Bifurcations of the Complex Dynamical System <I>Z</I><SUB><I>n</I>+1</SUB>=ln (<I>Zn</I>)+<I>C</I>

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  • Bifurcations of the Complex Dynamical System Zn+1=ln(Zn)+C
  • Bifurcations of the Complex Dynamical S
  • Bifurcations of the Complex Dynamical System<i>Z</i><sub><i>n</i>+1</sub>=ln (<i>Z</i><i>n</i>)+<i>C</i>

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Abstract

Mandelbrot sets of the complex logarithmic map (Remark: Graphics omitted.) have complicated and miscellaneous structures with various periods for 0≤arg(Zn)<2π, and they are displayed by computer graphics. Period-adding sequence can be clearly found near the boundary with fixed points region. The Hopf bifurcation occurs at the points on the boundary, and quasi-periodic chaos appears Cantor-like on a line segment of the complex C-plane where |Zn|=1 is satisfied. Lyapunov exponent is also shown by contour maps in the complex C-plane.

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