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

Mami Suzuki¹

¹ Department of Mathematics College of Liberal Arts J. F. Oberlin University 3758 Tokiwa-cho, Machida-City Tokyo, 194-0294, Japan

Annales Polonici Mathematici, Tome 102 (2011) no. 2, pp. 143-159

Résumé

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: $$ \begin{cases} x(t+1) = X(x(t),y(t)), \\ y(t+1) = Y(x(t), y(t)), \end{cases} $$ where $X(x,y)= \lambda_1 x+ \mu y +\sum_{i+j \geq 2} c_{ij} x^i y^j$, $Y(x,y)= \lambda_2 y+\sum_{i+j \geq 2}$ $d_{ij} x^i y^j$ satisfy some conditions. For these equations, we have obtained analytic solutions in the cases “$|\lambda_1| \ne 1$ or $|\lambda_2| \ne 1$” or “$\mu=0$” in earlier studies. In the present paper, we will prove the existence of an analytic solution for the case $\lambda_1 = \lambda_2 = 1$ and $\mu=1$.

Détail

DOI : 10.4064/ap102-2-4

Keywords: nonlinear difference equations difficult obtain analytic solutions especially eigenvalues equation absolute value nbsp consider second order nonlinear difference equation which transformed following simultaneous system nonlinear difference equations begin cases end cases where lambda sum geq j lambda sum geq j satisfy conditions these equations have obtained analytic solutions cases lambda lambda earlier studies present paper prove existence analytic solution lambda lambda