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Constructing chaotic polynomial maps. (English) Zbl 1170.37318

Summary: This paper studies the construction of one-dimensional real chaotic polynomial maps. Given an arbitrary nonzero polynomial of degree \(m (\geq 0)\), two methods are derived for constructing chaotic polynomial maps of degree \(m + 2\) by simply multiplying the given polynomial with suitably designed quadratic polynomials. Moreover, for \(m + 2\) arbitrarily given different positive constants, a method is given to construct a chaotic polynomial map of degree \(2m\) based on the coupled-expansion theory.
Furthermore, by multiplying a real parameter to a special kind of polynomial, which has at least two different non-negative or nonpositive zeros, the chaotic parameter region of the polynomial is analyzed based on the snap-back repeller theory. As a consequence, for any given integer \(n \geq 2\), at least one polynomial of degree \(n\) can be constructed so that it is chaotic in the sense of both Li–Yorke and Devaney. In addition, two natural ways of generalizing the logistic map to higher-degree chaotic logistic-like maps are given. Finally, an illustrative example is provided with computer simulations for illustration.

MSC:

37E05 Dynamical systems involving maps of the interval
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
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