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Bifurcation and chaotic burst in the dynamics of \(\lambda e^{bz+ce^z}\). (English) Zbl 1219.37033

Summary: Let \(S=\{f(z)=e^{bz+ce^z}: z\in\mathbb C\), \(c>0\), \(b\in\mathbb N\), \(b\) is odd with \(b\geq ce^{-2}\}\) be a class of functions and \(\mathcal E=\{f_\lambda(z)=\lambda f(z): f\in S\), \(\lambda\in\mathbb R\setminus \{0\}\}\) be a one-parameter family in \(S\). The dynamics of the functions \(f_\lambda\in\mathcal E\) is explored in this paper. Two critical parameters \(\lambda_q<0\) and \(\lambda_p>0\) are found and two sudden changes in the dynamics of functions are observed accordingly. For \(\lambda<\lambda_q\), the Fatou set of \(f_\lambda\) is a 2-cycle of attracting basins or parabolic domains corresponding to a real attracting or parabolic 2-periodic point while it is an invariant attracting basin of a real attracting fixed point for \(\lambda\in(\lambda_q,\lambda_p)\). Further, the Fatou set of \(f_\lambda\) is an invariant parabolic domain corresponding to a real parabolic fixed point for \(\lambda=\lambda_q\) and \(\lambda_p\). It is shown that the Fatou set of \(f_\lambda\) becomes empty for \(\lambda>\lambda_p\).

MSC:

37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
37F45 Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations (MSC2010)
37F50 Small divisors, rotation domains and linearization in holomorphic dynamics
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