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

Bayrakdar, Tuna; Ergin, Abdullah Aziz

doi:10.3906/mat-1708-33

Cited by 1 publication

(1 citation statement)

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“…In the light of generalized Hamiltonian system theory, different types of three-dimensional dynamical systems have been analysed, such as Ltrue0u¨ 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 [50–55]. 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.…”

Section: Introductionmentioning

confidence: 99%

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

Song,

Pan,

2024

R. Soc. Open Sci.

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Many behavioural interactions in real life involve three individuals. When each individual has two alternative strategies, they can be abstracted into mathematical models by means of 2 × 2 × 2 asymmetric games. In this paper, we explore a special class of 2 × 2 × 2 asymmetric games satisfying fixed conditions. Firstly, we analyse two solitary interior equilibrium points and provide the judgement condition for their instability based on the Jacobi matrix local stability analysis method. Secondly, we analyse the interior equilibrium points that are continuously distributed within a line and probe into their stability conditions based on generalized Hamiltonian systems theory. Under the circumstances, the stable interior equilibrium point is surrounded by closed orbits in phase space, which presents an observable stable state where two strategies coexist and fluctuate in each of the three game populations. This work enriches the study of 2 × 2 × 2 asymmetric games’ evolutionary dynamics.

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