The logistic map and the birth of period-3 cycle
The goal of this paper is to present a proof that for the logistic map the period-3 begins at . The third-iterate map is the key for understanding the birth of the period-3 cycle. Any point in a period-3 cycle repeats every three iterates by definition. Such points satisfy the condition ,and the...
| Main Authors: | , , |
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| Format: | Article |
| Language: | Russian |
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MIREA - Russian Technological University
2012-06-01
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| Series: | Тонкие химические технологии |
| Subjects: | |
| Online Access: | https://www.finechem-mirea.ru/jour/article/view/741 |
| _version_ | 1797882143950503936 |
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| author | Luis Alberto Toro Carlos Ariel Cardona Yu. A. Pisarenko |
| author_facet | Luis Alberto Toro Carlos Ariel Cardona Yu. A. Pisarenko |
| author_sort | Luis Alberto Toro |
| collection | DOAJ |
| description | The goal of this paper is to present a proof that for the logistic map the period-3 begins at . The third-iterate map is the key for understanding the birth of the period-3 cycle. Any point in a period-3 cycle repeats every three iterates by definition. Such points satisfy the condition ,and they are therefore fixed points of the third-iterate map. This fact and the so called tangent bifurcation for the logistic map, as well as the fixed points definition, are used for finding the value. The algebraic treatment utilizes some properties of symmetric polynomials in three variables. For the purposes of this paper, the bifurcation diagram for the logistic map is also presented, as well as a program in Mathematica for its construction. |
| first_indexed | 2024-04-10T03:31:20Z |
| format | Article |
| id | doaj.art-dd79551642c44a3f9eb8969295c79a61 |
| institution | Directory Open Access Journal |
| issn | 2410-6593 2686-7575 |
| language | Russian |
| last_indexed | 2024-04-10T03:31:20Z |
| publishDate | 2012-06-01 |
| publisher | MIREA - Russian Technological University |
| record_format | Article |
| series | Тонкие химические технологии |
| spelling | doaj.art-dd79551642c44a3f9eb8969295c79a612023-03-13T07:25:33ZrusMIREA - Russian Technological UniversityТонкие химические технологии2410-65932686-75752012-06-01737176735The logistic map and the birth of period-3 cycleLuis Alberto Toro0Carlos Ariel Cardona1Yu. A. Pisarenko2Instituto de Biotecnología y Agroindustria, ManizalesInstituto de Biotecnología y Agroindustria, ManizalesM.V. Lomonosov Moscow State University of Fine Chemical TechnologiesThe goal of this paper is to present a proof that for the logistic map the period-3 begins at . The third-iterate map is the key for understanding the birth of the period-3 cycle. Any point in a period-3 cycle repeats every three iterates by definition. Such points satisfy the condition ,and they are therefore fixed points of the third-iterate map. This fact and the so called tangent bifurcation for the logistic map, as well as the fixed points definition, are used for finding the value. The algebraic treatment utilizes some properties of symmetric polynomials in three variables. For the purposes of this paper, the bifurcation diagram for the logistic map is also presented, as well as a program in Mathematica for its construction.https://www.finechem-mirea.ru/jour/article/view/741<i>attractor</i><i>bifurcation diagram</i><i>dynamical systems</i><i>chaos</i><i>fixed points</i><i>logistic map</i><i>maps</i><i>symmetric polynomials</i><i>tangent condition</i><i>vector fields</i> |
| spellingShingle | Luis Alberto Toro Carlos Ariel Cardona Yu. A. Pisarenko The logistic map and the birth of period-3 cycle Тонкие химические технологии <i>attractor</i> <i>bifurcation diagram</i> <i>dynamical systems</i> <i>chaos</i> <i>fixed points</i> <i>logistic map</i> <i>maps</i> <i>symmetric polynomials</i> <i>tangent condition</i> <i>vector fields</i> |
| title | The logistic map and the birth of period-3 cycle |
| title_full | The logistic map and the birth of period-3 cycle |
| title_fullStr | The logistic map and the birth of period-3 cycle |
| title_full_unstemmed | The logistic map and the birth of period-3 cycle |
| title_short | The logistic map and the birth of period-3 cycle |
| title_sort | logistic map and the birth of period 3 cycle |
| topic | <i>attractor</i> <i>bifurcation diagram</i> <i>dynamical systems</i> <i>chaos</i> <i>fixed points</i> <i>logistic map</i> <i>maps</i> <i>symmetric polynomials</i> <i>tangent condition</i> <i>vector fields</i> |
| url | https://www.finechem-mirea.ru/jour/article/view/741 |
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