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The dynamics of the second order rational difference equation in the title with complex parameters and arbitrary complex initial conditions is investigated. Two associated difference equations are also studied. The solutions in the complex plane of such equations exhibit many rich and complicated asymptotic behavior. The analysis of the local stability of these three difference equations and periodicity have been carried out. We further exhibit several interesting characteristics of the solutions of this equation, using computations, which does not arise when we consider the same equation with positive real parameters and initial conditions. Many interesting observations led us to pose several open problems and conjectures of paramount importance regarding chaotic and higher order periodic solutions and global asymptotic convergence of such difference equations. It is our hope that these observations of these complex difference equations would certainly be new add-ons to the present art of research in rational difference equations in understanding the behavior in the complex domain.
We discuss an effective way for analyzing the system on the magnetized twisted orbifolds in operator formalism, especially in the complicated cases $T^{2}/Z_{3}$, $T^{2}/Z_{4}$ and $T^{2}/Z_{6}$. We can obtain the exact and analytical results which c
We study the ground-state properties of a class of $mathbb{Z}_n$ lattice gauge theories in 1 + 1 dimensions, in which the gauge fields are coupled to spinless fermionic matter. These models, stemming from discrete representations of the Weyl commutat
We give another proof for [ sum_{n=1}^{infty}frac{1}{n^2}=frac{pi^2}{6} ] that basically follows from the theory of difference equations.
The aim of this paper is to establish global Calder{o}n--Zygmund theory to parabolic $p$-Laplacian system: $$ u_t -operatorname{div}(| abla u|^{p-2} abla u) = operatorname{div} (|F|^{p-2}F)~text{in}~Omegatimes (0,T)subset mathbb{R}^{n+1}, $$ prov
Considered are ${cal N}=2, SU(N_c)$ or $U(N_c)$ SQCD with $N_F<2N_c-1$ equal mass quark flavors. ${cal N}=2$ supersymmetry is softly broken down to ${cal N}=1$ by the mass term $mu_{rm x}{rm Tr},(X^2)$ of colored adjoint scalar partners of gluons, $m