Abstract
We investigate the global behavior of a cubic second order difference equation $x_{n+1}=Ax_{n}^{3}+ Bx_{n}^{2}x_{n-1}+Cx_{n}x_{n-1}^{2}+Dx_{n-1}^{3}+Ex_{n}^{2} +Fx_{n}x_{n-1}+Gx_{n-1}^{2}+Hx_{n}+Ix_{n-1}+J$ , $n=0,1,\ldots$ , 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.
Highlights
1 Introduction and preliminaries In this paper we study the global dynamics of the following polynomial difference equation: xn+ = Ax n + Bx nxn– + Cxnx n– + Dx n– + Ex n + Fxnxn– + Gx n– + Hxn + Ixn– + J, ( )
One of the major problems in the dynamics of polynomial maps is determining the basin of attraction of the point at ∞ and in particular the boundary of that basin known as the Julia set
In [ ] we precisely determined the Julia set for a second order quadratic polynomial equation, that is, ( ) where A + B + C + D =, and we obtained the global dynamics in the interior of the Julia set, which includes all the point for which solutions are not asymptotic to the point at ∞
Summary
Proposition The zero equilibrium of ( ) is one of the following: (a) locally asymptotically stable if H + I < , (b) nonhyperbolic and locally stable if H + I = , (c) unstable if H + I > , (d) a saddle point if H > |I – |, (e) a repeller if – I < H < I –. Proposition The positive equilibrium solution of ( ) is one of the following: (a) locally asymptotically stable if p + q < , (b) nonhyperbolic and locally stable if p + q = , (c) unstable if p + q > , (d) a saddle point if p > |q – |, (e) a repeller if – q < p < q –.
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