Iteration of certain meromorphic functions with unbounded singular values

Nayak, Tarakanta; Prasad, M. G.

doi:10.1017/s0143385709000364

Cited by 14 publications

(10 citation statements)

References 17 publications

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“…By taking f 3 (z) = λ(z m /sinh m ), m or m/2 is an odd natural number and λ is any non-zero real number, it is observed that O f3 = {0}. A critical parameter λ * > 0 is found in [12] such that for |λ| > λ * , F(f 3 ) is the basin of attraction or parabolic basin corresponding to a 2-periodic point and 0 ∈ F(f 3 ). Further, it is proved that all the Fatou components are simply connected which means that J (f 3 ) is connected.…”

Section: Resultsmentioning

confidence: 99%

Omitted values and dynamics of meromorphic functions

Nayak¹,

Zheng²

2010

Journal of the London Mathematical Society

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Let M be the class of all transcendental meromorphic functions f : C → C ∪ {∞} with at least two poles or one pole that is not an omitted value, and M o = {f ∈ M : f has at least one omitted value}. Some dynamical issues of the functions in M o are addressed in this article. A complete classification in terms of forward orbits of all the multiply connected Fatou components is made. As a corollary, it follows that the Julia set is not totally disconnected unless all the omitted values are contained in a single Fatou component. Non-existence of both Baker wandering domains and invariant Herman rings are proved. Eventual connectivity of each wandering domain is proved to exist. For functions with exactly one pole, we show that Herman rings of period 2 also do not exist. A necessary and sufficient condition for the existence of a dense subset of singleton buried components in the Julia set is established for functions with two omitted values. The conjecture that a meromorphic function has at most two completely invariant Fatou components is confirmed for all f ∈ M o except in the case when f has a single omitted value, no critical value and is of infinite order. Some relevant examples are discussed.

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Section: Resultsmentioning

confidence: 99%

Omitted values and dynamics of meromorphic functions

Nayak¹,

Zheng²

2010

Journal of the London Mathematical Society

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“…The initial study of iterations of transcendental meromorphic functions is mainly found in [1,2,3,5]. Further, researches in this direction are pursued in [7,13,18,26]. Since the Julia sets (chaotic sets) occur as one of the crucial component in these investigations, their various characterizations and identification of their intrinsic properties are primarily developed in these studies.…”

Section: Introductionmentioning

confidence: 99%

Chaos in dynamics of a family of transcendental meromorphic functions

Sajid¹,

Kapoor²

2017

Journal of Nonlinear Analysis and Application

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In this paper, the chaotic behaviours in the real and complex dynamics of ζ λ (z) = λ z z+1 e −z , λ > 0, z ∈ C are investigated. The bifurcation in the dynamics of ζ λ (x), x ∈ R \ {−1}, occurs at several parameter values and the dynamics becomes chaotic when the parameter λ crosses certain values. The Lyapunov exponent of ζ λ (x) is computed for quantifying the chaos. The characterization of the Julia set of ζ λ (z) as complement of the basin of attraction is found and is applied to computationally simulate the images of the Julia sets. Finally, the results on the dynamics of ζ λ (z) are compared with the known results.

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“…And the dynamics of function λ e z −1 z , λ > 0 is investigated by Kapoor and Prasad [3]. Further, the dynamics of certain transcendental meromorphic functions with unbounded singular values was explored by Nayak and Prasad [5]. Especially in [8], Sajid and Alsuwaiyan studied chaotic behavior in the real dynamics of a one parameter family of non linear function λ xe x x−1 , λ > 0, x ∈ R {1}.…”

Section: Introductionmentioning

confidence: 99%

Fixed points and dynamics on generating function of Genocchi numbers

Lim¹

2016

J. Nonlinear Sci. Appl.

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Recently, there have been many works related with dynamics of various functions. In this paper, singular values and fixed points of generating function of Genocchi numbers, g λ (z) = λ 2z e z +1 , λ(∈ R) > 1, are investigated. It is shown that the function g λ (z) has infinitely many singular values and its critical values lie in the left half plane and one point on the real axis in the right half plane. Further, the real fixed points of g λ (z) and their nature are determined. Finally, we provide numerical evidence of the existence of chaotic phenomena by illustrating bifurcation diagrams of system and by calculating the Lyapunov exponent.

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Iteration of certain meromorphic functions with unbounded singular values

Cited by 14 publications

References 17 publications

Omitted values and dynamics of meromorphic functions

Omitted values and dynamics of meromorphic functions

Chaos in dynamics of a family of transcendental meromorphic functions

Fixed points and dynamics on generating function of Genocchi numbers

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