Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Quadratic recurrence equations - exact explicit solution of period four
  fixed points functions in bifurcation diagram   This article presents the exact solution of fixed points functions for the
cycle of period four of the quadratic recurrence equations. The solution is
demonstrated for the quadratic map and the logistic map. These recurrence
equations, presenting the real domain, as well as the Mandelbrot set,
presenting the complex domain, are at the very heart of dynamical systems and
chaos theory. Up to now, the closed explicit solutions of fixed points
functions have only been known for three bifurcation ranges: for the cycles of
period one, two and three. With the discovery of the solution for cycle four,
disclosed in this paper, further step has been made in our comprehension of
simultaneous complexity and simplicity which represents the beauty of nature.

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The categories of this paper are: physics
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The categories of this paper are: gr-qc
0
The categories of this paper are: hep-ph
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The categories of this paper are: q-bio
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The categories of this paper are: math
26
The categories of this paper are: bayes-an
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The categories of this paper are: patt-sol
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The categories of this paper are: adap-org
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The categories of this paper are: alg-geom
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The categories of this paper are: stat
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The categories of this paper are: q-fin
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The categories of this paper are: math-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: acc-phys
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The categories of this paper are: chem-ph
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The categories of this paper are: cond-mat