On global dynamics in a multi-dimensional discrete map

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2015, 2015(special): 652-659. doi: 10.3934/proc.2015.0652

On global dynamics in a multi-dimensional discrete map

1.	Department of Mathematics, Pennsylvania State University, PO Box PSU, Lehman, PA 18627, United States

Received September 2014 Revised October 2015 Published November 2015

Abstract
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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.

Keywords: Singular perturbations, Nonlinear differential delay equations, Global asymptotic stability, Reduction to multi-valued discrete maps, Interval maps..

Mathematics Subject Classification: Primary: 34K20, 34K26; Secondary: 37E0.

Citation: Anatoli F. Ivanov. On global dynamics in a multi-dimensional discrete map. Conference Publications, 2015, 2015 (special) : 652-659. doi: 10.3934/proc.2015.0652

References:

[1]	P. Collet and J. P. Eckmann, Iterated Maps on the Interval as Dynamical Systems,, Birkhäuser, (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 (1993).
[3]	R. L. Devaney, An Introduction to Chaotic Dynamical Systems., Second Edition. Addison-Wesley Publ. Co., (1989).
[4]	O. Diekmann, S. van Gils, S. Verdyn Lunel, and H. O. Walther, Delay Equations: Complex, Functional, and Nonlinear Analysis,, Springer-Verlag, (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.
[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., 730 (1978), 283.
[9]	A. N. Sharkovsky, S. F. Kolyada, A. G. Sivak and V. V. Fedorenko, Dynamics of One-dimensional Maps,, Kluwer Academic Publishers, (1997).

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References:

[1]	P. Collet and J. P. Eckmann, Iterated Maps on the Interval as Dynamical Systems,, Birkhäuser, (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 (1993).
[3]	R. L. Devaney, An Introduction to Chaotic Dynamical Systems., Second Edition. Addison-Wesley Publ. Co., (1989).
[4]	O. Diekmann, S. van Gils, S. Verdyn Lunel, and H. O. Walther, Delay Equations: Complex, Functional, and Nonlinear Analysis,, Springer-Verlag, (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.
[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., 730 (1978), 283.
[9]	A. N. Sharkovsky, S. F. Kolyada, A. G. Sivak and V. V. Fedorenko, Dynamics of One-dimensional Maps,, Kluwer Academic Publishers, (1997).

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