N1(2023)

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homepage: http://www.applied-business-solutions.eu
Type: Article
Title: Tool QUATTRO-20 for Examining of the Recurrent Sequencies Generated by Discrete Analogue of the Verhulst Equation PDF Article
Author: Jelena Kozmina
Orcid: https://orcid.org/0009-0005-5075-5190
Author: Alytis Gruodis
Orcid: https://orcid.org/0009-0008-2989-3283
On-line: 30-June-2023
Metrics: Applied Business: Issues & Solutions 1(2023)16-29 – ISSN 2783-6967.
DOI: 10.57005/ab.2023.1.3
URL: http://www.applied-business-solutions.eu/h23/2023_1_3.html
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Abstract. QUATTRO-20 as advanced tool for estimation of the recurrent sequences was created and tested. Discrete analogue of Verhulst equation x(t+1)=F(x(t)), F(x)=rx(1-x), t=0, 1, 2, ..., was selected as the model of recurrent sequence. Related mathematical material is presented in user-friendly form: convergence conditions, Lyapunov index, behaviour of the sequencies generated by second, third, fourth compositions of function F(x). QUATTRO-20 contains several visualization methods such as xy plot, Bifurcation diagram, distribution of Lyapunov index, CobWeb plot, graphical solution. Novel graphical technique of realization of the sequence convergence was presented.
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JEL: H1; M1; Z1.
Keywords: Discrete analogue of Verhulst equation; Logistic map; convergence condition; Bifurcation diagram; Lyapunov exponent; Lyapunov index; CobWeb plot.
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Citation: Jelena Kozmina, Alytis Gruodis (2023) Tool QUATTRO-20 for Examining of the Recurrent Sequencies Generated by Discrete Analogue of the Verhulst Equation. – Applied Business: Issues & Solutions 1(2023)16– 29 – ISSN 2783-6967.
https://doi.org/10.57005/ab.2023.1.3
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References.

1. Cencini, M.; Cecconi, F; Vulpiani, A. (2009) Chaos. From Simple Models to Complex Systems - World Scientific, Singapore, 2009.
2. Klages, R. (2008) Introduction to Dynamical Systems. Lecture Notes for MAS424/MTHM021. Version 1.2 - Queen Mary, University of London, 2008.
3. Quinn, T. (2013) Population Dynamics. - In: Encyclopedia of Environmetrics. 2nd Ed. - John Wiley & Sons, Ltd. - https://doi.org/10.1002/9780470057339.vap028.
4. Strogatz, S. H. (2015) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry and engineering. - CRC Press, 2015.
5. Uzokov, O.X. (2020) Chaos as the Basis of Order. Entropy as Measures of Chaos - International Journal of Advanced Research in Science, Engineering and Technology 7(12) (2020) 16149-16154.
6. Bertels, K.; Neuberg, L.; Vassiliadis, S.; Pechanek, D. (2001) On Chaos and Neural Networks: The Backpropagation Paradigm - Artificial Intelligence Review 15(2001)165-187.
7. Bacaer, N. (2011) A Short History of Mathematical Population Dynamics. - Springer-Verlag London Limited, 2011. - https://doi.org/10.1007/978-0- 85729-115-8-6.
8. May, R. (1974) Biological Populations with Nonoverlapping Generations: Stable Points, Stable Cycles, and Chaos - Science 186(1974)645-7. - https://doi.org/10.1126/science.186.4164.645.
9. Lorenz, E. N. (1990) Can chaos and intransitivity lead to interannual variability? - Tellus 42A (1990)378-389. - https://doi.org/10.1034/j.1600- 0870.1990.t01-2-00005.x
10. Heinz-Otto Peitgen, Hartmut Jurgens, Dietmar Saupe (1992) Chaos and Fractals. New Frontiers of Science. - Springer-Verlag.
11. Kozmina, Y. (2018) Discrete Analogue of the Verhulst Equation and Attractors. Methodological Aspects of Teaching - Innovative Infotechnologies for Science, Business and Education 1(24) (2018) 3-12.
12. Petropoulou, E. N.(2010) A Discrete Equivalent of the Logistic Equation - Advances in Difference Equations 2010 (2010) 57073-57088 - https://doi.org/10.1155/2010/457073.
13. Kalman, D. (2023) Verhulst Discrete Logistic Growth - Mathematics Magazine 96:3 (2023) 244-258 - https://doi.org/10.1080/0025570X.2023.2199676.
14. Conejero, J.A.; Garibo-i-Orts, O.; Lizama, C. (2023) Inferring the fractional nature of Wu Baleanu trajectories. - Nonlinear Dynamics 111(2023) 12421–12431 - https://doi.org/10.1007/s11071-023-08463-1.
15. Pikovsky A., Rosenblum M., and Kurths J. (2001) Synchronization. A Universal Concept in Nonlinear Sciences - Cambridge: Cambridge University Press.
16. Afsar, O.; Eroglu, D.; Marwan, N.; Kurths, J. (2015) Scaling behaviour for recurrence-based measures at the edge of chaos. - Europhysics Letters - 112(2015)10005 - https://doi.org/112.10.1209/0295-5075/112/10005.
17. Kozmina, J.; Gruodis A. (2020) QUATTRO-20 - WinApi program. - https://github.com/Alytis/QUATTRO-20.
18. Alligood, K. T.; Sauer, T. D.; Yorke. J. A. (1996) Chaos. An introduction to dynamical systems.- Springer-Verlag, 1996.
19. Holmgren, R.A. (2000) A first course in discrete dynamical systems. Sec. Edition - Springer-Verlag, 2000.
20. Schuster, H. G.; Just, W. (2005) Deterministic Chaos. An Introduction - WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim, 2005.
21. Robinson, C. (1995) Dynamical systems. convergence, symbolic dynamics, and chaos.- CRC Press, 1995.
22. Cvitanovic, P.; Artuso, R.; Mainieri, R.; Tanner, G.; Vattay, G. (2011) Chaos: Classical and Quantum. Volume I: Deterministic Chaos - Gone With the Wind press, Atlanta, 2011
23. Wiggins, S. (2000) Introduction to Applied Nonlinear Dynamical Systems and Chaos. Second Edition. - Springer, 2000. - 860 p.
24. Gesmann, M. (2012) Logistic map: Feigenbaum diagram in R. - https://magesblog.com/post/2012-03-17-logistic-map-feigenbaum-diagram/.
25. Kozmina, Y.; Gruodis, A. (2020) QUATTRO-20: advanced tool for estimation of the recurrent sequences - In: 18th International Conference "Information Tehnologies and Management", April 23-24, 2020, ISMA University of Applied Science, Riga, Latvia.
26. Misiurewitz, M. (1981) Absolutely continuous measures for certain maps of an interval - Publications mathématiques de l’I.H.É.S. 53 (1981) 17-51. - http://www.numdam.org/item?id=PMIHES_1981_53_17_0.
27. Huberman, B. A.; Rudnick, J. (1980) Scaling Behavior of Chaotic Flows - Phys. Rev. Lett. 45 (1980) 154. - https://doi.org/10.1103/PhysRevLett.45.154.
28. Layek, G. (2015) An Introduction to Dynamical Systems and Chaos. - https://doi.org/10.1007/978-81-322-2556-0.
29. Kozmina, Y.; Gruodis, A. (2019) Number generation based on the chaotic sequences - In: The 17th International Scientific Conference "Information Technologies and Management - 2019", April 25-26, 2019, ISMA, Riga, Latvia - Nano Technologies and Computer Modelling (2019)17-18.