4 JULIA SET
In this part, we calculate for the Julia set by
constructing two iterative functions (eq.13 and eq.14)
on the complex plane.
𝑧
= 𝑧
ln𝑧
c
(13)
𝑧
= 𝑧
lnz
c
(14)
Fig.9 and fig.10 display the Julia sets for eq.12
and eq.13 with different values of c, respectively.
(a) (b)
(c) (d)
(e) (f)
Figure 9: Julia set of eq.13. (a) c = 0; (b) c=0.75; (c) c = -
0.15; (d) c=1; (e) c = 0.8+0.6i; (f) c=0.7i.
(a) (b)
(c) (d)
Figure 10: Julia set of eq.14. (a) c = -0.42i; (b) c=3; (c) c =
4; (d) c=2+1i.
5 CONCLUSIONS AND
DISCUSSION
In this study, we propose two new chaotic maps,
which are inspired by information entropy. Test and
analysis results suggest that they are chaotic, with
relatively small positive Lyapunov exponents around
0.014. In addition, we extend the chaotic maps to the
complex plane and obtain the Julia sets.
In the distribution of eq.7, asymmetry seems to
arise from a symmetry map. This might be caused by
the computational software, or the map itself. This
special Frobenius-Perron question remains unknown.
Future work can attempt to calculate the exact
distribution to answer this question and apply these
chaotic maps and Julia sets to new applications in
image encryption, finance, random number
generation and other applications.
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