Non-Hyperbolic Points of 2-D Discrete (DE) Homogeneous Polynomial Systems

Abstract

Let 𝑂(0,0) be an isolated equilibrium point of the two-dimensional (2D) discrete system: ' 𝑥#$% = 𝑥# + ℎ𝑃&(𝑥#, 𝑦#) , 𝑦#$% = 𝑦# + ℎ𝑄&(𝑥, 𝑦) (1) where 𝑃&(𝑥, 𝑦),𝑄&(𝑥, 𝑦) are homogenous polynomials of order 𝑚 ≥ 1 𝑃&( 𝑥, 𝑦) =3𝑝&'(,( 𝑥# &'(𝑦# ( & (+, 𝑄&( 𝑥, 𝑦) =3𝑞&'(,( 𝑥# &'(𝑦# ( & (+, for some integer 𝑚 ≥ 1 and 𝑝&'(,( , 𝑞&'(,( ∈ ℝ, 𝑘 = 0,…,𝑚. Assume ℎ is a small positive constant, and, 𝑃(0,0) = 𝑄(0,0) = 0. We can consider (1) to be the Euler’s approximation of the ODE system