American Institute of Mathematical Sciences

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

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, United States

Received  September 2014 Revised  October 2015 Published  November 2015

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, 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.

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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.

show all references

References:
[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.

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