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Nonlinear dynamics in the relativistic field equation

Author

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  • Tanaka, Yosuke
  • Mizuno, Yuji
  • Kado, Tatsuhiko
  • Zhao, Hua-An

Abstract

We have investigated relativistic equations and chaotic behaviors of the gravitational field with the use of general relativity and nonlinear dynamics. The space component of the Friedmann equation shows chaotic behaviors in case of the inflation (h=G˙/G>0) and open (ζ=−1) universe. In other cases (h≦0, ζ=0 and ζ=+1), there occurs non-chaotic behaviors. We have shown the following properties of the Friedmann chaos: (1) the sensitive dependence of solutions on the initial values (x0andx˙0) and the parameters (a, b, c and d); (2) the self-similarity of solutions in the x–x˙ plane and the x–ρ plane. We carried out the numerical calculations with the use of the microsoft EXCEL. The self-similarity and the hierarchy structure of the universe have been also discussed on the basis of E-infinity theory.

Suggested Citation

  • Tanaka, Yosuke & Mizuno, Yuji & Kado, Tatsuhiko & Zhao, Hua-An, 2007. "Nonlinear dynamics in the relativistic field equation," Chaos, Solitons & Fractals, Elsevier, vol. 31(5), pages 1054-1075.
    • Handle: RePEc:eee:chsofr:v:31:y:2007:i:5:p:1054-1075
    DOI: 10.1016/j.chaos.2005.11.077
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    1. Tanaka, Yosuke & Mizuno, Yuzi & Kado, Tatsuhiko, 2005. "Chaotic dynamics in the Friedmann equation," Chaos, Solitons & Fractals, Elsevier, vol. 24(2), pages 407-422.
    2. El Naschie, M.S., 2005. "The two-slit experiment as the foundation of E-infinity of high energy physics," Chaos, Solitons & Fractals, Elsevier, vol. 25(3), pages 509-514.
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    1. Tanaka, Yosuke & Mizuno, Yuji & Ohta, Shigetoshi & Mori, Keisuke & Horiuchi, Tanji, 2009. "Nonlinear dynamics in the Einstein–Friedmann equation," Chaos, Solitons & Fractals, Elsevier, vol. 41(2), pages 533-549.
    2. Tanaka, Yosuke & Shudo, Takefumi & Yosinaga, Tetsutaro & Kimura, Hiroshi, 2008. "Relativistic field equations and nonlinear dynamics," Chaos, Solitons & Fractals, Elsevier, vol. 37(4), pages 941-949.
    3. Tanaka, Yosuke & Nakano, Shingo & Ohta, Shigetoshi & Mori, Keisuke & Horiuchi, Tanji, 2009. "Einstein–Friedmann equation, nonlinear dynamics and chaotic behaviours," Chaos, Solitons & Fractals, Elsevier, vol. 42(4), pages 2159-2173.
    4. Ćirić, Ljubomir B. & Ješić, Siniša N. & Ume, Jeong Sheok, 2008. "The existence theorems for fixed and periodic points of nonexpansive mappings in intuitionistic fuzzy metric spaces," Chaos, Solitons & Fractals, Elsevier, vol. 37(3), pages 781-791.

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