Article Contents
Article Contents

# On global dynamics in a multi-dimensional discrete map

• We derive preliminary results on global dynamics of the multi-dimensional discrete map $$F:\; (x_1,x_2,\dots,x_{k-1},x_k)\mapsto (x_1+af(x_k),x_1,x_2,\dots,x_{k-1})$$ where the continuous real-valued function $f$ is one-sided bounded and satisfying the negative feedback condition, $x\cdot f(x)<0, x\ne0$, $a$ is a positive parameter. We show the existence of a compact global attractor for map $F$, and derive a condition for the global attractivity of the zero fixed point.
Mathematics Subject Classification: Primary: 34K20, 34K26; Secondary: 37E05.

 Citation:

•  [1] P. Collet and J. P. Eckmann, Iterated Maps on the Interval as Dynamical Systems, Birkhäuser, Boston, 1980. [2] W. de Melo and S. van Strien, One-dimensional dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete 3 [Results in Mathematics and Related Areas 3], 25, Springer-Verlag, Berlin, 1993, 605 pp. [3] R. L. Devaney, An Introduction to Chaotic Dynamical Systems. Second Edition. Addison-Wesley Publ. Co., 1989, 336 pp. [4] O. Diekmann, S. van Gils, S. Verdyn Lunel, and H. O. Walther, Delay Equations: Complex, Functional, and Nonlinear Analysis, Springer-Verlag, New York, 1995. [5] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer Applied Mathematical Sciences, 99, 1993. [6] B. Hasselblatt and A. B. Katok, Handbook of dynamical systems, North Holland, 2002. [7] A. F. Ivanov and S. I. Trofimchuk, On global dynamics in a periodic differential equation with deviating argument, Applied Mathematics and Computation, 252 (2015), 446-456. [8] R. D. Nussbaum, Periodic solutions of nonlinear autonomous functional differential equations. Functional differential equations and approximation of fixed points (Proc. Summer School and Conf., Univ. Bonn, Bonn, 1978), pp. 283-325, Lecture Notes in Math., 730, Springer, Berlin, 1979. [9] A. N. Sharkovsky, S. F. Kolyada, A. G. Sivak and V. V. Fedorenko, Dynamics of One-dimensional Maps, Kluwer Academic Publishers, Ser.: Mathematics and Its Application, vol. 407, 1997, 261 pp.
Open Access Under a Creative Commons license