American Institute of Mathematical Sciences

Advanced
October  2011, 16(3): 927-944. doi: 10.3934/dcdsb.2011.16.927

The logistic map of matrices

Zenonas Navickas 1, , Rasa Smidtaite 1, , Alfonsas Vainoras 2, and  Minvydas Ragulskis 3,

1. 

Department of Applied Mathematics, Kaunas University of Technology, Studentu 50-325, Kaunas LT-51368, Lithuania, Lithuania
2. 

Institute of Cardiology, Kaunas University of Medicine, Sukileliu av. 17, LT-50009, Kaunas, Lithuania
3. 

Research Group for Mathematical and Numerical Analysis of Dynamical Systems, Kaunas University of Technology, Studentu 50-222, Kaunas LT-51368, Lithuania

Received  September 2010 Revised  March 2011 Published  June 2011

The standard iterative logistic map is extended by replacing the scalar variable by a square matrix of variables. Dynamical properties of such an iterative map are explored in detail when the order of matrices is 2. It is shown that the evolution of the logistic map depends not only on the control parameter but also on the eigenvalues of the matrix of initial conditions. Several computational examples are used to demonstrate the convergence to periodic attractors and the sensitivity of chaotic processes to initials conditions.
Keywords: idempotent, The logistic map, attractor, nilpotent..
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: Zenonas Navickas, Rasa Smidtaite, Alfonsas Vainoras, Minvydas Ragulskis. The logistic map of matrices. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 927-944. doi: 10.3934/dcdsb.2011.16.927
References:
[1]

R. M. May, Simple mathematical models with very complicated dynamics,, Nature, 261 (1976), 459.  doi: 10.1038/261459a0.  Google Scholar

[2]

S. H. Strogatz, "Nonlinear Dynamics and Chaos: with applications to physics, biology, chemistry, and engineering,", Perseus Publishing, (2000).   Google Scholar

[3]

R. M. B. Young and P. L. Read, Flow transitions resembling bifurcations of the logistic map in simulations of the baroclinic rotating annulus,, Physica D: Nonlinear Phenomena, 237 (2008), 2251.  doi: 10.1016/j.physd.2008.02.014.  Google Scholar

[4]

A. Díaz-Méndez, J. V. Marquina-Pérez, M. Cruz-Irisson, R. Vázquez-Medina and J. L. Del-Río-Correa, Chaotic noise MOS generator based on logistic map,, Microelectron. J., 40 (2009), 638.  doi: 10.1016/j.mejo.2008.06.042.  Google Scholar

[5]

A. Ferretti and N. K. Rahman, A study of coupled logistic map and its applications in chemical physics,, Chem. Phys., 119 (1988), 275.  doi: 10.1016/0301-0104(88)87190-8.  Google Scholar

[6]

A. A. Hnilo, Chaotic (as the logistic map) laser cavity,, Opt. Commun., 53 (1985), 194.  doi: 10.1016/0030-4018(85)90330-X.  Google Scholar

[7]

N. Singh and A. Sinha, Optical image encryption using Hartley transform and logistic map,, Opt. Commun., 282 (2009), 1104.  doi: 10.1016/j.optcom.2008.12.001.  Google Scholar

[8]

V. Patidar, N. K. Pareek and K. K. Sud, A new substitution-diffusion based image cipher using chaotic standard and logistic maps,, Commun. Nonlinear Sci., 14 (2009), 3056.  doi: 10.1016/j.cnsns.2008.11.005.  Google Scholar

[9]

T. Nagatani, Vehicular motion through a sequence of traffic lights controlled by logistic map,, Phys. Lett. A, 372 (2008), 5887.  doi: 10.1016/j.physleta.2008.07.063.  Google Scholar

[10]

J. Miskiewicz and M. Ausloos, A logistic map approach to economic cycles I. The best adapted companies,, Physica A: Statistical and Theoretical Physics, 336 (2004), 206.  doi: 10.1016/j.physa.2004.01.026.  Google Scholar

[11]

