# Bifurcations of Some Planar Discrete Dynamical Systems with Applications

#### Abstract

The focus of this thesis is on some contemporary problems in the field of difference equations and discrete dynamical systems. The problems that I worked on range global attractivity results to all types of bifurcations for systems of difference equations in the plane.^ The major goal was to investigate the impact of nonlinear perturbation and the introduction of quadratic terms on linear fractional difference equations such as the Beverton-Holt as well as the Sigmoid Beverton-Holt Model with delay that describes the growth or decay of single species.^ The first Manuscript was on the study of the following equation: xn+1=p+x^{2}nx^{ 2}n-1 1 Which was an open problem suggested by Dr. Kulenović. It is a perturbation of the linear fractional difference equation: xn+1=p+x^{2}nx^{ 2}n-1 ^ The solutions of Equation(1) undergo very interesting dynamics as I showed that the variation of the parameter *p* can cause the exhibition of the Naimark-Sacker bifurcation. I compute the direction of the Naimark-Sacker bifurcation for the difference equation (1) and I provide an asymptotic approximation of the closed invariant curve which comes to existence as the unique positive equilibrium point loses its stability. Moreover tools and global stability result to provide a region of the parameter where local stability implies global stability of the equilibrium.^ In my second Manuscript, I considered the difference equation: xn+1=xnCx^{2} n-1+Dxn+F 2 where *C*, *D* and *F* are positive numbers and the initial conditions *x*_{-1 } and *x*_{0} are non-negative numbers. Equation \eqref{open2} which is also a non-linear perturbation of the Beverton Holt model, belongs to the category of difference equations with a unique positive equilibrium that exhibit the Naimark-Sacker bifurcation. The investigation of the dynamics of such equation is very challenging as it depends on more than one parameter. However I give a method for proving that its dynamics undergoes the Naimark-Sacker bifurcation. Moreover I compute the direction of the Neimark-Sacker bifurcation for this difference equation and provide the asymptotic approximation of the invariant closed curve. Furthermore I give the necessary and sufficient conditions for global asymptotic stability of the zero equilibrium as well as sufficient conditions for global asymptotic stability of the positive equilibrium.^ The following theorem is the major result that I relied on to prove global asymptotic stability of the equilibria in my first two Manuscripts:^ **Theorem 1** *Let I be a compact interval of the real numbers and assume that f* : *I*^{3} → * I* *is a continuous function satisfying the following properties: 1. f*(*x*, *y*, *z*) * is non-decreasing in x and non-increasing in y and z; 2. The system* fM,m,m =Mfm,M,M =m *has a unique solution M = m in I. Then the equation x _{ n}*

_{+1}=

*f*(

*x*,

_{n}*x*

_{n}_{-1},

*x*

_{n}_{-2 })

*has a unique equilibrium x¯ in I and every solution of it that enters I must converge to x¯. In addition, x¯ is globally asymptotically stable.*^ As of my third manuscript, I focused on providing some possible scenarios for general discrete competitive dynamical systems in the plane. I applied the results achieved to a class of second order difference equations of the form: xn+1=fxn,xn-1 ,n=0,1,... where the function

*f*(

*x*,

*y*) is decreasing in the variable

*x*and increasing in the variable

*y.*In my proofs I relied on a collection of well established theorems and results. Furthermore I illustrate my results with an application to equation: xn+1=x

^{2}n-1c x

^{2}n-1+dxn+f, n=0,1,...3 With initial conditions

*x*

_{-1}and

*x*

_{0}arbitrary nonnegative numbers and parameters

*c*,

*d*,

*f*>0. Equation. (3) is a special case of: xn+1=Cx

^{2}n-1+Dxn +Fcx

^{2}n-1+dxn+ f,n=0,1,... which of great interest to the field of difference equation and special cases of it were considered by different scholars. It also turns out to be a non-linear perturbation of the Sigmoid Beverton-Holt model. I characterize completely the global bifurcations and dynamics of equation.(3) with the basins of attraction of all its equilibria and periodic solutions. Moreover I provide techniques to investigates cases that are not covered by the established theorems in the theory of competitive maps.^ Finally in my fourth manuscript I considered extending some existing theorems and proving some new global stability results, namely for difference equations that are of the form xn+1=fxn,xn-1 where

*f*(

*x*,

*y*) is either increasing in the first and decreasing in the second variable, or decreasing in both variables. In addition I illustrate my results with examples and applications. I also provide a new proof for Pielou's equation (a mathematical model in population dynamics).^

#### Subject Area

Mathematics

#### Recommended Citation

Toufik Khyat,
"Bifurcations of Some Planar Discrete Dynamical Systems with Applications"
(2017).
*Dissertations and Master's Theses (Campus Access).*
Paper AAI10266921.

http://digitalcommons.uri.edu/dissertations/AAI10266921