Strange Attractors
- Fluid Mechanics 3.3

in Physics on Fluid Mechanics · 14 min

Attractors in a Dissipative Flow

For a viscous fluid, the state at time $t$ may be regarded as a point $X(t)$ in a phase space. Formally $X$ contains the velocity field, and possibly temperature or other fields, subject to the constraints and boundary conditions. The Navier-Stokes equation defines an evolution

\[ X(t)=S_tX(0). \]

An invariant set $\mathcal{A}$ is called an attractor if

\[ S_t\mathcal{A}=\mathcal{A} \]

and if all states in some neighborhood of $\mathcal{A}$ approach it as $t\to\infty$. In hydrodynamics this is natural because viscosity dissipates mechanical energy. Even when the phase space is infinite-dimensional, the long-time motion may be confined to a much smaller set.

The simplest attractors are:

  1. A fixed point, corresponding to a steady flow.
  2. A limit cycle, corresponding to a periodic flow.
  3. An invariant torus, corresponding to quasi-periodic flow.

A strange attractor is an attracting invariant set on which the motion is aperiodic and sensitive to initial conditions. It is not merely a high dimensional torus with many frequencies. Its geometry is folded and fractal-like, and nearby trajectories separate exponentially before being folded back by dissipation.

Lyapunov Exponents

Let $X(t)$ and $X(t)+\delta X(t)$ be two nearby trajectories. Linearizing the evolution about $X(t)$ gives

\[ \delta X(t)=DS_t(X(0))\delta X(0). \]

The largest Lyapunov exponent is defined by

\[ \lambda_1 =\lim_{t\to\infty} \frac{1}{t} \log\frac{|\delta X(t)|}{|\delta X(0)|}, \]

for a generic small perturbation direction. If

\[ \lambda_1>0, \]

then nearby trajectories separate exponentially:

\[ |\delta X(t)|\sim |\delta X(0)|e^{\lambda_1t}. \]

This is sensitive dependence on initial conditions. The motion is still deterministic, but long-time prediction from finite-precision initial data is impossible.

For a full spectrum of Lyapunov exponents

\[ \lambda_1\ge\lambda_2\ge\cdots, \]

a dissipative system has negative total volume growth in phase space. In a finite-dimensional truncation this means

\[ \sum_i\lambda_i<0. \]

Thus a strange attractor has both stretching and contraction: some directions expand, while the total phase volume contracts. The combination produces folding.

Stretching and Folding

A useful model is a map which stretches a region, folds it, and places it back into itself. Stretching separates nearby points, while folding keeps the motion bounded. Repetition creates fine-scale structure.

For a Poincare map $P$, a strange attractor is an invariant set satisfying

\[ P(\mathcal{A})=\mathcal{A}, \]

but the map on $\mathcal{A}$ is not equivalent to a rigid rotation on a circle or torus. The dynamics may have infinitely many unstable periodic orbits embedded in the attractor. A typical trajectory does not settle onto one of them, but wanders among their neighborhoods.

This is the essential difference from quasi-periodic motion. On a torus, nearby trajectories neither separate exponentially nor converge. On a strange attractor, nearby trajectories separate exponentially along unstable directions, but the dissipative dynamics keeps them inside the same bounded set.

Dimension of an Attractor

The dimension of a strange attractor need not be an integer. One practical estimate is the Kaplan-Yorke dimension. Suppose the Lyapunov exponents are ordered and let $m$ be the largest integer such that

\[ \sum_{i=1}^m\lambda_i\ge0. \]

Then

\[ D_{\mathrm{KY}} =m+\frac{\sum_{i=1}^m\lambda_i}{|\lambda_{m+1}|}. \]

This number measures how many effective degrees of freedom are active on the attractor. In a hydrodynamic problem the actual phase space is very large, but at moderate $\mathrm{Re}$ the attractor dimension may still be small enough for a reduced description to be meaningful.

For developed turbulence, many Lyapunov exponents may be positive. Then the motion is not only chaotic in time but also complicated in space. The effective number of degrees of freedom grows with $\mathrm{Re}$.

Lorenz System

The Lorenz system is not the full Navier-Stokes equation, but it is a classical low-dimensional model obtained from a truncation of thermal convection. It has the form

\[ \begin{align*} \dot X&=\sigma(Y-X), \nl \dot Y&=rX-Y-XZ, \nl \dot Z&=XY-bZ. \end{align*} \]

Here $r$ is proportional to the Rayleigh number, and $\sigma,b$ are positive parameters. For certain parameter values the system has a strange attractor. It illustrates several features relevant to hydrodynamics:

  1. The equations are deterministic.
  2. The phase volume contracts.
  3. The motion is aperiodic.
  4. The largest Lyapunov exponent is positive.

Thus complicated irregular motion does not require random forcing. It may arise from deterministic nonlinear equations.

Relation with Transition to Turbulence

The Landau-Hopf picture suggests that turbulence is obtained by adding more and more incommensurable frequencies. The discovery of strange attractors shows a different possibility. After only a few instabilities, an invariant torus may lose stability and be replaced by a strange attractor. This is the Ruelle-Takens scenario.

In fluid mechanics this means that a flow can become irregular before a large number of independent frequencies have appeared. The observed signal may have a broadband spectrum, but this does not necessarily mean that it is a linear superposition of many independent oscillations. It may instead be the spectrum of deterministic chaotic motion on a strange attractor.

The distinction is important. Quasi-periodic motion has a discrete Fourier spectrum. Chaotic motion has sensitive dependence and usually produces a continuous component in the spectrum. In experiments the difference is seen by studying return maps, Lyapunov exponents, and the geometry of the reconstructed attractor.

For fully developed turbulence the attractor is expected to be very high-dimensional. Nevertheless, the idea of a strange attractor clarifies the onset of irregular motion: randomness is not required at the level of the equations. The apparent randomness is produced by nonlinear deterministic dynamics together with sensitivity to initial conditions.