Difference equations on $\mathbb{R}_*^+$, of the form $u_{n+2}=\frac{f(u_{n+1})}{u_n+\lambda}$, $\lambda>0$, with applications to perturbations of dynamical systems
DOI:
https://doi.org/10.5644/SJM.18.01.05Keywords:
dynamical systems, QRT-maps, perturbationsAbstract
There is classical difference equation on $\mathbb{R}_*^+$
\begin{equation*}\label{premequat}
u_{n+2}u_n=f(u_{n+1}), \tag{1}
\end{equation*}
that is in particular applied to several symmetric special QRT-applications.
We study perturbations of this equation (1) by
\begin{equation*}\label{deuxequat}
u_{n+2}(u_n+\lambda)=f(u_{n+1}),\hskip 4mm \lambda>0, \tag{2}
\end{equation*}
showing general theorems of regarding permanence, convergence or divergence, and attraction of the fixed point. \vspace*{-.6cm}
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References
Amleh A. M., Camouzis E, Ladas G., On the dynamics of a rational difference equation, part I., {Int. J. Difference Equ., 3(1), p. 1-35, (2008).
Bastien G. and Rogalski M., Results and problems about solutions of perturbed Lyness' type order $k$ différence équations in $R_*^+$, $u_{n+k}(u_n+l)=f(u_{n+k-1},dots,u_{n+1})$, with examples, and test of the efficiency of a quasi--Lyapounov function method, J. of Diff. Eq. and Appl., vol 19, Issue 8, 2013.
Bastien G. and Rogalski M., On some algebraic difference equations $u_{n+2}u_n=psi(u_{n+1})$ in $R_*^{+2}$, related to families of conics or cubics: generalization of the Lyness' sequence, J. Math. Anal. Appl. 300 (2004), p. 303-333.
Bastien G. and Rogalski M., New method for the study of solutions of the difference equation $u_{n+2}=frac{u_{n+1}+a}{u_n+b}$, {Radovi Matematički, vol.12 (2004), p. 135-152.
Bastien G. and Rogalski M., On the Algebraic Difference Equation $u_{n+2}u_n=psi(u_{n+1})$ in $R_*^+$, Related to a Family of Elliptic Quartics in the Plane, Advances in Difference Equations, 2005: 3, p. 227-261, (2005).
Bastien G. and Rogalski M., Global behavior of the solutions of Lyness' difference equation $u_{n+2}u_n=u_{n+1}+a$, J. Difference Equations and Appl., 10, p. 977-1003, (2004).
Duistermaat J., Discrete Integrable Systems. QRT Maps ans Elliptic Surfaces, Springer, (2010).
El-Morshedy H.-A., The global attractivity of difference equation of non increasing nonlinearities with applications, Comput. Math. Appl., 45 (2004), p. 749-758.
Kalabuv si' c S. and al., Stability analysis of a certain class of difference equations by using KAM theory, Advances in Difference Equations, (2019).
Kocic V. L. and Ladas G., Global Behaviour of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic Publishers, Dordrecht, Holland, (1993).
Kocic V. L., Ladas G. and Rodrigues W., On the rationel recursive sequences, J. Math. Anal. Appl. 173 (1993), p. 127-157.
Kulenović M. R. S. and Merino O., A global attractivity result for maps with invariant boxes, Discrete and Continuous Dynamical Systems, Series B, (2005).
Kulenović M. R. S. and Merino O., Discrete Dynamical Systems and Difference Equations with Mathematica, Chapman and Hall / CRC, Boca Raton, London, (2002).
Kulenović M. R. S., Ladas G. and Sizer W. S. On the recursive sequence $x_{n+1}=frac{a x_n+beta x_{n+1}}{gamma x_n+delta x_{n-1}}$, MATH. SCI. RES. HOT-LINE 2(5), 1-16,(1998).
Ladas G., Tzanetopoulos G., Tovbis A., On May's host parasitoid model, J. Differ. Equ.Appl. 2, p. 185-204 (1996).
Merino O., Global Attractivity of the Equilibrium of a Difference Equation: An Elementary Proof Assisted by Computer Algebra System, J. Difference. Eq. and Appl., vol. 17, issue 1, (2011).
Quispel G. R. W., Roberts J. A. G. and Thompson C. J., Integrable mappings and soliton equations, Physics Letters A: 126}, p. 419-421, (1988).
Zeeman E. C., Geometric Unfolding of a Difference Equation, Herftord College, Oxford, (1996).
