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                <text>Agricultura sostenible</text>
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                <text>Dominio científico: Agricultura sostenible</text>
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              <text>The logistic map and the birth of period-3 cycle</text>
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              <text>Luis Alberto Toro, Carlos Ariel Cardona, Yu. A. Pisarenko</text>
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              <text>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.</text>
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              <text>2012</text>
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              <text>&lt;i&gt;attractor&lt;/i&gt;, &lt;i&gt;bifurcation diagram&lt;/i&gt;, &lt;i&gt;chaos&lt;/i&gt;, &lt;i&gt;dynamical systems&lt;/i&gt;, &lt;i&gt;fixed points&lt;/i&gt;, &lt;i&gt;logistic map&lt;/i&gt;, &lt;i&gt;maps&lt;/i&gt;, &lt;i&gt;symmetric polynomials&lt;/i&gt;, &lt;i&gt;tangent condition&lt;/i&gt;, &lt;i&gt;vector fields&lt;/i&gt;</text>
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              <text>Тонкие химические технологии</text>
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              <text>MIREA - Russian Technological University</text>
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          <name>Coverage</name>
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              <text>Chemistry</text>
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              <text>&lt;a href="https://www.finechem-mirea.ru/jour/article/view/741" target="_blank" rel="noreferrer noopener"&gt;https://www.finechem-mirea.ru/jour/article/view/741&lt;/a&gt;</text>
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