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THREE DIMENSIONAL

SYSTEMS

Lecture 6: The Lorenz

Equations

6. The Lorenz (1963) Equations

The Lorenz equations were originally derived by Saltzman (1962) as a ‘minimalist’ model of thermal convection in a box

x˙ = σ(y − x) (1) y˙ = rx − y − xz (2) z˙ = xy − bz (3)

where σ (“Prandtl number”), r (“Rayleigh number”) and b are parameters (> 0). These equations also arise in studies of convection and instability in planetary atmospheres, mod- els of lasers and dynamos etc. Willem Malkus also devised a water-wheel demonstration...

8.1 Simple properties of the Lorenz Equations

• Nonlinearity - the two nonlinearities are xy and xz
• Symmetry - Equations are invariant un- der (x, y) → (−x, −y). Hence if (x(t), y(t), z(t)) is a solution, so is (−x(t), −y(t), z(t))
• Volume contraction - The Lorenz system is dissipative i.e. volumes in phase-space contract under the flow
• Fixed points - (x∗, y∗, z∗) = (0, 0 , 0) is a fixed point for all values of the param- eters. For r > 1 there is also a pair of fixed points C±^ at x∗^ = y∗^ = ±

√ b(r − 1), z∗^ = r −1. These coalesce with the origin as r → 1 +^ in a pitchfork bifurcation

Linear stability of the origin

Linearization of the original equations about the origin yields

x˙ = σ(y − x) y˙ = rx − y z ˙ = −bz

Hence, the z-motion decouples, leaving ( x ˙ y ˙

)

( −σ σ r − 1

) ( x y

)

with trace τ = −σ − 1 < 0 and determinant ∆ = σ(1 − r).

For r > 1, origin is a saddle point since ∆ < 0 (see Lecture 4)

For r < 1, origin is a sink since τ 2 − 4∆ = (σ + 1)^2 − 4 σ(1 − r) = (σ − 1)^2 + 4στ > 0 → a stable node.

Actually for r < 1 it can be shown that every trajectory approaches the origin as t → ∞ the origin is globally stable, hence there can be no limit cycles or chaos for r < 1.

6.2 Chaos on a Strange Attractor

Lorenz considered the case σ = 10, b = 8/ 3 , r = 28 with (x 0 , y 0 , z 0 ) = (0, 1 , 0).

NB rH = σ(σ + b + 3)/(σ − b − 1) ' 24 .74, hence r > rH. The resulting solution y(t) looks like...

Fig. 6.2.

After an initial transient, the solution settles into an irregular oscillation that persists as t → ∞ but never repeats exactly. The motion is aperiodic.

Lorenz discovered that a wonderful structure emerges if the solution is visualized as a tra- jectory in phase space. For instance, when x(t) is plotted against z(t), the famous but- terfly wing pattern appears

Fig. 6.2.

• The trajectory appears to cross itself re- peatedly, but that’s just an artifact of projecting the 3-dimensional trajectory onto a 2-dimensional plane. In 3-D no cross- ings occur!

What is the geometric structure of the strange attractor?

The uniqueness theorem means that trajec- tories cannot cross or merge, hence the two surfaces of the strange attractor can only ap- pear to merge.

Lorenz concluded that “there is an infinite complex of surfaces” where they appear to merge. Today this “infinite complex of sur- faces” would be called a FRACTAL.

A fractal is a set of points with zero volume but infinite surface area.

Fractals will be discussed later after a closer look at chaos...

Exponential divergence of nearby trajectories

The motion on the attractor exhibits sensi- tive dependence on initial conditions. Two trajectories starting very close together will rapidly diverge from each other, and there- after have totally different futures. The prac- tical implication is that long-term prediction becomes impossible in a system like this, where small uncertainties are amplified enormously fast.

Suppose we let transients decay so that the trajectory is “on” the attractor. Suppose

x(t) is a point on the attractor at time t,

and consider a nearby point, say x(t) + δ(t),

where δ is a tiny separation vector of initial length ||δ 0 || = 10−^15 , say

Fig. 6.2.

In numerical studies of the Lorenz attractor, one finds that ||δ(t)|| ∼ ||δ 0 ||eλt, where λ ' 0 .9.

Hence neighbouring trajectories separate ex- ponentially fast!

Equivalently, if we plot ln ||δ(t)|| versus t, we find a curve that is close to a straight line with a positive slope λ.

Fig. 6.2.

Note...

• The curve is never straight, but has wig- gles since the strength of exponential di- vergence varies somewhat along the at- tractor.
• The exponential divergence must stop when the separation is comparable to the “di- ameter” of the attractor - the trajecto- ries cannot get any further apart! (curve saturates for large t)
• The number λ is often called the Lya- punov exponent, though this is somewhat sloppy terminology....

When a system has a positive Lyapunov ex- ponent, there is a time horizon beyond which prediction will break down....

Fig. 6.2.

Suppose we measure the initial conditions of an experimental system very accurately. Of course no measurement is perfect - there is always some error ||δ 0 || between our estimate and the true initial state. After a time t the discrepancy grows to ||δ(t)|| ∼ ||δ 0 ||eλt.

Let a be a measure of our tolerance, i.e. if a prediction is within a of the true state, we consider it acceptable. Then our prediction becomes intolerable when ||δ(t)|| ≥ a, i.e. af- ter a time

thorizon ∼ O

( 1 λ

ln a ||δ||

)

The logarithmic dependence on ||δ 0 || is what hurts....!

Example Suppose a = 10−^3 , ||δ 0 || = 10−^7.

⇒ thorizon = 4 ln 10 λ

If we improve the initial error to ||δ 0 || = 10−^13 ,

⇒ thorizon = 10 ln 10 λ i.e. only 10/4 = 2.5 times longer!

3. Sensitive dependence on initial conditions means that nearby trajectories diverge ex- ponentially fast, i.e. the system has at least one positive Lyapunov exponent.

Some people think that chaos is just a fancy word for instability. For example, the system x˙ = x is deterministic and shows exponen- tial separation of nearby trajectories. How- ever, we should not consider this system to be chaotic!

Trajectories are repelled to infinity, and never return. Hence infinity is a fixed point of the system, and ingredient 1. above specifically excludes fixed points!

Defining “attractor” and “strange attractor”

The term attractor is also difficult to define in a rigorous way. Loosely, an attractor is a set of points to which all neighbouring tra- jectories converge. Stable fixed points and stable limit cycles are examples.

More precisely, we define an attractor to be a closed set A with the following properties:

1. A is an invariant set: any trajectory x(t)

that starts in A stays in A for all time.

2. A attracts an open set of initial condi- tions: there is an open set U containing

A such that if x(0) ∈ U , then the dis-

tance from x(t) to A tends to zero as

t → ∞. Hence A attracts all trajectories that start sufficiently close to it. The largest such U is called the basin of at- traction of A.