Lecture 6

● Chaos● Lorenz equations and Malkus' waterwheel● Some properties of the Lorenz Eq.'s● Lorenz Map● Towards definitions of:

– Chaos, – Attractors and strange attractors– Transient chaos

Lorenz Equations● Lorenz (1963), drastically simplified model of

convection rolls in the atmosphere

● Parameters: σ (Prandtl number),r (Rayleigh number), b >0

● Lorenz found a “paradox”:● Ruled out all known possibilities for long-term

behaviour (no stable FP, no stable limit cycles)● Also showed: all trajectories confined to bounded

region and are attracted to set of zero measure, so what could this set be?

x = σ( y− x)y = rx− y−xzz = xy−bz

Malkus' Waterwheel

● Very slow inflow: gravity never overcomes friction, no motion● Faster inflow: top cup heavy enough to get motion starting, steady motion in one direction or other (symmetry)

● Even faster inflow: Can destabilize rotation. Wheel rotates for a few turns in one direction, then some cups get too full and wheel has not enough inertia to carry them over the top. Wheel slows down for a while and may even reverse direction -> (deterministic) chaos.

Waterwheel (2)

● Illustration of Malkus' waterwheel

( http://www.youtube.com/watch?v=JxKAxojliKY )

Properties of the Lorenz Eq's. (1)

● Two non-linearities ~ xz and xy● Symmetry: (x,y) -> (-x, -y)

● If (x,y,z) is solution, so is (-x,-y,z)

● Lorenz system is dissipative, Volumes in phase space contract

x = σ( y− x)y = rx− y−xzz = xy−bz

Dissipative Systems (1) ● How do volumes in phase space evolve for 3d

systems dx/dt=f(x)?

V(t+dt)=V(t)+(volume covered by patches of surface, integrated over all patches)V (t+dt)=V (t)+∫S

f dt ∘n dA

Dissipative Systems (2)

V =V (t+dt)−V (t )

dt=∫S

f ∘ndA

V =∫V∇∘ f dV

● If div f < 0 a 3d system dx/dt=f(x) is dissipative

● Lorenz system:

∇ ∘f =∂

∂ x(σ ( y−x ))+

∂

∂ y(rx− y− xz)+

∂

∂ z(xy−bz )

=−σ−1−b<0

V (t )=V (0)exp((−σ−1−b) t)

● Since div f = const.

Dissipative Systems (3)

● All trajectories are attracted to limiting sets of zero measure

● There cannot be any quasiperiodic solutions● They would have to be on the surface of a torus

and this torus would have to be invariant under the flow, i.e. have constant volume in phase space. -> contradiction.

● Lorenz system cannot have repelling FP's or repelling closed orbits● Repellers are sources

of phase space volume

-> contradiction.

Lorenz System – Fixed Points

● Two types of fixed points● Origin (x*,y*,z*)=(0,0,0) ~ motionless state of

waterwheel● For r>1 symmetric pair C+ and C- ~ left and right

rotations of waterwheel

● Pitchfork bifurcation for r->1

● Linear stability of origin

(x* , y* , z*)=(±√b(r−1) ,±√b(r−1) ,r−1)

(xyz )=(

−σ σ 0r −1 00 0 −b)(

xyz )

Lorenz System – Fixed Points

● r>1: origin is saddle (2 stable, 1 unstable direction)● r<1: origin is stable node● r<1 origin is globally stable, can construct

Liapunov function

λ=−σ+1

2±√(σ+1)2

4−1+r

V (x , y , z)=x2/σ+ y2

+z2

1/2 V=1 /σ x x+ y y+z z

=( yx−x2)+(rxy− y2

−xzy )+(zxy−bz 2)

=(r+1)xy−x2− y2

−bz 2

=−[ x−r+1

2y ]

2

−[1−(r+1

2)

2

] y2−bz2

Lorenz System – Fixed Points (2)

● Stability of C+ and C-

● Stable for● Lose stability in a subcritical Hopf bifurcation

● Below rH: unstable limit cycles (saddle cycles)

1<r<rH=σ(σ+b+3)/ (σ−b−1)

Situation so far ...

