On the spectral density of a class of chaotic time series

The purpose of this paper is to show explicitly the spectral density function of the stationary stochastic process determined by a certain class of two‐dimensional maps F α defined below (α is a parameter in (0, 1)), the random variable φ( x , y ) = x and the invariant probability described below. We first define the transformation T α : [0, 1]←[0, 1] given by Tα ( x ) = { x /α if 0 ≤ x < α and (α( x −α)/1 −α) if α≤ x ≤ 1 where α∈ (0, 1) is a constant. The map T α describes a model for a particle (or the probability of a certain kind of element in a given population) that moves around, in discrete time, in the interval [0, 1]. The results presented here can be stated either for T α or for F α but we prefer the latter. The results for T α can be obtained from the more general setting described by F α . The map F α is defined from K = ([0, 1]× (0, α)) ∨ ([0, α]×[α, 1]) ⊂ ;R; 2 to itself and is given by F α (x, y) = (T α (x), G α (x, y)) for (x, y) ∈K , where Gα ( x , y ) = {α y if 0 ≤ x < α and α + ((1 −α)/α) y if α≤ x < 1. The spectral density function of the stationary process with probability ν (invariant for F α and absolutely continuous with respect to the Lebesgue measure) Z t = X t + ξ t = φ{ F t α ( X 0 , Y 0 )} + ξ t for t ∈ Z where ( X 0 , Y 0 ) ∈ R 2 and ξ t } t∈Z is a white noise process, is given explicitly (Theorem 1) by fZ(λ) = f X (λ) + (σ 2 ξ /2π) = (1/2πvar( X t ))[γ{exp( i λ)}− C (0)] + (σ 2 /2π) for all λ∈[0, 2π), where var( X t ) = (α 2 −α + 1)(α 2 − 5α + 5){12(2 −α) 2 } −1 , γ is given by Equation (2.10) of Proposition 5 and C (0) = (1 + α 2 −α 3 ){3(2 −α)} −1 . We also estimate the parameter α based on a time series.