5.1: Windows of periodicity scaling, the "linear" approximation
- Page ID
- 102234
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Windows of periodicity scaling
Windows of periodicity
It is a commonly observed feature of chaotic dynamical systems [1] that, as a system parameter is varied, a stable period-n orbit appears (by a tangent bifurcation) which then undergoes a period-doubling cascade to chaos and finally terminates via a crisis. This parameter range between the tangent bifurcation and the final crisis is called a period-n window. Note, that the central part of the picture to the left is similar to the whole bifurcation diagram (see also at the bottom of the page).
For c = -1.75 in the period-3 window stable and unstable period-3 orbits appear by a tangent bifurcation. The stable period-3 orbit is shown to the left below. If N = 3 is set (see to the right), we get 8 intersections (fixed points of fco3) which correspond to two unstable fixed points and 6 points of the stable and unstable period-3 orbits of fc .
On the left picture the stable period-3 orbit goes through two "linear" and one central quadratic regions of the blue parabola. Therefore in the vicinity of x = 0 the map fco3 is "quadratic-like" and iterations of the map repeat bifurcations of the original quadratic map fc . This sheds light on the discussed similarity of windows of periodicity.
fcon map renormalization. The "linear" approximation
Consider a period-n window. Under iterations the critical orbit consecutively cycles through n narrow intervals S1 → S2 → S3 → ... → S1 each of width sj (we choose S1 to include the critical point x = 0).
Following we expand fcon(x) for small x (in the narrow central interval S1 ) and c near its value cc at superstability of period-n attracting orbit. We see that the sj are small and the map in the intervals S2 , S2 , ... Sn may be regarded as approximately linear; the full quadratic map must be retained for the central interval. One thus obtains
xj+n ~ Λn [xj2 + β(c - cc )] ,
where Λn = λ2 λ3 ...λn is the product of the map slopes, λj = 2xj in (n-1) noncentral intervals and
β = 1 + λ2-1 + (λ2 λ3 )-1 + ... + Λn-1 ~ 1
for large Λn . We take Λn at c = cc and treat it as a constant in narrow window.
Introducing X = Λn x and C = β Λn2 (c - cc ) we get quadratic map
Xj+n ~ Xn2 + C
Therefore the window width is ~ (9/4β)Λn-2 while the width of the central interval scales as Λn-1. This scaling is called fcon map renormalization.
Numbers
For the biggest period-3 window Λ3 = -9.29887 and β = 0.60754. So the central band is reduced ~ 9 times and reflected with respect to the x = 0 line as we have seen before. The width of the window is reduced β Λ32 = 52.5334 times. On the left picture below you see the whole bifurcation diagram of fc . Similar image to the right is located in the centeral band of the biggest period-3 window and is stretched by 9 times in the horizontal x and by 54 times in the vertical c directions.
