Lecture 6. Lorenz equations and Malkus' waterwheel Some properties of the Lorenz Eq.'s Lorenz Map Towards definitions of:

Size: px

Start display at page:

Download "Lecture 6. Lorenz equations and Malkus' waterwheel Some properties of the Lorenz Eq.'s Lorenz Map Towards definitions of:"

Brian Prosper Wood
5 years ago
Views:

1 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

2 Lorenz Equations Lorenz (1963), drastically simplified model of convection rolls in the atmosphere ẋ = σ ( y x) ẏ = rx y xz ż = xy bz 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?

3 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.

4 Waterwheel (2) Illustration of Malkus' waterwheel ( )

5 Properties of the Lorenz Eq's. (1) ẋ = σ ( y x) ẏ = rx y xz ż = xy bz 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

6 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

7 Dissipative Systems (2) V (t+dt) V (t ) V = = dt S f nda V = V f dv If div f < 0 a 3d system dx/dt=f(x) is dissipative Lorenz system: f = (σ ( y x))+ x y = σ 1 b< 0 Since div f = const. V (t )=V (0)exp(( σ 1 b)t) (rx y xz)+ z (xy bz)

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

8 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.

9 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 (x *, y *, z * )=(± b(r 1),± b(r 1),r 1) Pitchfork bifurcation for r->1 Linear stability of origin ( ẋ ẏ ) ( σ σ 0 = r 1 0 ż b)( x ) y 0 0 z

10 Lorenz System Fixed Points λ= σ +1 2 ± (σ+1)2 1+r 4 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 V (x, y, z)=x 2 /σ+ y 2 +z 2 1/2 V =1/σ x ẋ+ y ẏ+z ż =( yx x 2 )+(rxy y 2 xzy)+(zxy bz 2 ) =(r+1)xy x 2 y 2 bz 2 = [ x r+1 2 y] 2 [1 ( r+1 2 ) 2 ] y 2 bz 2

11 Lorenz System Fixed Points (2) Stability of C + and C - Stable for 1<r<r H =σ (σ +b+3)/(σ b 1) Lose stability in a subcritical Hopf bifurcation Below r H : unstable limit cycles (saddle cycles)

12 Situation so far... What happens for r>r H? No stable objects? Trajectories repelled to infinity? (No, remain inside a certain 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...

13 Lorenz Attractor Numerical integration for r>r H just above r H Aperiodic Seems to settle on very thin set in 3d -> Strange Attractor Two trajectories with ICs That differed by one 100 billionth in Z direction!

14 Lorenz Map z n+1 z n Why are there no periodic solutions for r>r H? z n 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 unstable!

15 Dependence on Initial Conditions Attractor exhibits sensitive dependence on ICs Consider x(t) and x(t)+δ(t) on attractor and δ 0 is tiny separation vector Numerically one finds: δ(t) ~ δ 0 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

16 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 ln a δ 0 )

17 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 quasiperiodic 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?

18 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(t 1 ) 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)

19 Attractors (2) Consider ẋ= x x 3 ẏ= y Is I={x -1<=x<=1} an invariant set? Does I attract an open set of ICs? Is I an attractor?

20 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)

21 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...)

22 Summary What you should remember: Chaos Liapunov exponents Attractors + Strange attractors Transient chaos Seminar: Talks! Using synchronized chaos to send messages

THREE DIMENSIONAL SYSTEMS. Lecture 6: The Lorenz Equations

THREE DIMENSIONAL SYSTEMS. Lecture 6: The Lorenz Equations 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