14,465 research outputs found
Asymptotic behavior of a competitive system of linear fractional difference equations
We investigate the global asymptotic behavior of solutions of the system of difference equations xn+1 = (a+x n )/(b+y n ), yn+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 x0 and y0 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
Dynamics of a rational system of difference equations in the plane
We consider a rational system of first order difference equations in the
plane with four parameters such that all fractions have a common denominator.
We study, for the different values of the parameters, the global and local
properties of the system. In particular, we discuss the boundedness and the
asymptotic behavior of the solutions, the existence of periodic solutions and
the stability of equilibria
Asymptotic solutions of forced nonlinear second order differential equations and their extensions
Using a modified version of Schauder's fixed point theorem, measures of
non-compactness and classical techniques, we provide new general results on the
asymptotic behavior and the non-oscillation of second order scalar nonlinear
differential equations on a half-axis. In addition, we extend the methods and
present new similar results for integral equations and Volterra-Stieltjes
integral equations, a framework whose benefits include the unification of
second order difference and differential equations. In so doing, we enlarge the
class of nonlinearities and in some cases remove the distinction between
superlinear, sublinear, and linear differential equations that is normally
found in the literature. An update of papers, past and present, in the theory
of Volterra-Stieltjes integral equations is also presented
On competitive discrete systems in the plane. I. Invariant Manifolds
Let be a competitive map on a rectangular region . The main results of this paper give conditions which guarantee
the existence of an invariant curve , which is the graph of a continuous
increasing function, emanating from a fixed point . We show that
is a subset of the basin of attraction of and that the set consisting
of the endpoints of the curve in the interior of is forward invariant.
The main results can be used to give an accurate picture of the basins of
attraction for many competitive maps.
We then apply the main results of this paper along with other techniques to
determine a near complete picture of the qualitative behavior for the following
two rational systems in the plane.
with
and arbitrary nonnegative initial conditions so
that the denominator is never zero.
with
and arbitrary nonnegative initial conditions.Comment: arXiv admin note: text overlap with arXiv:0905.1772 by other author
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