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Instability of powers of the golden mean

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  • Manchein, C.
  • Beims, M.W.

Abstract

In this paper we determine the Lyapunov exponents (LEs) for some Lebesgue measure zero periodic orbits from the Gauss map. This map generates the integers of a simple continued fractions representation (CFR). Only periodic orbits related to powers of the golden mean ϕ=(5-1)/2 are considered. It is shown that the LE from the CFR of any power (1/ϕi) (i=±1,±2,…) can be written as a multiple of λϕ, which is the LE related to the golden mean. When i is odd, the LEs are given by λG(xi)=iλϕ, and when i is even the LEs are λG(xi)=iλϕ/2. In general, the LE from the CFR of (1/ϕi) increases as i increases. Additionally, the LE is determined when (1/ϕi) is multiplied by an integer. We also present some examples of the instability of the CFRs related to quark’s mass ratio.

Suggested Citation

  • Manchein, C. & Beims, M.W., 2008. "Instability of powers of the golden mean," Chaos, Solitons & Fractals, Elsevier, vol. 35(2), pages 246-251.
    • Handle: RePEc:eee:chsofr:v:35:y:2008:i:2:p:246-251
    DOI: 10.1016/j.chaos.2007.07.008
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    References listed on IDEAS

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    1. Tanaka, Yosuke, 2007. "The mass spectrum of heavier hadrons and E-infinity theory," Chaos, Solitons & Fractals, Elsevier, vol. 32(3), pages 996-1007.
    2. Marek Crnjac, L., 2005. "Periodic continued fraction representations of different quark’s mass ratios," Chaos, Solitons & Fractals, Elsevier, vol. 25(4), pages 807-814.
    3. Tanaka, Yosuke, 2006. "Elementary particle mass, subquark model and E-infinity theory," Chaos, Solitons & Fractals, Elsevier, vol. 28(2), pages 290-305.
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