Non-Hyperbolic Points of 2-D Discrete (DE) Homogeneous Polynomial Systems
No Thumbnail Available
Date
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
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