4.6: Period doubling bifurcations
- Page ID
- 101796
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Period doubling bifurcation for real quadratic maps
For c < 1/4, after the tangent bifurcation, the x1 fixed point of the quadratic map (the left intersection of fc and the green line) stays attracting while its multiplier |λ1 | < 1 . For c < -3/4 it is λ1 = 1 - (1 - 4c)½ < -1 so the fixed point becomes repelling and an attracting period-2 orbit appears. This phenomenon is called the period doubling bifurcation.
To watch the birth of the period-2 orbit consider the two-fold iterate f o2(x) = f(f(x)) of the quadratic map. It is evident that two fixed points of fc are the fixed points of fco2. Moreover as since for these points (fco2)' = (fc')2, therefore for -3/4 < c < 1/2 the left fixed point of fco2 is attracting (to the left below). At c = -3/4 it loses stability and two new intersections of fco2 with the green line simultaneously appear (see to the right). These x3 , x4 points make a period-2 orbit. When this cycle appears f o2(x3 )' = f o2(x4 )' = 1 and the slope decrease as c is decreased.
A period doubling bifurcation is also known as a flip bifurcation, as since the period two orbit flips from side to side about its period one parent orbit. This is because f' = -1 .
At c =-5/4 the cycle becomes unstable and a stable period-4 orbit appears. Period doubling bifurcation is called also the pitchfork bifurcation (see below).
Period doubling bifurcation on complex plane
These pictures illustrate this process on complex plane. While c is changed from c = 0 to c = -3/4 (inside the main cardioid) attractor z1 moves from 0 to the parabolic point p = -1/2 (with multiplier λ = -1 ). Two points of an unstable period 2 orbit are
z3,4 = -1/2 ± i t, t = (3/4 + c)½ > 0.
Therefore they move towards the point p too from above and below. At c = -3/4 attractor meets the repelling orbit and they merge into one parabolic point. For c < -3/4, since c leaves the main cardioid and gets into the biggest (1/2) bulb, attractor turns into repeller and the unstable period-2 orbit becomes attracting with two points
z3,4 = -1/2 ± t, t = (-3/4 - c)½.
The right picture above is disconnected Cantor dust. We get it if we will go up after crossing the p point.