K. P. Harikrishnan and V. M. Nandakumaran, An analogue of the logistic map in two dimensions,, Phys. Lett. A, 142 (1989), 483.  doi: 10.1016/0375-9601(89)90519-7.  Google Scholar

[12]

M. McCartney, A discrete time car following model and the bi-parameter logistic map,, Commun. Nonlinear Sci., 14 (2009), 233.  doi: 10.1016/j.cnsns.2007.06.012.  Google Scholar

[13]

M. Rani and R. Agarwal, Generation of fractals from complex logistic map,, Chaos Soliton. Fract., 42 (2009), 447.  doi: 10.1016/j.chaos.2009.01.011.  Google Scholar

[14]

J. J. Dai, A result regarding convergence of random logistic maps,, Stat. Probabil. Lett., 47 (2000), 11.  doi: 10.1016/S0167-7152(99)00131-5.  Google Scholar

[15]

K. Erguler and M. P. Stumpf, Statistical interpretation of the interplay between noise and chaos in the stochastic logistic map,, Math. Biosci., 216 (2008), 90.  doi: 10.1016/j.mbs.2008.08.012.  Google Scholar

[16]

A. L. Lloyd, The coupled logistic map: a simple model for the effects of spatial heterogeneity on population dynamics,, J. Theor. Biol., 173 (1995), 217.  doi: 10.1006/jtbi.1995.0058.  Google Scholar

[17]

L. Xu, G. Zhang, B. Han, L. Zhang, M. F. Li and Y. T. Han, Turing instability for a two-dimensional logistic coupled map lattice,, Phys. Lett. A, 374 (2010), 3447.  doi: 10.1016/j.physleta.2010.06.065.  Google Scholar

[18]

R. Bedient and M. Frame, Carrying surfaces for return maps of averaged logistic maps,, Comput. Graph., 31 (2007), 887.  doi: 10.1016/j.cag.2007.06.001.  Google Scholar

[19]

X. Wang and Q. Liang, Reverse bifurcation and fractal of the compound logistic map,, Commun. Nonlinear Sci., 13 (2008), 913.  doi: 10.1016/j.cnsns.2006.08.007.  Google Scholar

[20]

D. S. Bernstein, "Matrix Mathematics: Theory, Facts, and Formulas with Application to Linear Systems Theory,", Princeton University Press, (2005).   Google Scholar

[21]

E. W. Weisstein, Logistic Map, MathWorld - A Wolfram Web Resource,, 25 August, (2010).   Google Scholar

[22]

M. Ragulskis and Z. Navickas, The rank of a sequence as an indicator of chaos in discrete nonlinear dynamical systems,, Commun. Nonlinear Sci., 16 (2011), 2894.  doi: 10.1016/j.cnsns.2010.10.008.  Google Scholar

show all references

References:
[1]

R. M. May, Simple mathematical models with very complicated dynamics,, Nature, 261 (1976), 459.  doi: 10.1038/261459a0.  Google Scholar

[2]

S. H. Strogatz, "Nonlinear Dynamics and Chaos: with applications to physics, biology, chemistry, and engineering,", Perseus Publishing, (2000).   Google Scholar

[3]

R. M. B. Young and P. L. Read, Flow transitions resembling bifurcations of the logistic map in simulations of the baroclinic rotating annulus,, Physica D: Nonlinear Phenomena, 237 (2008), 2251.  doi: 10.1016/j.physd.2008.02.014.  Google Scholar

[4]

A. Díaz-Méndez, J. V. Marquina-Pérez, M. Cruz-Irisson, R. Vázquez-Medina and J. L. Del-Río-Correa, Chaotic noise MOS generator based on logistic map,, Microelectron. J., 40 (2009), 638.  doi: 10.1016/j.mejo.2008.06.042.  Google Scholar

[5]

A. Ferretti and N. K. Rahman, A study of coupled logistic map and its applications in chemical physics,, Chem. Phys., 119 (1988), 275.  doi: 10.1016/0301-0104(88)87190-8.  Google Scholar

[6]

