Quasi-periodic Flow and Frequency Locking
- Fluid Mechanics 3.2
- From Periodic to Quasi-periodic Motion
- Poincare Map
- Frequency Locking
- Arnold Tongues
- Relation with the Landau Picture
From Periodic to Quasi-periodic Motion
After the first instability of a steady flow, the new motion is often periodic. If the first unstable mode has frequency $\omega_1$, then the velocity may be written schematically as
\[ \b{v}(\b{r},t) =\b{V}(\b{r},\omega_1t), \qquad \b{V}(\b{r},\phi+2\pi)=\b{V}(\b{r},\phi). \]
The motion is a limit cycle in the phase space of the fluid. Increasing $\mathrm{Re}$ further may destabilize this periodic motion. A second independent frequency $\omega_2$ may then appear. The corresponding flow has the form
\[ \b{v}(\b{r},t) =\b{V}(\b{r},\phi_1,\phi_2), \qquad \phi_i=\omega_it+\phi_i^0, \]
where
\[ \b{V}(\b{r},\phi_1+2\pi,\phi_2) =\b{V}(\b{r},\phi_1,\phi_2+2\pi) =\b{V}(\b{r},\phi_1,\phi_2). \]
Thus the trajectory lies on a two-dimensional torus. More generally, after $m$ independent oscillatory instabilities one obtains
\[ \b{v}(\b{r},t) =\b{V}(\b{r},\phi_1,\cdots,\phi_m), \qquad \phi_i=\omega_it+\phi_i^0, \]
with $\b{V}$ periodic in each phase. The motion is then on an $m$-torus.
If the frequencies are rationally independent, i.e. if
\[ n_1\omega_1+\cdots+n_m\omega_m=0,\qquad n_i\in\mathbb{Z}, \]
implies $n_1=\cdots=n_m=0$, the orbit never closes. It is dense on the torus. The motion is not periodic, but it is not random: every value is still determined by the phases.
The Fourier spectrum of such a motion is discrete. Since $\b{V}$ is periodic in each phase, it may be expanded as
\[ \b{V}(\b{r},\phi_1,\cdots,\phi_m) =\sum_{\b{n}\in\mathbb{Z}^m} \b{V}_{\b{n}}(\b{r}) e^{i(n_1\phi_1+\cdots+n_m\phi_m)}. \]
Hence
\[ \b{v}(\b{r},t) =\sum_{\b{n}\in\mathbb{Z}^m} \b{V}_{\b{n}}(\b{r}) e^{i(n_1\omega_1+\cdots+n_m\omega_m)t}. \]
The observed frequencies are all integer linear combinations of the basic frequencies.
Poincare Map
A periodic flow can be studied by looking once per period. If the period is $T$, the time-$T$ map
\[ P:X(0)\mapsto X(T) \]
is the Poincare map of the motion, where $X$ denotes the state of the fluid. A stable periodic motion is a stable fixed point of $P$.
When this fixed point loses stability through a complex pair of multipliers, the Poincare map may acquire an invariant closed curve. In the continuous flow this corresponds to a two-torus. Motion on this invariant curve is described by an angle variable
\[ \theta_{n+1}=\theta_n+2\pi\rho \pmod{2\pi}, \]
where $\rho$ is the rotation number. If $\rho$ is rational, the orbit is periodic. If $\rho$ is irrational, the orbit is quasi-periodic and dense on the invariant curve.
This gives another way to understand the appearance of a second frequency: the original period gives one phase, and the rotation on the Poincare section gives the second phase.
Frequency Locking
The preceding discussion assumes that the frequencies remain independent. In nonlinear hydrodynamics this need not be true. Suppose two frequencies are close to a rational relation
\[ q\omega_2-p\omega_1\simeq0, \qquad p,q\in\mathbb{Z}. \]
Then the slowly varying resonant phase is
\[ \theta=q\phi_2-p\phi_1. \]
Nonlinear coupling may produce an equation of the form
\[ \dot{\theta} =\Delta-K\sin\theta+\cdots, \qquad \Delta=q\omega_2-p\omega_1. \]
This is the standard phase-locking equation. If
\[ |\Delta|<|K|, \]
there is a stable fixed value of $\theta$. Then
\[ q\omega_2-p\omega_1=0 \]
in the observed motion, even if the uncoupled frequencies were only close to this relation. The flow becomes periodic rather than quasi-periodic. This is called frequency locking.
If $|\Delta|>|K|$, the phase $\theta$ drifts, and the two frequencies remain independent. Thus the transition between quasi-periodicity and locking is a competition between detuning and nonlinear coupling.
Arnold Tongues
Let $\lambda$ be a parameter measuring the strength of the nonlinear coupling. Near a rational ratio $p/q$, the locked region in parameter space has the shape of a tongue:
\[ |q\omega_2-p\omega_1|<K(\lambda). \]
These regions are called Arnold tongues. Their widths are small for weak coupling and grow as the nonlinearity increases. Low-order resonances such as $1:1$, $1:2$, and $2:3$ are usually the most visible, because their coupling terms occur at relatively low order in the amplitudes.
This explains why a hydrodynamic system whose linear modes have many possible frequencies may nevertheless show a simple periodic response. The nonlinear terms can select a rational relation between the phases and suppress the independent drift.
Relation with the Landau Picture
The Landau picture of transition to turbulence may be described as a successive appearance of more and more independent frequencies:
\[ \omega_1,\qquad \omega_1,\omega_2,\qquad \omega_1,\omega_2,\omega_3,\qquad \cdots . \]
With many incommensurable frequencies, the motion becomes very complicated and its spectrum contains many lines of the form
\[ n_1\omega_1+n_2\omega_2+\cdots+n_m\omega_m. \]
However, this construction alone does not fully explain developed turbulence. First, frequency locking can reduce the number of independent frequencies by forcing rational relations. Second, nonlinear dynamics can destroy invariant tori and produce a strange attractor. Thus quasi-periodic motion is one possible route away from laminar flow, but it is not the only route to turbulence.
