American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Mean square approximation of multi dimensional reflecting fractional Brownian motion via penalty method
  • PROC Home
  • This Issue
  • Next Article
    Existence of a stable set for some nonlinear parabolic equation involving critical Sobolev exponent
2005, 2005(Special): 453-462. doi: 10.3934/proc.2005.2005.453

Nim-induced dynamical systems over Z2

Michael A. Jones 1, and  Diana M. Thomas 2,

1. 

Montclair State University, Department of Mathematical Sciences, Upper Montclair, NJ 07043, United States
2. 

Department of Mathematical Sciences, Montclair State University, Upper Montclair, NJ 07043, United States

Received  September 2004 Revised  April 2005 Published  September 2005

Winning and losing positions in the well-known two-player game, Nim, are defined recursively as a two symbol sequence depending on a $k$-parameter set known as the subtraction set. In this paper, we write the recursion as a nonlinear dynamical system defined on the phase space $\mathbb Z_2^{s_k}$ with the binary sequence for Nim generated by the appropriate initial conditions. The transient dynamics and Garden of Eden points are completely determined for arbitrary-sized subtraction sets. A characterization of cycle lengths for two parameter subtraction sets is determined. Extensions of the two parameter case to an arbitrary-sized subtraction set are explored.
Keywords: Dynamical systems, nonlinear, finite field..
Mathematics Subject Classification: Primary: 91A46, 94A55; Secondary: 65Q0.
Citation: Michael A. Jones, Diana M. Thomas. Nim-induced dynamical systems over Z2. Conference Publications, 2005, 2005 (Special) : 453-462. doi: 10.3934/proc.2005.2005.453
[1]

Igor E. Shparlinski. On some dynamical systems in finite fields and residue rings. Discrete and Continuous Dynamical Systems, 2007, 17 (4) : 901-917. doi: 10.3934/dcds.2007.17.901
[2]

Jianjun Paul Tian. Finite-time perturbations of dynamical systems and applications to tumor therapy. Discrete and Continuous Dynamical Systems - B, 2009, 12 (2) : 469-479. doi: 10.3934/dcdsb.2009.12.469
[3]

Xavier Cabré, Amadeu Delshams, Marian Gidea, Chongchun Zeng. Preface of Llavefest: A broad perspective on finite and infinite dimensional dynamical systems. Discrete and Continuous Dynamical Systems, 2018, 38 (12) : i-iii. doi: 10.3934/dcds.201812i
[4]

María J. Garrido-Atienza, Oleksiy V. Kapustyan, José Valero. Preface to the special issue "Finite and infinite dimensional multivalued dynamical systems". Discrete and Continuous Dynamical Systems - B, 2017, 22 (5) : i-iv. doi: 10.3934/dcdsb.201705i
[5]

Tianhu Yu, Jinde Cao, Chuangxia Huang. Finite-time cluster synchronization of coupled dynamical systems with impulsive effects. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3595-3620. doi: 10.3934/dcdsb.2020248
[6]

Felix X.-F. Ye, Hong Qian. Stochastic dynamics Ⅱ: Finite random dynamical systems, linear representation, and entropy production. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 4341-4366. doi: 10.3934/dcdsb.2019122
[7]

Roumen Anguelov, Jean M.-S. Lubuma, Meir Shillor. Dynamically consistent nonstandard finite difference schemes for continuous dynamical systems. Conference Publications, 2009, 2009 (Special) : 34-43. doi: 10.3934/proc.2009.2009.34
[8]

Mahdi Khajeh Salehani. Identification of generic stable dynamical systems taking a nonlinear differential approach. Discrete and Continuous Dynamical Systems - B, 2018, 23 (10) : 4541-4555. doi: 10.3934/dcdsb.2018175
[9]

Laura Gardini, Iryna Sushko. Preface: Special issue on nonlinear dynamical systems in economic modeling. Discrete and Continuous Dynamical Systems - B, 2021, 26 (11) : i-iv. doi: 10.3934/dcdsb.2021241
[10]

Thomas Westerbäck. Parity check systems of nonlinear codes over finite commutative Frobenius rings. Advances in Mathematics of Communications, 2017, 11 (3) : 409-427. doi: 10.3934/amc.2017035
[11]

Mingyou Zhang, Qingsong Zhao, Yu Liu, Wenke Li. Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition. Electronic Research Archive, 2020, 28 (1) : 369-381. doi: 10.3934/era.2020021
[12]

Xuemei Li, Zaijiu Shang. On the existence of invariant tori in non-conservative dynamical systems with degeneracy and finite differentiability. Discrete and Continuous Dynamical Systems, 2019, 39 (7) : 4225-4257. doi: 10.3934/dcds.2019171
[13]

Ian Melbourne, Dalia Terhesiu. Mixing properties for toral extensions of slowly mixing dynamical systems with finite and infinite measure. Journal of Modern Dynamics, 2018, 12: 285-313. doi: 10.3934/jmd.2018011
[14]

Ahmed Bonfoh, Ibrahim A. Suleman. Robust exponential attractors for singularly perturbed conserved phase-field systems with no growth assumption on the nonlinear term. Communications on Pure and Applied Analysis, 2021, 20 (10) : 3655-3682. doi: 10.3934/cpaa.2021125
[15]

Grégory Berhuy, Jean Fasel, Odile Garotta. Rank weights for arbitrary finite field extensions. Advances in Mathematics of Communications, 2021, 15 (4) : 575-587. doi: 10.3934/amc.2020083
[16]

Caixia Gao, Enmin Feng, Zongtao Wang, Zhilong Xiu. Nonlinear dynamical systems of bio-dissimilation of glycerol to 1,3-propanediol and their optimal controls. Journal of Industrial and Management Optimization, 2005, 1 (3) : 377-388. doi: 10.3934/jimo.2005.1.377
[17]

Masatoshi Shiino, Keiji Okumura. Control of attractors in nonlinear dynamical systems using external noise: Effects of noise on synchronization phenomena. Conference Publications, 2013, 2013 (special) : 685-694. doi: 10.3934/proc.2013.2013.685
[18]

Abderrahim Azouani, Edriss S. Titi. Feedback control of nonlinear dissipative systems by finite determining parameters - A reaction-diffusion paradigm. Evolution Equations and Control Theory, 2014, 3 (4) : 579-594. doi: 10.3934/eect.2014.3.579
[19]

Ta T.H. Trang, Vu N. Phat, Adly Samir. Finite-time stabilization and $H_\infty$ control of nonlinear delay systems via output feedback. Journal of Industrial and Management Optimization, 2016, 12 (1) : 303-315. doi: 10.3934/jimo.2016.12.303
[20]

Evelyn Lunasin, Edriss S. Titi. Finite determining parameters feedback control for distributed nonlinear dissipative systems -a computational study. Evolution Equations and Control Theory, 2017, 6 (4) : 535-557. doi: 10.3934/eect.2017027

 Impact Factor: 

Tools

Metrics

  • PDF downloads (26)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy