More on Poincaré’s and Perron’s Theorems for Difference Equations∗

We investigate the global asymptotic behavior of solutions of the system of difference equations x n+1 = (a + x n )/(b + y n ), y n+1 = (d + y n )/(e + x n ), n = 0,1,..., where the parameters a,b,d, and e are positive numbers and the initial conditions x 0 and y 0 are arbitrary nonnegative numbers. In certain range of parameters, we prove the existence of the global stable manifold of the unique positive equilibrium of this system which is the graph of an increasing curve. We show that the stable manifold of this system separates the positive quadrant of initial conditions into basins of attraction of two types of asymptotic behavior. In the case where a = d and b = e, we find an explicit equation for the stable manifold to be y = x.

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“…The following Perron-type theorem established in [8] is a result of this type. It says that the growth rates of the solutions of both (2.1) and (2.3) are equal to the moduli of the roots of the characteristic equation…”

Section: Asymptotic Expansions For Poincaré Difference Equationsmentioning

confidence: 99%

Asymptotic Expansions for Higher-Order Scalar Difference Equations

Agarwal

Pituk

2007

Adv. Differ Equ.

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We give an asymptotic expansion of the solutions of higher-order Poincaré difference equation in terms of the characteristic solutions of the limiting equation. As a consequence, we obtain an asymptotic description of the solutions approaching a hyperbolic equilibrium of a higher-order nonlinear difference equation with sufficiently smooth nonlinearity. The proof is based on the inversion formula for the z-transform and the residue theorem.

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More on Poincaré’s and Perron’s Theorems for Difference Equations∗

Cited by 109 publications

References 20 publications

Rate of convergence of solutions of rational difference equation of second order

Rate of convergence of solutions of rational difference equation of second order

Asymptotic behavior of a competitive system of linear fractional difference equations

Asymptotic Expansions for Higher-Order Scalar Difference Equations

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