Asymptotic Approximations of the Stable and Unstable Manifolds of Fixed Points of a Two-dimensional Cubic Map

We find the asymptotic approximations of the stable and unstable manifolds of the saddle equilibrium solutions of the following difference equation xn+1 = a x3n + bx3n-1 + cxn + dxn-1, n = 0,1... where the parameters a, b, c and d are positive numbers and the initial conditions x-1 and x0 are arbitrary numbers. these manifolds determine completely the global dynamics of this equation

Basins of Attraction of Period-Two Solutions of Monotone Difference Equations

We investigate the global character of the difference equation of the form xn+1 = f (xn, xn–1), n = 0,1, . . . 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. MSC: 39A10; 39A20; 37B25; 37D1

Global Dynamics of Cubic Second Order Difference Equation in the First Quadrant

We investigate the global behavior of a cubic second order difference equation xn+1=Ax3n+ Bx 2nxn−1 + Cxnx2n−1 + Dx3n−1 + Ex2n + Fxnxn−1 + Gx2n−1 + Hxn + Ixn−1 + J, n = 0, 1, …, with nonnegative parameters and initial conditions. We establish the relations for the local stability of equilibriums and the existence of period-two solutions. We then use this result to give global behavior results for special ranges of the parameters and determine the basins of attraction of all equilibrium points. We give a class of examples of second order difference equations with quadratic terms for which a discrete version of the 16th Hilbert problem does not hold. We also give the class of second order difference equations with quadratic terms for which the Julia set can be found explicitly and represent a planar quadratic curve

Local dynamics and global attractivity of a certain second-order quadratic fractional difference equation

We investigate the local and global character of the equilibrium and the local stability of the period-two solution of the difference equation [Mathematical equations cannot be displayed here, refer to PDF] where the parameters β, γ, δ, B, C, D are nonnegative numbers which satisfy B + C + D \u3e 0 and the initial conditions x-1 and x0 are arbitrary nonnegative numbers such that Bxnxn-1 + Cx2n-1 + Dxn \u3e 0 for all n ≥ 0

Birkhoff Normal Forms, KAM Theory and Symmetries for Certain Second Order Rational Difference Equation with Quadratic Term

By using the KAM theory, we investigate the stability of the equilibrium solution of a certain difference equation. We also use the symmetries to find effectively the periodic solutions with feasible periods. The specific feature of this difference equation is the fact that we were not able to use the invariant to prove stability or to find feasible periods of the solutions

Dynamics of a two-dimensional competitive system of rational difference equations with quadratic terms

We investigate global dynamics of the following systems of difference equations: [Mathematical equations cannot be displayed here, refer to PDF] where the parameters b1, a2, A1, c2, are positive numbers and the initial condition y0 is an arbitrary nonnegative number and x0 is a positive number. We show that this system has rich dynamics which depends on the part of the parametric space. We find precisely the basins of attraction of all attractors including the points at ∞

Birkhoff Normal Forms, KAM Theory and Time Reversal Symmetry for Certain Rational Map

By using the KAM(Kolmogorov-Arnold-Moser) theory and time reversal symmetries, we investigate the stability of the equilibrium solutions of the system: x n + 1 = 1 y n , y n + 1 = β x n 1 + y n , n = 0 , 1 , 2 , … , where the parameter β > 0 , and initial conditions x 0 and y 0 are positive numbers. We obtain the Birkhoff normal form for this system and prove the existence of periodic points with arbitrarily large periods in every neighborhood of the unique positive equilibrium. We use invariants to find a Lyapunov function and Morse’s lemma to prove closedness of invariants. We also use the time reversal symmetry method to effectively find some feasible periods and the corresponding periodic orbits

Global Dynamics of a Cooperative Discrete System in the Plane

Kulenovic, Mustafa R. S./0000-0002-2936-7098WOS: 000439096400002In this paper, we consider the cooperative system x(n+1) = ax(n) + by(n)(2)/1 + y(n)(2), y(n+1) = cx(n)(2)/1 + x(n)(2) + dy(n), n = 0, 1, ... , where all parameters a, b, c, d are positive numbers and the initial conditions x(0), y(0) are non-negative numbers. We describe the global dynamics of this system in a number of cases. An interesting feature of this system is that it exhibits a coexistence of locally stable equilibrium and locally stable periodic solutions as well as the Allee effect