A. A. Hnilo, Chaotic (as the logistic map) laser cavity,, Opt. Commun., 53 (1985), 194.  doi: 10.1016/0030-4018(85)90330-X.  Google Scholar

[7]

N. Singh and A. Sinha, Optical image encryption using Hartley transform and logistic map,, Opt. Commun., 282 (2009), 1104.  doi: 10.1016/j.optcom.2008.12.001.  Google Scholar

[8]

V. Patidar, N. K. Pareek and K. K. Sud, A new substitution-diffusion based image cipher using chaotic standard and logistic maps,, Commun. Nonlinear Sci., 14 (2009), 3056.  doi: 10.1016/j.cnsns.2008.11.005.  Google Scholar

[9]

T. Nagatani, Vehicular motion through a sequence of traffic lights controlled by logistic map,, Phys. Lett. A, 372 (2008), 5887.  doi: 10.1016/j.physleta.2008.07.063.  Google Scholar

[10]

J. Miskiewicz and M. Ausloos, A logistic map approach to economic cycles I. The best adapted companies,, Physica A: Statistical and Theoretical Physics, 336 (2004), 206.  doi: 10.1016/j.physa.2004.01.026.  Google Scholar

[11]

K. P. Harikrishnan and V. M. Nandakumaran, An analogue of the logistic map in two dimensions,, Phys. Lett. A, 142 (1989), 483.  doi: 10.1016/0375-9601(89)90519-7.  Google Scholar

[12]

M. McCartney, A discrete time car following model and the bi-parameter logistic map,, Commun. Nonlinear Sci., 14 (2009), 233.  doi: 10.1016/j.cnsns.2007.06.012.  Google Scholar

[13]

M. Rani and R. Agarwal, Generation of fractals from complex logistic map,, Chaos Soliton. Fract., 42 (2009), 447.  doi: 10.1016/j.chaos.2009.01.011.  Google Scholar

[14]

J. J. Dai, A result regarding convergence of random logistic maps,, Stat. Probabil. Lett., 47 (2000), 11.  doi: 10.1016/S0167-7152(99)00131-5.  Google Scholar

[15]

K. Erguler and M. P. Stumpf, Statistical interpretation of the interplay between noise and chaos in the stochastic logistic map,, Math. Biosci., 216 (2008), 90.  doi: 10.1016/j.mbs.2008.08.012.  Google Scholar

[16]

A. L. Lloyd, The coupled logistic map: a simple model for the effects of spatial heterogeneity on population dynamics,, J. Theor. Biol., 173 (1995), 217.  doi: 10.1006/jtbi.1995.0058.  Google Scholar

[17]

L. Xu, G. Zhang, B. Han, L. Zhang, M. F. Li and Y. T. Han, Turing instability for a two-dimensional logistic coupled map lattice,, Phys. Lett. A, 374 (2010), 3447.  doi: 10.1016/j.physleta.2010.06.065.  Google Scholar

[18]

R. Bedient and M. Frame, Carrying surfaces for return maps of averaged logistic maps,, Comput. Graph., 31 (2007), 887.  doi: 10.1016/j.cag.2007.06.001.  Google Scholar

[19]

X. Wang and Q. Liang, Reverse bifurcation and fractal of the compound logistic map,, Commun. Nonlinear Sci., 13 (2008), 913.  doi: 10.1016/j.cnsns.2006.08.007.  Google Scholar

[20]

D. S. Bernstein, "Matrix Mathematics: Theory, Facts, and Formulas with Application to Linear Systems Theory,", Princeton University Press, (2005).   Google Scholar

[21]

E. W. Weisstein, Logistic Map, MathWorld - A Wolfram Web Resource,, 25 August, (2010).   Google Scholar

[22]

M. Ragulskis and Z. Navickas, The rank of a sequence as an indicator of chaos in discrete nonlinear dynamical systems,, Commun. Nonlinear Sci., 16 (2011), 2894.  doi: 10.1016/j.cnsns.2010.10.008.  Google Scholar

[1]

C. Bonanno, G. Menconi. Computational information for the logistic map at the chaos threshold. Discrete & Continuous Dynamical Systems - B, 2002, 2 (3) : 415-431. doi: 10.3934/dcdsb.2002.2.415
[2]

