2014/09/07 by Mohammad Sajid, Sajid, M., G. P. Kapoor +1
Mathematics · #26A18 #30D05 #37C25 #37D45 #37F50 #58K05 #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Meromorphic and Entire Functions #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.1409.2166
openalex publication_date 2014/09/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The characterization and properties of Julia sets of one parameter family of transcendental meromorphic functions ζλ(z)=λ(z)/(z+1) e-z, λ>0, z∈ ℂ is investigated in the present paper. It is found that bifurcations in the dynamics of ζλ(x), x∈ ℝ∖ \-1\, occur at several parameter values and the dynamics of the family becomes chaotic when the parameter λ crosses certain values. The Lyapunov exponent of ζλ(x) for certain values of the parameter λ is computed for quantifying the chaos in its dynamics. The characterization of the Julia set of the function ζλ(z) as complement of the basin of attraction of an attracting real fixed point of ζλ(z) is found here and is applied to computationally simulate the images of the Julia sets of ζλ(z). Further, it is established that the Julia set of ζλ(z) for λ>(√(2)+1) e√(2) contains the complement of attracting periodic orbits of ζλ(x). Finally, the results on the dynamics of functions λtan z, λ∈ \mathbbC∖\0\, Eλ(z) = λ\fracez -1z, λ> 0 and fλ=λf(z), λ>0, where f(z) has certain properties, are compared with the results found in the present paper.