Instability of powers of the golden mean

C. Manchein and M.W. Beims

Chaos, Solitons & Fractals, 2008, vol. 35, issue 2, 246-251

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.

Date: 2008
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Persistent link: https://EconPapers.repec.org/RePEc:eee:chsofr:v:35:y:2008:i:2:p:246-251

DOI: 10.1016/j.chaos.2007.07.008

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