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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 and 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), 459-467. doi: 10.1038/261459a0.

[2]

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

[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), 2251-2262. doi: 10.1016/j.physd.2008.02.014.

[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), 638-640. doi: 10.1016/j.mejo.2008.06.042.

[5]

A. Ferretti and N. K. Rahman, A study of coupled logistic map and its applications in chemical physics, Chem. Phys., 119 (1988), 275-288. doi: 10.1016/0301-0104(88)87190-8.

[6]

A. A. Hnilo, Chaotic (as the logistic map) laser cavity, Opt. Commun., 53 (1985), 194-196. doi: 10.1016/0030-4018(85)90330-X.

[7]

N. Singh and A. Sinha, Optical image encryption using Hartley transform and logistic map, Opt. Commun., 282 (2009), 1104-1109. doi: 10.1016/j.optcom.2008.12.001.

[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), 3056-3075. doi: 10.1016/j.cnsns.2008.11.005.

[9]

T. Nagatani, Vehicular motion through a sequence of traffic lights controlled by logistic map, Phys. Lett. A, 372 (2008), 5887-5890. doi: 10.1016/j.physleta.2008.07.063.

[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), 206-214. doi: 10.1016/j.physa.2004.01.026.

[11]

K. P. Harikrishnan and V. M. Nandakumaran, An analogue of the logistic map in two dimensions, Phys. Lett. A, 142 (1989), 483-489. doi: 10.1016/0375-9601(89)90519-7.

[12]

M. McCartney, A discrete time car following model and the bi-parameter logistic map, Commun. Nonlinear Sci., 14 (2009), 233-243. doi: 10.1016/j.cnsns.2007.06.012.

[13]

M. Rani and R. Agarwal, Generation of fractals from complex logistic map, Chaos Soliton. Fract., 42 (2009), 447-452. doi: 10.1016/j.chaos.2009.01.011.

[14]

J. J. Dai, A result regarding convergence of random logistic maps, Stat. Probabil. Lett., 47 (2000), 11-14. doi: 10.1016/S0167-7152(99)00131-5.

[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), 90-99. doi: 10.1016/j.mbs.2008.08.012.

[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), 217-230. doi: 10.1006/jtbi.1995.0058.

[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), 3447-3450. doi: 10.1016/j.physleta.2010.06.065.

[18]

R. Bedient and M. Frame, Carrying surfaces for return maps of averaged logistic maps, Comput. Graph., 31 (2007), 887-895. doi: 10.1016/j.cag.2007.06.001.

[19]

X. Wang and Q. Liang, Reverse bifurcation and fractal of the compound logistic map, Commun. Nonlinear Sci., 13 (2008), 913-927. doi: 10.1016/j.cnsns.2006.08.007.

[20]

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

[21]

E. W. Weisstein, Logistic Map, MathWorld - A Wolfram Web Resource, 25 August, 2010. Available from: http://mathworld.wolfram.com/LogisticMap.html.

[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), 2894-2906. doi: 10.1016/j.cnsns.2010.10.008.

show all references

References:
[1]

R. M. May, Simple mathematical models with very complicated dynamics, Nature, 261 (1976), 459-467. doi: 10.1038/261459a0.

[2]

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

[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), 2251-2262. doi: 10.1016/j.physd.2008.02.014.

[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), 638-640. doi: 10.1016/j.mejo.2008.06.042.

[5]

A. Ferretti and N. K. Rahman, A study of coupled logistic map and its applications in chemical physics, Chem. Phys., 119 (1988), 275-288. doi: 10.1016/0301-0104(88)87190-8.

[6]

A. A. Hnilo, Chaotic (as the logistic map) laser cavity, Opt. Commun., 53 (1985), 194-196. doi: 10.1016/0030-4018(85)90330-X.

[7]

N. Singh and A. Sinha, Optical image encryption using Hartley transform and logistic map, Opt. Commun., 282 (2009), 1104-1109. doi: 10.1016/j.optcom.2008.12.001.

[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), 3056-3075. doi: 10.1016/j.cnsns.2008.11.005.

[9]

T. Nagatani, Vehicular motion through a sequence of traffic lights controlled by logistic map, Phys. Lett. A, 372 (2008), 5887-5890. doi: 10.1016/j.physleta.2008.07.063.

[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), 206-214. doi: 10.1016/j.physa.2004.01.026.

[11]

K. P. Harikrishnan and V. M. Nandakumaran, An analogue of the logistic map in two dimensions, Phys. Lett. A, 142 (1989), 483-489. doi: 10.1016/0375-9601(89)90519-7.

[12]

M. McCartney, A discrete time car following model and the bi-parameter logistic map, Commun. Nonlinear Sci., 14 (2009), 233-243. doi: 10.1016/j.cnsns.2007.06.012.

[13]

M. Rani and R. Agarwal, Generation of fractals from complex logistic map, Chaos Soliton. Fract., 42 (2009), 447-452. doi: 10.1016/j.chaos.2009.01.011.

[14]

J. J. Dai, A result regarding convergence of random logistic maps, Stat. Probabil. Lett., 47 (2000), 11-14. doi: 10.1016/S0167-7152(99)00131-5.

[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), 90-99. doi: 10.1016/j.mbs.2008.08.012.

[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), 217-230. doi: 10.1006/jtbi.1995.0058.

[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), 3447-3450. doi: 10.1016/j.physleta.2010.06.065.

[18]

R. Bedient and M. Frame, Carrying surfaces for return maps of averaged logistic maps, Comput. Graph., 31 (2007), 887-895. doi: 10.1016/j.cag.2007.06.001.

[19]

X. Wang and Q. Liang, Reverse bifurcation and fractal of the compound logistic map, Commun. Nonlinear Sci., 13 (2008), 913-927. doi: 10.1016/j.cnsns.2006.08.007.

[20]

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

[21]

E. W. Weisstein, Logistic Map, MathWorld - A Wolfram Web Resource, 25 August, 2010. Available from: http://mathworld.wolfram.com/LogisticMap.html.

[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), 2894-2906. doi: 10.1016/j.cnsns.2010.10.008.

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