and tournaments based on replicator equations [49]. In recent years, research on the Hamiltonian characteristics of three-dimensional dynamical systems has also been going deep [50][51][52][53][54][55]. Based on generalized Hamiltonian system theory, we probe into the dynamic properties of a special class of 2 × 2 × 2 asymmetric evolutionary games that meet certain conditions, that is, to discuss the existence and stability of interior equilibrium points, and the stability here refers to Lyapunov stability.…”