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The logistic map of matrices
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 |
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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