Dalibor Pražák. Exponential attractor for the delayed logistic equation with a nonlinear diffusion. Conference Publications, 2003, 2003 (Special) : 717-726. doi: 10.3934/proc.2003.2003.717
[3]

Àngel Jorba, Pau Rabassa, Joan Carles Tatjer. Period doubling and reducibility in the quasi-periodically forced logistic map. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1507-1535. doi: 10.3934/dcdsb.2012.17.1507
[4]

Percy Fernández-Sánchez, Jorge Mozo-Fernández, Hernán Neciosup. Dicritical nilpotent holomorphic foliations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3223-3237. doi: 10.3934/dcds.2018140
[5]

A. Cibotarica, Jiu Ding, J. Kolibal, Noah H. Rhee. Solutions of the Yang-Baxter matrix equation for an idempotent. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 347-352. doi: 10.3934/naco.2013.3.347
[6]

Michelle Nourigat, Richard Varro. Conjectures for the existence of an idempotent in $\omega $-polynomial algebras. Discrete & Continuous Dynamical Systems - S, 2011, 4 (6) : 1543-1551. doi: 10.3934/dcdss.2011.4.1543
[7]

Tracy L. Payne. Anosov automorphisms of nilpotent Lie algebras. Journal of Modern Dynamics, 2009, 3 (1) : 121-158. doi: 10.3934/jmd.2009.3.121
[8]

Isaac A. García, Douglas S. Shafer. Cyclicity of a class of polynomial nilpotent center singularities. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2497-2520. doi: 10.3934/dcds.2016.36.2497
[9]

Mark Pollicott. Ergodicity of stable manifolds for nilpotent extensions of Anosov flows. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 599-604. doi: 10.3934/dcds.2002.8.599
[10]

P. De Maesschalck. Gevrey normal forms for nilpotent contact points of order two. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 677-688. doi: 10.3934/dcds.2014.34.677
[11]

Luca Capogna. Optimal regularity for quasilinear equations in stratified nilpotent Lie groups and applications. Electronic Research Announcements, 1996, 2: 60-68.

[12]

Antonio Algaba, María Díaz, Cristóbal García, Jaume Giné. Analytic integrability around a nilpotent singularity: The non-generic case. Communications on Pure & Applied Analysis, 2020, 19 (1) : 407-423. doi: 10.3934/cpaa.2020021
[13]

Juan J. Nieto, M. Victoria Otero-Espinar, Rosana Rodríguez-López. Dynamics of the fuzzy logistic family. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 699-717. doi: 10.3934/dcdsb.2010.14.699
[14]

Luis Caffarelli, Serena Dipierro, Enrico Valdinoci. A logistic equation with nonlocal interactions. Kinetic & Related Models, 2017, 10 (1) : 141-170. doi: 10.3934/krm.2017006
[15]

Daniel Fusca. The Madelung transform as a momentum map. Journal of Geometric Mechanics, 2017, 9 (2) : 157-165. doi: 10.3934/jgm.2017006
[16]

Lluís Alsedà, Michał Misiurewicz. Semiconjugacy to a map of a constant slope. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3403-3413. doi: 10.3934/dcdsb.2015.20.3403
[17]

Richard Evan Schwartz. Outer billiards and the pinwheel map. Journal of Modern Dynamics, 2011, 5 (2) : 255-283. doi: 10.3934/jmd.2011.5.255
[18]

Valentin Ovsienko, Richard Schwartz, Serge Tabachnikov. Quasiperiodic motion for the pentagram map. Electronic Research Announcements, 2009, 16: 1-8. doi: 10.3934/era.2009.16.1
[19]

John Erik Fornæss, Brendan Weickert. A quantized henon map. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 723-740. doi: 10.3934/dcds.2000.6.723
[20]

Tomás Caraballo, Francisco Morillas, José Valero. Asymptotic behaviour of a logistic lattice system. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4019-4037. doi: 10.3934/dcds.2014.34.4019

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences