Discussion of the Lorenz Equations

Discussion of the Lorenz Equations Leibniz Universität Hannover Proseminar Theoretische Physik SS/2015 July 22, 2015 (LUH) Lorenz Equations July 22, 2015 1 / 42

Outline 1 2 3 4 5 6 7 8 (LUH) Lorenz Equations July 22, 2015 2 / 42

May 23 1917 Studied Mathematics After 1946: Focused on Meterology and introduced non linear statistical models to weather theory 1962 Deterministic Nonperiodic Flow 1972 Predictability: Does the Flap of a Butterfly s Wings in brazil set of a Tornado in Texas? April 16 2008 (LUH) Lorenz Equations July 22, 2015 3 / 42

a mechanical model of the Lorenz equations was invented by Willem Malkus and Lou Howard the idealized model obeys a special case of the Lorenz equations (LUH) Lorenz Equations July 22, 2015 4 / 42

the simplest version consists of a few leaky cups single stream of water flows in the cup from the top when the top cup gets heavy enough the wheel starts to rotate (LUH) Lorenz Equations July 22, 2015 5 / 42

nearly horizontal so that the cups form a continuous ring without gaps the series of water streams reduces the propability of overflow a brake on the wheel can add more or less friction the tilt of the wheel can be varied to alter the effective strength of gravity (LUH) Lorenz Equations July 22, 2015 6 / 42

the water is pumped in by the rate Q the leakage occurs at a rate proportional to the mass continuity equation m t = Q Km ω m θ (LUH) Lorenz Equations July 22, 2015 7 / 42

Torque Balance two sources of damping: viscous damping and inertial damping the gravitational torque tends to increase ω when sin (θ) > 0 Torque Balance I ω = vω + gr 2π 0 m(θ, t)sin(θ) dθ where v is the damping rate and g the effective gravitational constant (LUH) Lorenz Equations July 22, 2015 8 / 42

Amplitude Equations since m(θ, t) ist periodic in θ the equations can be written as a Fourier series m(θ, t) = [a n (t)sin(nθ) + b n (t)cos(nθ)] n=0 no sin(nθ) terms because the water is added symmetrically Q(θ) = q n cos(nθ) n=0 (LUH) Lorenz Equations July 22, 2015 9 / 42

Amplitude Equations matching coefficients of sin (nθ) and cos (nθ) yields to ȧ n = nωb n Ka n ḃ n = nωa n Kb n + q n rewriting the torque balance equation eliminates all but one term by orthogonality I ω = vω + πgra 1 (LUH) Lorenz Equations July 22, 2015 10 / 42

waterwheel equations ȧ = ωb 1 Ka 1 ḃ = ωa 1 Kb 1 + q 1 ω = ( vω + πgra 1 ) /I (LUH) Lorenz Equations July 22, 2015 11 / 42

Substitution Substitutions: Lorenz equations a 1 = αy b 1 = βz + q 1 /K ω = γx t = T τ Ẋ = σ (Y X) Ẏ = rx Y XZ Ż = XY bz (LUH) Lorenz Equations July 22, 2015 12 / 42

Fixed Points Setting the derivatives equal to zero: Fixed Points solving for b 1 a 1 = ωb 1 /K ωa 1 = q 1 Kb 1 a 1 = vω/πgr ωb 1 K = vω πgr ω = 0 or b 1 = Kv/πgr (LUH) Lorenz Equations July 22, 2015 13 / 42

Fixed Points there are two kinds of fixed points: if ω = 0, then a 1 = 0 and b 1 = q 1 /K (a 1, b 1, ω) = (0, q 1 /K, 0) no rotation with inflow balanced by leakage if ω 0, then b 1 = K/q 1 / ( ω 2 + K 2) = Kv/πgr (ω) 2 = πgrq 1 v K 2 when πgrq1 v > K 2 there are two solutions ±ω steady rotation in either direction (LUH) Lorenz Equations July 22, 2015 14 / 42

Rayleigh number the solutions only exist if πgrq 1 K 2 v > 1 this dimensionless group is called the Rayleigh number expresses a competition between a spinning and a stoping of the wheel (LUH) Lorenz Equations July 22, 2015 15 / 42

Convection crash course Geometry of the Lorenz Model z T 1 H T 2 δt x (LUH) Lorenz Equations July 22, 2015 16 / 42

Convection crash course Geometry of the Lorenz Model z T 1 H dz F B d δt δt T 2 F v x (LUH) Lorenz Equations July 22, 2015 16 / 42

Heat Equation T t = D T T (1) Approximate the Laplacian operator by δt dz T h 3 D T Thermal diffusivity H Height of the Box α Thermal Expansion Coefficient ρ 0 Initial density δt Temperature difference g Gravitational acceleration µ Dynamic Viscosity (LUH) Lorenz Equations July 22, 2015 17 / 42

Find a Value for the Thermal displacement Time: t T T t! = dt = t T D T T t T D T dz H 3 Calculate the bouyancy force: F B = αρ 0 gdt = αρ 0 g δt H dz D T Thermal diffusivity H Height of the Box α Thermal Expansion Coefficient ρ 0 Initial density δt Temperature difference g Gravitational acceleration µ Dynamic Viscosity (LUH) Lorenz Equations July 22, 2015 17 / 42

Determine the viscous dissipation force: F v = µv H 2 D T Thermal diffusivity H Height of the Box α Thermal Expansion Coefficient ρ 0 Initial density δt Temperature difference g Gravitational acceleration µ Dynamic Viscosity (LUH) Lorenz Equations July 22, 2015 17 / 42

Set them equal: F v = F B v = αρ 0gHδT µ dz From this one obtains the displacement Time: t d = µ αρ 0 ghδt D T Thermal diffusivity H Height of the Box α Thermal Expansion Coefficient ρ 0 Initial density δt Temperature difference g Gravitational acceleration µ Dynamic Viscosity (LUH) Lorenz Equations July 22, 2015 17 / 42

The ration between the displacement Times is the Rayleigh number: R = t T = αρ 0gH 3 δt t D D T µ D T Thermal diffusivity H Height of the Box α Thermal Expansion Coefficient ρ 0 Initial density δt Temperature difference g Gravitational acceleration µ Dynamic Viscosity (LUH) Lorenz Equations July 22, 2015 17 / 42

Derivation of the Equations Navier-Stokes-Equations (Convective Form) u t + u( u) µ 2 u = p + g (1) ρ 0 p is the Pressure, g is the gravitational acceleration, u is the stream field. Apply Bousinessq Approximation Substitute to dimensionless variables Introduce a Streamfunction Ψ Saltzmann Equations (LUH) Lorenz Equations July 22, 2015 18 / 42

Saltzmann Equations ( 1 σ x Make the Ansatz: ( Ψ z t ( Ψ) 2 Ψ z 2 xz Ψ x ( Ψ 2 Ψ z 2 x Ψ 2 )) Ψ x 2 zx = R T x + 2 Ψ 2 ) Ψ 2 z ψ X sin( πa x) sin(πz H H ) θ (Y cos( πa H x) Z sin(2π H z)) Where as: σ = µ r = R D T R c and R c = 27 π4 4 is the critical (LUH) Lorenz Equations July 22, 2015 19 / 42

Lorenz Ansatz: Fourier series truncation { ) {x, z} X a cos sin ( πa H ( πz H Convection Rolls ), sin ( πa ) cos H ( πz )} H H z x (LUH) Lorenz Equations July 22, 2015 20 / 42

(LUH) Lorenz Equations July 22, 2015 21 / 42

The actual Equations From using the Lorenz Ansatz in the Saltzmann Equations: Lorenz Equations Ẋ = σ(y X), Ẏ = X(r Z) Y, Ż = XY bz. (LUH) Lorenz Equations July 22, 2015 22 / 42

Properties of the Lorenz equations Lorenz equations Ẋ = σ (Y X) Ẏ = rx Y XZ Ż = XY bz σ ist the Prandtl number, r is the Rayleigh number and b has no specific name the system has two nonlinearities, the quadratic terms XZ and XY (LUH) Lorenz Equations July 22, 2015 23 / 42

Symmetry important symmetry in the Lorenz equations all solutions are either symmetric or have a symmetric partner X Y Z = X Y Z (LUH) Lorenz Equations July 22, 2015 24 / 42

Volume Contraction V (t + dt) = V (t) + (f ndt) da S V = f n da = f dv S f = σ 1 b < 0 V (t) = V (0) e (σ+1+b)t V volumes in phase space shrink exponentially fast shrinks to a limiting set of zero volume the Lorenz system is dissipative (LUH) Lorenz Equations July 22, 2015 25 / 42

Fixed Points the origin (x, y, z) = (0, 0, 0) is a fixed point for all values of the parameters for r > 1 there is also a symmetric pair of fixed points x ± b (r 1) y = ± b (r 1) z r 1 Lorenz called them C + and C they represent left- or right turning convection rolls or the steady rotation of the waterwheel as r 1 +, C + and C fuse to a pitchfork bifurcation (LUH) Lorenz Equations July 22, 2015 26 / 42

Stability of the Origin the linearization is obtained by omitting the nonlinearities Jacobian at the Origin: σ σ 0 J = r 1 0 det(j) = bσ(r 1) 0 0 b Z is decoupled and Z (t) 0 if r > 1, the origin is a saddle point because det(j) < 0 if r > 1, the origin is a sink for all incoming directions (LUH) Lorenz Equations July 22, 2015 27 / 42

(LUH) Lorenz Figure: Equations Lorenz 1963 July 22, 2015 28 / 42 Lorenz Equations Numerical Integration for different Rayleigh numbers Low s r < 40

Low s For r < 1, the origin is globally stable For 1 < r < r a 14 there are two stable fixed points C + and C 15 10 5 0 5 10 15 8 6 4 20 18 16 14 12 10 8 6 2 0 2 4 6 8 (LUH) Lorenz Equations July 22, 2015 29 / 42

Lorenz Equations Bifurcation Cloud - (LUH) Lorenz Equations July 22, 2015 29 / 42

Low s For r a < r < r H = σ(σ+b+3) σ b 1 there are also unstable limit cycles. 24 22 20 15 10 5 0 5 10 18 16 14 12 15 15 10 5 0 5 10 15 (LUH) Lorenz Equations July 22, 2015 29 / 42

Low s For r a < r < r H = σ(σ+b+3) σ b 1 there are also unstable limit cycles. 35 30 25 20 15 15 10 5 Semi-Chaotic 0 5 10 10 5 0 20 15 10 5 10 5 0 15 15 20 (LUH) Lorenz Equations July 22, 2015 29 / 42

Low s A subcritical Hopf bifurcation occurs at r > r H. The fixed points disappear, leading to seemingly chaotic behaviour 20 15 10 5 0 5 45 40 35 30 25 20 15 10 5 30 20 10 10 20 10 0 15 20 30 (LUH) Lorenz Equations July 22, 2015 29 / 42

Lorenz Equations Bifurcation Cloud - (LUH) Lorenz Equations July 22, 2015 29 / 42

High s For r > 99, there exist windows with stable limit cycles. For r > 313, there is only one global stable limit cycle 260 240 220 200 180 160 30 20 10 0 10 20 60 40 20 0 20 40 Figure: Solution for r = 212, Noisy Periodicity (LUH) Lorenz Equations July 22, 2015 30 / 42

High s For r > 99, there exist windows with stable limit cycles. For r > 313, there is only one global stable limit cycle 500 400 300 200 200 100 40 20 0 20 40 60 80 100 200 100 0 Figure: Solution for r = 350, Globally Stable limit Cycle (LUH) Lorenz Equations July 22, 2015 30 / 42

Exponential Divergence of Nearby Trajectories sensitive dependence on initial conditions long-term prediction is impossible (LUH) Lorenz Equations July 22, 2015 31 / 42

Exponential Divergence of Nearby Trajectories x (t) is a point on the attractor at time t x (t) + δ (t) is a nearby point with a tiny separation δ in numerical studies was found that δ (t) δ 0 e λt where λ 0.9 the divergence stops when the separation is comparable to the diameter of the attractor λ is often called the Ljapunov exponent (LUH) Lorenz Equations July 22, 2015 32 / 42

Time Horizon when a system has a positive Ljapunov exponent, there is a time horizon a is a measure of tolerance our prediction becomes intolerable when δ (t) a ( 1 t horizon O λ ln a ) δ 0 example: original prediction: improved prediction: t horizon 1 4 ln10 λ ln10 3 = 10 7 λ t horizon 1 10 3 10 ln10 ln = λ 10 13 λ after a millionfold improvement in our measurement we can predict only 2.5 times longer! big problem in weather prediction (LUH) Lorenz Equations July 22, 2015 33 / 42

Lorenz Equations Can we really tell if the solutions are aperiodic? Figure: Royal McBee LGP-30, used by lorenz for numerical integration (LUH) Lorenz Equations July 22, 2015 34 / 42

z Lorenz Equations Consider a certain perspective of the trajectory: Return Map 50 45 40 35 30 25 20 15 10 5 30 20 10 0 10 20 30 y Figure: Projection of the Attractor on the x-z plane (LUH) Lorenz Equations July 22, 2015 35 / 42

Return Map A approach to reduce the systems dynamics to a discrete 1D map. z n +1 46 44 42 40 38 36 34 32 30 30 32 34 36 38 40 42 44 46 48 z n Figure: Return Map for r=28, Left: Actual numerical results. Right: Idealized Function from Lorenz (LUH) Lorenz Equations July 22, 2015 36 / 42

The absolute value of the first derivative is f (z n ) > 1 For z n = z n + 1 the orbit is stable Pertubating the orbit by z, = z n+1 = z n + zf (z + z) z = zf (z + z) > z This implies: The Cycle is unstable z n +1 Return Map 46 44 42 40 38 36 34 32 30 30 32 34 36 38 40 42 44 46 48 z n (LUH) Lorenz Equations July 22, 2015 37 / 42

The absolute value of the first derivative is f (z n ) > 1 For z n = z n + 1 the orbit is stable Pertubating the orbit by z, = z n+1 = z n + zf (z + z) z = zf (z + z) > z This implies: The Cycle is unstable z n +1 Return Map 46 44 42 40 38 36 34 32 30 30 32 34 36 38 40 42 44 46 48 z n (LUH) Lorenz Equations July 22, 2015 37 / 42

The absolute value of the first derivative is f (z n ) > 1 For z n = z n + 1 the orbit is stable Pertubating the orbit by z, = z n+1 = z n + zf (z + z) z = zf (z + z) > z This implies: The Cycle is unstable z n +1 Return Map 46 44 42 40 38 36 34 32 30 30 32 34 36 38 40 42 44 46 48 z n (LUH) Lorenz Equations July 22, 2015 37 / 42

The absolute value of the first derivative is f (z n ) > 1 For z n = z n + 1 the orbit is stable Pertubating the orbit by z, = z n+1 = z n + zf (z + z) z = zf (z + z) > z This implies: The Cycle is unstable z n +1 Return Map 46 44 42 40 38 36 34 32 30 30 32 34 36 38 40 42 44 46 48 z n (LUH) Lorenz Equations July 22, 2015 37 / 42

The absolute value of the first derivative is f (z n ) > 1 For z n = z n + 1 the orbit is stable Pertubating the orbit by z, = z n+1 = z n + zf (z + z) z = zf (z + z) > z This implies: The Cycle is unstable z n +1 Return Map 46 44 42 40 38 36 34 32 30 30 32 34 36 38 40 42 44 46 48 z n (LUH) Lorenz Equations July 22, 2015 37 / 42

The last argument does not hold for all values of r Return Map Figure: Return Map, Left: r < 24.06 Right: r > 30.2, Sparrow[1980] (LUH) Lorenz Equations July 22, 2015 38 / 42

Strangeness of the Lorenz Attractor Do the planes of attraction intersect at the origin? 32 30 28 26 24 22 20 18 10 5 0 5 10 5 0 5 Figure: Lorenz Attractor at the origin for r = 28 (LUH) Lorenz Equations July 22, 2015 39 / 42

Strangeness of the Lorenz Attractor Do the planes of attraction intersect at the origin? Figure: The Lorenz attractor manifold at the origin, Sparrow[1980] (LUH) Lorenz Equations July 22, 2015 39 / 42

if the flap of a butterfly s wings can be instrumental in generating a tornado, it can equally well be instrumental in preventing a tornado Lorenz 1972 (LUH) Lorenz Equations July 22, 2015 40 / 42

Resources I Robert C Hilborn et al. and nonlinear dynamics: an introduction for scientists and engineers. In: Computers in Physics 8.6 (1994), pp. 689 689. Edward Lorenz. Predictability: does the flap of a butterfly s wing in Brazil set off a tornado in Texas? na, 1972. Edward N Lorenz. Deterministic nonperiodic flow. In: Journal of the atmospheric sciences 20.2 (1963), pp. 130 141. na. Bifurcation of the Lorenz Equation. 2008. url: https: //perlman.wordpress.com/2008/09/30/biffurcationof-the-lorenz-equation/ (visited on 07/10/2015). na. Lorenz Attractor: Increasing. 2007. url: https://www.youtube.com/watch?v=d284y1x-9o4 (visited on 07/10/2015). (LUH) Lorenz Equations July 22, 2015 41 / 42

Resources II na. Lorenz Attractor spreading into chaos. 2012. url: https://www.youtube.com/watch?v=5xu-9d4ahvu (visited on 07/10/2015). Felix Otto. Rayleigh-Benard Convection. 1999. url: http: //www.mis.mpg.de/applan/research/rayleigh.html (visited on 07/10/2015). Barry Saltzman. Finite amplitude free convection as an initial value problem-i. In: Journal of the Atmospheric Sciences 19.4 (1962), pp. 329 341. Colin Sparrow. The Lorenz equations: bifurcations, chaos, and strange attractors. Vol. 41. Springer Science & Business Media, 2012. Steven H Strogatz. Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering. Westview press, 2014. (LUH) Lorenz Equations July 22, 2015 42 / 42