Abstract:
The work belongs to the direction of experimental mathematics, which investigates the properties of mathematical objects by the computing facilities of a computer. The base is an exponential map, its topological properties (Cantor's bouquets) differ from properties of polynomial and rational complex-valued functions. The subject of the study are the character and features of the Fatou and Julia sets, as well as the equilibrium points and orbits of the zero of three iterated complex-valued mappings: $f:z\to (1+\mu)exp(iz)$, $g:z\to (1+\mu |z-z^{*}|)exp(iz)$, $h:z\to (1+\mu(z-z^{*}))exp(iz)$, with $z, \mu\in\mathbb{C}$, $z^{*}:exp(iz^{*})=z^{*}$. For a quasilinear map $g$ having no analyticity characteristic, two bifurcation transitions were discovered: the creation of a new equilibrium point (for which the critical value of the linear parameter was found and the bifurcation consists of “fork” type and “saddle”-node transition) and the transition to the radical transformation of the Fatou set. A nontrivial character of convergence to a fixed point is revealed, which is associated with the appearance of “valleys” on the graph of convergence rates. For two other maps, the monoperiodicity of regimes is significant, the phenomenon of “period doubling” is noted (in one case along the path $39 \to 3$, in the other along the path $17 \to 2$), and the coincidence of the period multiplicity and the number of sleeves of the Julia spiral in a neighborhood of a fixed point is found. A rich illustrative material, numerical results of experiments and summary tables reflecting the parametric dependence of maps are given. Some questions are formulated in the paper for further research using traditional mathematics methods.
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