Dynamics of two families of meromorphic functions involving hyperbolic cosine function

Madhusudan Bera () and M. Guru Prem Prasad ()
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Madhusudan Bera: Kalinga Institute of Industrial Technology (KIIT) Deemed to be University
M. Guru Prem Prasad: Indian Institute of Technology Guwahati

Indian Journal of Pure and Applied Mathematics, 2021, vol. 52, issue 2, 384-394

Abstract: Abstract In this paper, one-parameter families $${\mathcal {F}}\equiv \left\{ f_{\lambda }(z)=\lambda \left( \cosh z+\frac{1}{\cosh z}\right) \;\text{ for }\; z\in {\mathbb {C}}: \lambda >0\right\} $$ F ≡ f λ ( z ) = λ cosh z + 1 cosh z for z ∈ C : λ > 0 and $${\mathcal {G}}\equiv \left\{ g_{\lambda }(z)=\lambda \left( \cosh z-\frac{1}{\cosh z}\right) \;\text{ for }\; z\in {\mathbb {C}}: \lambda >0\right\} $$ G ≡ g λ ( z ) = λ cosh z - 1 cosh z for z ∈ C : λ > 0 are considered and the dynamics of functions $$f_{\lambda }\in {\mathcal {F}}$$ f λ ∈ F and $$g_{\lambda }\in {\mathcal {G}}$$ g λ ∈ G are investigated. It is shown that both the functions $$f_{\lambda }$$ f λ and $$g_{\lambda }$$ g λ have finite number of singular values and the origin is always an attracting fixed point of $$g_{\lambda }(z)$$ g λ ( z ) . The dynamics of $$f_{\lambda }(z)$$ f λ ( z ) and $$g_{\lambda }(z)$$ g λ ( z ) on the extended complex plane are studied by investigating the nature of the real fixed points and the singular values of $$f_{\lambda }$$ f λ and $$g_{\lambda }$$ g λ . It is shown that a bifurcation and chaotic burst occur at a certain parameter value of $$\lambda $$ λ for the functions $$f_{\lambda }$$ f λ in the family $${\mathcal {F}}$$ F but there is no bifurcation in the family $${\mathcal {G}}$$ G .

Keywords: Fatou sets; Julia sets; Transcendental meromorphic functions; 37F45; 37F50; 37F10; 37C25 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s13226-021-00143-3

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