Computational Complex Dynamics of $\displaystyle{f_{α, β, γ, δ}(z)=\frac{αz + β}{γz^2 +δz}}$

The dynamics of the family of maps $\displaystyle{f_{α, β, γ, δ}(z)=\frac{αz + β}{γz^2 +δz}}$ in complex plane is investigated computationally. This dynamical system $z_{n+1}=f_{α, β, γ, δ}(z_n)=\frac{αz_n + β}{γz_n^2 +δz_n}$ has periodic solutions with higher periods which was absent in the real line scenario. It is also found that there are chaotic fractal and non-fractal like solutions of the dynamical systems. A few special cases of parameters are also have been taken care.

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