On global dynamics in a multi-dimensional discrete map

Anatoli F.  Ivanov

doi:10.3934/proc.2015.0652

Article Contents

2015, Volume 2015: 652-659. Doi: 10.3934/proc.2015.0652 open access

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On global dynamics in a multi-dimensional discrete map

Anatoli F. Ivanov^1,

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

Received: August 31, 2014

Revised: September 30, 2015

Published: October 2015

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Abstract

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:

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

Mathematics Subject Classification: Primary: 34K20, 34K26; Secondary: 37E05.

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References

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