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Visual presentation of dynamic systems with hyperbolic planar symmetry

Author

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  • Chen, Ning
  • Li, Zichuan
  • Jin, Yuanyuan

Abstract

Hyperbolic symmetric mappings defined on hyperbolic tilings are investigated. Ljapunov exponents of the dynamic systems are computed with the Euclidean distance. The parameter combinations with great impact on the characteristics of the dynamic systems were chosen as the window coordinates for construction of generalized Mandelbrot sets. The accelerated direct search algorithm is used to search for the set of the critical points in the fundamental region. The parameter space is separated into chaotic, periodic and mixed regions by the Ljapunov exponents of the critical points. The generalized Mandelbrot sets (M-set), which are the cross-sections of the parameter space, were constructed. Three different types of hyperbolic symmetry patterns, which are chaotic attractors, filled-in Julia sets and mixed images composed of an attractor and a filled-in Julia set from the same set of parameters, were created by using parameters from this kind of M-sets.

Suggested Citation

  • Chen, Ning & Li, Zichuan & Jin, Yuanyuan, 2009. "Visual presentation of dynamic systems with hyperbolic planar symmetry," Chaos, Solitons & Fractals, Elsevier, vol. 40(2), pages 621-634.
    • Handle: RePEc:eee:chsofr:v:40:y:2009:i:2:p:621-634
    DOI: 10.1016/j.chaos.2007.08.020
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    References listed on IDEAS

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    1. Chen, Ning & Meng, Fan Yu, 2007. "Critical points and dynamic systems with planar hexagonal symmetry," Chaos, Solitons & Fractals, Elsevier, vol. 32(3), pages 1027-1037.
    2. El Naschie, M.S., 2006. "Holographic dimensional reduction: Center manifold theorem and E-infinity," Chaos, Solitons & Fractals, Elsevier, vol. 29(4), pages 816-822.
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    1. Chen, Ning & Hao, Ding & Tang, Ming, 2009. "Automatic generation of symmetric IFSs contracted in the hyperbolic plane," Chaos, Solitons & Fractals, Elsevier, vol. 41(2), pages 829-842.

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