● What happens for r>rH? No stable objects?

● Trajectories repelled to infinity? (No, remain inside acertain ellipsoid, seminar)

● Stable limit cycles we are not aware of? (Lorenz gave argument that such cycles are unstable)Trajectories confined to bounded set of zero volume, repelled from various unstable objects, and move on this set forever without intersecting ...

Lorenz Attractor

● Numerical integration for r>rH just above r

H

● Aperiodic● Seems to settle on

“very thin” set in 3d

-> Strange Attractor

Two trajectories with ICsThat differed by one 100 billionth in Z direction!

Lorenz Map

zn+1

zn

● Why are there no periodic so- lutions for r>r

H ?

● zn nth maximum of z coordinate

on attractor● Plot is not actually a curve, does have some “thickness” (works because attractor has dim. ~2)

● Graph satisfies |f'(z)|>1 -> previous lecture about maps, any limit cycle must be un- stable!

Dependence on Initial Conditions

● Attractor exhibits sensitive dependence on ICs

● Consider x(t) and x(t)+δ(t) on attractor and δ0 is tiny

separation vector

● Numerically one finds: ||δ(t)||~||δ0|| exp(λ t)

● Plot ln ||δ|| vs. t -> “straight line”– Never exactly straight, strength of exp. Divergence varies along

attractor– Upper limit given by dimensions of attractor

Liapunov Exponents● λ is often called the Liapunov exponent

● For an n-dimensional system we could define n Liapunov exponents for perturbations along the n different axis; for large t divergence is dominated by largest exponent λ

● λ depends on position on attractor (should average over all trajectories)

● λ defines a time horizon over which a system can be predicted– Let's assume a is a tolerance of measurement, if ||δ||<a

we say a prediction is acceptable, ||δ||>a after

t predict O(1/λ lna

∥δ0∥)

Defining Chaos● No universally accepted definition, but three

important ingredients:● DEF: Chaos is aperiodic long-term behaviour

in a deterministic system that exhibits sensitive dependence on initial conditions.● Aperiodic long-term behaviour: a set of trajectories

exists that does not settle on FPs, periodic or quasi-periodic orbits; should not be too “rare”

● Deterministic: no random or noisy inputs, behaviour arises from system's nonlinearities

● Sensitive dependence ...: Systems has a positive Liapunov exponent

● What about dx/dt=x ?

Attractors

● Loosely speaking: “an attractor is a set to which all neighbouring trajectories converge”

● DEF: An attractor is a closed set A with

● A is invariant ( x(t1) in A -> x(t) in A for all t)

● A attracts an open set of ICs– There exists an open set U⊃A such that distance

between x(t) and A tends to zero for any x ∈U; largest U is called the basin of attraction of A

● A is minimal (no proper subset that fulfills above conditions)

Attractors (2)

● Consider

● Is I={x|-1<=x<=1} an invariant set?● Does I attract an open set of ICs?● Is I an attractor?

x= x−x3

y=− y

Attractors (3)

● Actually ... it has not yet been shown that the Lorenz attractor is truly an attractor (i.e. minimal)in this sense!

● Strange attractors● = attractors with sensitive dependence on initial

conditions● (used to be “strange” because they are often fractal,

but today more emphasis on sensitive dependence on ICs)

Transient Chaos

● System may initially exhibit “chaos” (i.e. aperiodic behaviour and sensitivity to initial conditions) and settle down on a periodic orbit or fixed point later on.

● Deterministic systems can be unpredictable even though final states are very simple!● Games of “chance”, i.e. rolling dice (always ends up

in one of the six equilibrium positions, but final position depends sensitively on initial conditions ...)

Summary

● What you should remember:● Chaos● Liapunov exponents● Attractors + Strange attractors● Transient chaos

● Seminar:● Talks!● Using synchronized chaos to send messages