The logistic map of matrices

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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

Abstract
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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

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