2025/05/01 by Kumar, Gaurav, Prasaad, M. Guru Prem
#Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.2505.00305
In this article, the dynamics of a one-parameter family of functions fλ(z) = \fracsinzz2 + λ, λ>0, are studied. It shows the existence of parameters 0< λ1< λ2 such that bifurcations occur at λ1 and λ2 for fλ. It is proved that the Fatou set F(fλ) is the union of basins of attraction in the complex plane for λ∈ (λ1, λ2) ∪ (λ2, ∞). Further, every Fatou component of fλ is simply connected for λ≥ λ1. The boundary of the Fatou set F(fλ) is the Julia set J(fλ) in the extended complex plane for λ> 1. Interestingly, it is found that fλ has only one completely invariant Fatou component, say Uλ such that F(fλ) = Uλ for λ>λ2. Moreover, the characterization of the Julia set of fλ is seen for λ∈ (λ1, ∞)∖ \λ2\.