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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2505.00305 |
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| _version_ | 1866913814016950272 |
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| author | Kumar, Gaurav Prasaad, M. Guru Prem |
| author_facet | Kumar, Gaurav Prasaad, M. Guru Prem |
| contents | In this article, the dynamics of a one-parameter family of functions $f_λ(z) = \frac{\sin{z}}{z^2 + λ},$ $λ>0$, are studied. It shows the existence of parameters $0< λ_{1}< λ_{2}$ such that bifurcations occur at $λ_1$ and $λ_2$ for $f_λ$. It is proved that the Fatou set $\mathcal{F}(f_λ)$ is the union of basins of attraction in the complex plane for $λ\in (λ_1, λ_2) \cup (λ_2, \infty)$. Further, every Fatou component of $f_λ$ is simply connected for $λ\geq λ_1$. The boundary of the Fatou set $\mathcal{F}(f_λ)$ is the Julia set $\mathcal{J}(f_λ)$ in the extended complex plane for $λ> 1$. Interestingly, it is found that $f_λ$ has only one completely invariant Fatou component, say $U_λ$ such that $\mathcal{F}(f_λ) = U_λ$ for $λ>λ_2$. Moreover, the characterization of the Julia set of $f_λ$ is seen for $λ\in (λ_1, \infty)\setminus \{λ_2\}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_00305 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Iterations of Meromorphic Functions involving Sine Kumar, Gaurav Prasaad, M. Guru Prem Dynamical Systems In this article, the dynamics of a one-parameter family of functions $f_λ(z) = \frac{\sin{z}}{z^2 + λ},$ $λ>0$, are studied. It shows the existence of parameters $0< λ_{1}< λ_{2}$ such that bifurcations occur at $λ_1$ and $λ_2$ for $f_λ$. It is proved that the Fatou set $\mathcal{F}(f_λ)$ is the union of basins of attraction in the complex plane for $λ\in (λ_1, λ_2) \cup (λ_2, \infty)$. Further, every Fatou component of $f_λ$ is simply connected for $λ\geq λ_1$. The boundary of the Fatou set $\mathcal{F}(f_λ)$ is the Julia set $\mathcal{J}(f_λ)$ in the extended complex plane for $λ> 1$. Interestingly, it is found that $f_λ$ has only one completely invariant Fatou component, say $U_λ$ such that $\mathcal{F}(f_λ) = U_λ$ for $λ>λ_2$. Moreover, the characterization of the Julia set of $f_λ$ is seen for $λ\in (λ_1, \infty)\setminus \{λ_2\}$. |
| title | Iterations of Meromorphic Functions involving Sine |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2505.00305 |