Analytic solutions of a nonlinear two variables difference system whose eigenvalues are both 1

Mami Suzuki. "Analytic solutions of a nonlinear two variables difference system whose eigenvalues are both 1." Annales Polonici Mathematici 102.2 (2011): 143-159. <http://eudml.org/doc/286086>.

@article{MamiSuzuki2011,
abstract = {For nonlinear difference equations, it is difficult to obtain analytic solutions, especially when all the eigenvalues of the equation are of absolute value 1. We consider a second order nonlinear difference equation which can be transformed into the following simultaneous system of nonlinear difference equations: ⎧ x(t+1) = X(x(t),y(t)) ⎨ ⎩ y(t+1) = Y(x(t), y(t)) where $X(x,y) = λ₁x + μy + ∑_\{i+j≥2\} c_\{ij\}x^\{i\}y^\{j\}$, $Y(x,y) = λ₂y + ∑_\{i+j≥2\} d_\{ij\}x^\{i\}y^\{j\}$ satisfy some conditions. For these equations, we have obtained analytic solutions in the cases "|λ₁| ≠ 1 or |λ₂| ≠ 1" or "μ = 0" in earlier studies. In the present paper, we will prove the existence of an analytic solution for the case λ₁ = λ₂ = 1 and μ = 1.},
author = {Mami Suzuki},
journal = {Annales Polonici Mathematici},
keywords = {analytic solutions; functional equations; nonlinear difference equations},
language = {eng},
number = {2},
pages = {143-159},
title = {Analytic solutions of a nonlinear two variables difference system whose eigenvalues are both 1},
url = {http://eudml.org/doc/286086},
volume = {102},
year = {2011},
}

TY - JOUR
AU - Mami Suzuki
TI - Analytic solutions of a nonlinear two variables difference system whose eigenvalues are both 1
JO - Annales Polonici Mathematici
PY - 2011
VL - 102
IS - 2
SP - 143
EP - 159
AB - For nonlinear difference equations, it is difficult to obtain analytic solutions, especially when all the eigenvalues of the equation are of absolute value 1. We consider a second order nonlinear difference equation which can be transformed into the following simultaneous system of nonlinear difference equations: ⎧ x(t+1) = X(x(t),y(t)) ⎨ ⎩ y(t+1) = Y(x(t), y(t)) where $X(x,y) = λ₁x + μy + ∑_{i+j≥2} c_{ij}x^{i}y^{j}$, $Y(x,y) = λ₂y + ∑_{i+j≥2} d_{ij}x^{i}y^{j}$ satisfy some conditions. For these equations, we have obtained analytic solutions in the cases "|λ₁| ≠ 1 or |λ₂| ≠ 1" or "μ = 0" in earlier studies. In the present paper, we will prove the existence of an analytic solution for the case λ₁ = λ₂ = 1 and μ = 1.
LA - eng
KW - analytic solutions; functional equations; nonlinear difference equations
UR - http://eudml.org/doc/286086
ER -