Transition to Turbulence by Period Doubling
- Fluid Mechanics 3.4
- Period Doubling
- Poincare Map Formulation
- Cascade
- Spectrum
- Logistic Map as a Model
- Hydrodynamical Interpretation
Period Doubling
Another route from regular motion to irregular motion is period doubling. Suppose that after the first instability of a steady flow the observed motion is periodic with period $T$:
\[ \b{v}(\b{r},t+T)=\b{v}(\b{r},t). \]
As $\mathrm{Re}$ is increased, this periodic motion may lose stability and be replaced by a new periodic motion with period $2T$:
\[ \b{v}(\b{r},t+2T)=\b{v}(\b{r},t), \qquad \b{v}(\b{r},t+T)\ne\b{v}(\b{r},t). \]
The original frequency
\[ \omega=\frac{2\pi}{T} \]
is then accompanied by the subharmonic frequency $\omega/2$. Further increase of $\mathrm{Re}$ may produce periods
\[ 4T,\;8T,\;16T,\;\cdots . \]
Thus the spectrum contains successive subharmonics
\[ \frac{\omega}{2},\quad \frac{\omega}{4},\quad \frac{\omega}{8},\quad\cdots . \]
After infinitely many doublings, the period is no longer finite. This gives a possible transition to aperiodic deterministic motion.
Poincare Map Formulation
Let $P_{\mathrm{Re}}$ be a Poincare map of the flow. A periodic solution of period $T$ is a fixed point
\[ P_{\mathrm{Re}}(X_\ast)=X_\ast. \]
Linear stability of this periodic orbit is determined by the eigenvalues of the derivative
\[ DP_{\mathrm{Re}}(X_\ast). \]
These eigenvalues are the Floquet multipliers. A period-doubling bifurcation occurs when one real multiplier crosses $-1$:
\[ \lambda=-1. \]
After the crossing, the fixed point of $P_{\mathrm{Re}}$ is unstable, but there may be a stable orbit of period two:
\[ P_{\mathrm{Re}}(X_1)=X_2,\qquad P_{\mathrm{Re}}(X_2)=X_1. \]
In the original continuous-time flow this is a periodic orbit of period $2T$.
Near the bifurcation, the dynamics along the critical direction can often be reduced to a one-dimensional normal form
\[ x_{n+1}=-(1+\epsilon)x_n+ax_n^3+\cdots . \]
The fixed point $x=0$ changes stability at $\epsilon=0$. A nonzero period-two orbit satisfies
\[ f_{\epsilon}(x)=-x. \]
Using the normal form,
\[ -(1+\epsilon)x+ax^3=-x, \]
and hence
\[ x^2=\frac{\epsilon}{a}. \]
Thus, when $\epsilon/a>0$, the amplitude of the doubled oscillation grows as
\[ |x|\propto \sqrt{|\epsilon|}. \]
This square-root law is analogous to the amplitude scaling near a soft hydrodynamic instability.
Cascade
Let the successive period-doubling thresholds be
\[ \mathrm{Re}_1,\mathrm{Re}_2,\mathrm{Re}_3,\cdots . \]
At $\mathrm{Re}_1$ the period becomes $2T$, at $\mathrm{Re}_2$ it becomes $4T$, and so on. In many one-dimensional maps these thresholds accumulate at a finite value
\[ \mathrm{Re}_{\infty} =\lim_{n\to\infty}\mathrm{Re}_n. \]
For $\mathrm{Re}<\mathrm{Re}_\infty$, the motion is still periodic, although the period may be very large. At $\mathrm{Re}=\mathrm{Re}_\infty$, the period-doubling cascade has no finite period left.
The intervals between successive thresholds shrink geometrically. For a wide class of one-hump maps,
\[ \delta =\lim_{n\to\infty} \frac{\mathrm{Re}_n-\mathrm{Re}_{n-1}} {\mathrm{Re}_{n+1}-\mathrm{Re}_n} \simeq4.6692016. \]
This number is the Feigenbaum constant. Its universality is not a direct consequence of the Navier-Stokes equation; it belongs to the reduced return map when that map has the appropriate structure. Nevertheless, it explains why very different physical systems can show similar period-doubling scaling.
There is also a spatial scaling constant
\[ \alpha\simeq2.5029079, \]
which describes the relative size of structures in the return map. These constants are useful diagnostics when experimental data show a clear period-doubling sequence.
Spectrum
Before the first doubling, a measured quantity $u(t)$ has a Fourier expansion with frequencies
\[ n\omega,\qquad n\in\mathbb{Z}. \]
After one doubling, the fundamental frequency is $\omega/2$, so the spectrum contains
\[ \frac{n\omega}{2}. \]
After $m$ doublings, the fundamental frequency is
\[ \frac{\omega}{2^m}. \]
Thus increasingly low subharmonics appear. Near the accumulation point the spectral lines become very dense on finite frequency intervals. Beyond the accumulation point the motion may become chaotic, and the spectrum usually develops a continuous component.
This distinguishes period doubling from quasi-periodic transition. In quasi-periodic motion, new independent frequencies appear. In period doubling, the new frequencies are subharmonics of the old one.
Logistic Map as a Model
A simple model which shows the period-doubling cascade is the logistic map
\[ x_{n+1}=rx_n(1-x_n). \]
For small $r$ the attracting set is a fixed point. As $r$ increases, this fixed point loses stability and a stable period-two orbit appears. Further increase produces period-four, period-eight, and so on.
This map is not a fluid equation. Its role is to model the return map near a low-dimensional transition. If a hydrodynamic Poincare map can be reduced to a similar one-dimensional map, the same cascade may appear in the fluid signal.
Hydrodynamical Interpretation
In a fluid experiment, period doubling is observed by measuring a quantity such as velocity at a fixed point, pressure drop, or heat flux. If the original oscillation has period $T$, then the time series begins to alternate between two unequal cycles. This is the period-$2T$ state. Later it may alternate among four cycles, then eight, and so on.
Such behavior has been observed in several hydrodynamic systems, including thermal convection, Taylor-Couette flow, and wakes. The exact thresholds are not universal; they depend on geometry, boundary conditions, and the chosen control parameter. What may be universal is the scaling of the return map once the dynamics has reduced to the appropriate form.
Period doubling is therefore a route to turbulence, but not the only one. Other flows pass through quasi-periodic motion, intermittency, or a direct transition to a strange attractor. In high-dimensional fluid systems several routes may coexist. The useful point is that complicated turbulent motion can arise from a definite sequence of deterministic bifurcations rather than from external randomness.
