Basins of attraction of period-two solutions of monotone difference equations

Arzu Bilgin,Mustafa Rs Kulenović,Esmir Pilav

doi:10.1186/s13662-016-0801-y

Arzu Bilgin, Mustafa Rs Kulenović + Show 1 more

Open Access

https://doi.org/10.1186/s13662-016-0801-y

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Abstract

We investigate the global character of the difference equation of the form $$x_{n+1} = f(x_{n}, x_{n-1}),\quad n=0,1, \ldots $$ with several period-two solutions, where f is increasing in all its variables. We show that the boundaries of the basins of attractions of different locally asymptotically stable equilibrium solutions or period-two solutions are in fact the global stable manifolds of neighboring saddle or non-hyperbolic equilibrium solutions or period-two solutions. As an application of our results we give the global dynamics of three feasible models in population dynamics which includes the nonlinearity of Beverton-Holt and sigmoid Beverton-Holt types.

Highlights

Let I be some interval of real numbers and let f ∈ C [I × I, I] be such that f (I × I) ⊆ K where K ⊆ I is a compact set
See [ ] for higher order versions of Theorems and. None of these results provide any information as regards the basins of attraction of different equilibrium solutions or period-two solutions when there exist several equilibrium solutions and periodtwo solutions
As we have shown in [ ] if the first order difference equation xn+ = f has two equilibrium points the corresponding delay difference equation xn+ = f has one period-two solution

Summary

Introduction

Let I be some interval of real numbers and let f ∈ C [I × I, I] be such that f (I × I) ⊆ K where K ⊆ I is a compact set. There are several global attractivity results for equation ( ) which give the sufficient conditions for all solutions to approach a unique equilibrium. In this paper we consider equation ( ) which has three equilibrium points and up to three minimal period-two solutions which are in North-East ordering. If T is differentiable map on a nonempty set S, a sufficient condition for T to be strongly monotonic with respect to the NE ordering is that the Jacobian matrix at all points x has the sign configuration. Theorem For a nonempty set R ⊂ Rn and a partial order on Rn, let T : R → R be an order-preserving map, and let a, b ∈ R be such that a ≺ b and ta, bu ⊂ R.

To check injectivity of T we set

Proof It can be seen that