Real cubic polynomials with a fixed point of multiplicity two

Abstract

The aim of this paper is to study the dynamics of the real cubic polynomials that have a fixed point of multiplicity two. Such polynomials are conjugate to f a (x) = ax 2 (x − 1) + x, a ̸ = 0. We will show that when a > 0 and x ̸ = 1, then | f n a (x)| converges to 0 or ∞ and, if a < 0 and a belongs to a special subset of the parameter space, then there is a closed invariant subset Λ a of R on which f a is chaotic.

Chaotic behavior in real dynamics and singular values of family of generalized generating function of Apostol-Genocchi numbers

“…For 1 2 < λ < 1, by Equation (2.1), f λ,a (x) > 1 since x λ < 0. Therefore, the fixed point x λ of f λ,a (x) is repelling fixed point for 1 2 < λ < 1. For 1 < λ < λ * , since the function x f a (x) is increasing on R + and λ…”

“…Theorem 2.1. Let f λ,a ∈ F. Then, the function f λ,a (x) has one fixed point 0 for all λ, one nonzero real fixed point x λ for λ > 1 2 , and f λ,a (x) has no nonzero real fixed points for λ 1 2 . For λ > 1, if a > 0, then the fixed point x λ of f λ,a (x) is positive and if a < 0, then x λ is negative.…”

“…The exploration of dynamics of a real function has become an important topic, partially due to the fact that it deduce the iterations of the function in the complex plane which is mainly influenced by its real dynamics [6,12,16,23]. The real dynamics of the cubic polynomials, generalized logistic maps and oneparameter family of transcendental functions are given in [1,10,15,21] respectively. The real fixed points are described for one-parameter family of function x b x −1 in [17] and for a two-parameter family λ( x b x −1 ) n in [8].…”

Bifurcation and Chaos in Real Dynamics of a Two-Parameter Family Arising from Generating Function of Generalized Apostol-Type Polynomials

“…The dynamics of the real cubic polynomials is a little more complicated than that of the quadratic polynomials. The real dynamics of the cubic polynomials are given in [5,6]. Generally, the dynamics of transcendental functions is more complicated than polynomials.…”

Real fixed points and dynamics of one parameter family of function

“…The investigation of the dynamical properties of this exponential family is simpler than other kinds of families which include exponential map, (Fagella and Garijo, 2003;Kuroda and Jang, 1997;Petek and Rugelj, 1998;Yanagihara and Gotoh, 1998). The real dynamics of the cubic polynomials, generalized logistic maps and one parameter family of transcendental functions was found in Akbari and Rabii (2015); Magrenan and Gutierrez (2015); Radwan (2013); and Sajid and Alsuwaiyan (2014) respectively. The real dynamics of functions has become an important research area, partially due to the dynamics in the complex plane which was induced using the real dynamics by Kapoor and Prasad (1998), Nayak and Prasad (2014), Sajid (2012), and Sajid and Kapoor (2007).…”