American Institute of Mathematical Sciences

Advanced
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, (1980).   Google Scholar

[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).   Google Scholar

[3]

R. L. Devaney, An Introduction to Chaotic Dynamical Systems., Second Edition. Addison-Wesley Publ. Co., (1989).   Google Scholar

[4]

O. Diekmann, S. van Gils, S. Verdyn Lunel, and H. O. Walther, Delay Equations: Complex, Functional, and Nonlinear Analysis,, Springer-Verlag, (1995).   Google Scholar

[5]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations,, Springer Applied Mathematical Sciences, 99 (1993).   Google Scholar

[6]

B. Hasselblatt and A. B. Katok, Handbook of dynamical systems,, North Holland, (2002).   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[9]

A. N. Sharkovsky, S. F. Kolyada, A. G. Sivak and V. V. Fedorenko, Dynamics of One-dimensional Maps,, Kluwer Academic Publishers, (1997).   Google Scholar

show all references

References:
[1]

P. Collet and J. P. Eckmann, Iterated Maps on the Interval as Dynamical Systems,, Birkhäuser, (1980).   Google Scholar

[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).   Google Scholar

[3]

R. L. Devaney, An Introduction to Chaotic Dynamical Systems., Second Edition. Addison-Wesley Publ. Co., (1989).   Google Scholar

[4]

O. Diekmann, S. van Gils, S. Verdyn Lunel, and H. O. Walther, Delay Equations: Complex, Functional, and Nonlinear Analysis,, Springer-Verlag, (1995).   Google Scholar

[5]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations,, Springer Applied Mathematical Sciences, 99 (1993).   Google Scholar

[6]

B. Hasselblatt and A. B. Katok, Handbook of dynamical systems,, North Holland, (2002).   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[9]

A. N. Sharkovsky, S. F. Kolyada, A. G. Sivak and V. V. Fedorenko, Dynamics of One-dimensional Maps,, Kluwer Academic Publishers, (1997).   Google Scholar

[1]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450
[2]

Stefan Ruschel, Serhiy Yanchuk. The spectrum of delay differential equations with multiple hierarchical large delays. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 151-175. doi: 10.3934/dcdss.2020321
[3]

Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217
[4]

Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020468
[5]

Javier Fernández, Cora Tori, Marcela Zuccalli. Lagrangian reduction of nonholonomic discrete mechanical systems by stages. Journal of Geometric Mechanics, 2020, 12 (4) : 607-639. doi: 10.3934/jgm.2020029
[6]

Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345
[7]

Hongguang Ma, Xiang Li. Multi-period hazardous waste collection planning with consideration of risk stability. Journal of Industrial & Management Optimization, 2021, 17 (1) : 393-408. doi: 10.3934/jimo.2019117
[8]

Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020107
[9]

Feifei Cheng, Ji Li. Geometric singular perturbation analysis of Degasperis-Procesi equation with distributed delay. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 967-985. doi: 10.3934/dcds.2020305
[10]

Maoding Zhen, Binlin Zhang, Vicenţiu D. Rădulescu. Normalized solutions for nonlinear coupled fractional systems: Low and high perturbations in the attractive case. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020379
[11]

Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264
[12]

Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020050
[13]

Eduard Feireisl, Elisabetta Rocca, Giulio Schimperna, Arghir Zarnescu. Weak sequential stability for a nonlinear model of nematic electrolytes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 219-241. doi: 10.3934/dcdss.2020366
[14]

A. M. Elaiw, N. H. AlShamrani, A. Abdel-Aty, H. Dutta. Stability analysis of a general HIV dynamics model with multi-stages of infected cells and two routes of infection. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020441
[15]

Xin-Guang Yang, Lu Li, Xingjie Yan, Ling Ding. The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay. Electronic Research Archive, 2020, 28 (4) : 1395-1418. doi: 10.3934/era.2020074
[16]

Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020047
[17]

Zhenzhen Wang, Tianshou Zhou. Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020323
[18]

Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380
[19]

Mokhtar Bouloudene, Manar A. Alqudah, Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad. Nonlinear singular $ p $ -Laplacian boundary value problems in the frame of conformable derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020442
[20]

Thomas Bartsch, Tian Xu. Strongly localized semiclassical states for nonlinear Dirac equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 29-60. doi: 10.3934/dcds.2020297

 Impact Factor: 

Tools

Metrics

  • PDF downloads (84)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences