LECTURE 3: FLOWS NONLINEAR DYNAMICS AND CHAOS Patrick E McSharr Sstems Analsis, Modelling & Prediction Group www.eng.o.ac.uk/samp patrick@mcsharr.net Tel: +44 83 74 Numerical integration Stabilit analsis Linear flows in one and two dimensions Harmonic oscillator Damped harmonic oscillator van Der Pol oscillator Rössler sstem Moore-Spiegel sstem Lorenz sstem Trinit Term 7, Weeks 3 and 4 Mondas, Wednesdas & Fridas 9: - : Seminar Room Mathematical Institute Universit of Oford Nonlinear dnamics and chaos c 7 Patrick McSharr p. Nonlinear dnamics and chaos c 7 Patrick McSh Numerical integration Stabilit analsis Equation of motion: ẋ = f[(t), t] Man numerical methods for solving ODEs Assume t is a small step (effectivel converting the flow into a map) Z t+ t (t + t) = (t) + f[(t), t]dt t Euler method: (t + t) = (t) + tf[(t), t] + O( t ) Second order Runge-Kutta method (uses trial step): k = tf[(t), t] k = tf [(t) + k /, t + t/] (t + t) = (t) + k + O( t 3 ) Advisable to use fourth order Runge-Kutta method Equation for a flow: Linearising about ields where J((t)) is the Jacobian of f at (t) The perturbation ɛ(t + τ) after a τ is ẋ = f() ɛ(t) = J((t))ɛ(t) ɛ(t + τ) = M((t), τ)ɛ(t) where M((t), τ) is the linear propagator defined b»z t+τ M((t), τ) = ep J((t))dt t A particular perturbation ɛ(t + τ) grows if ɛ(t + τ) / ɛ(t) > There eists some perturbation that will grow if the largest eigenvalue λ of the matri M((t), τ) satisfies λ >. Nonlinear dnamics and chaos c 7 Patrick McSharr p.3 Nonlinear dnamics and chaos c 7 Patrick McSh
One-dimensional stabilit analsis Two-dimensional stabilit analsis Equation of motion: ẋ = f((t)) Consider a fied point such that ẋ = f( ) = Linearising about ields ɛ(t) = ep[j( )t]ɛ(t) Consider the growth of a perturbation to the fied point: Stabilit is determined b the sign of λ = J( ) = df/d at Equations of motion: ẋ = f(, ) ẏ = g(, ) Consider a fied point (, ) such that ẋ = ẏ = Calculate the Jacobian matri J at (, ) Define f = f/ etc. The eigenvalues of J are found b solving the characteristic equation: (f λ)(g λ) = f g Solutions are of the form: = ep(λt), = ep(λt) This gives two possible solutions for λ (real or comple conjugates) Nonlinear dnamics and chaos c 7 Patrick McSharr p. Nonlinear dnamics and chaos c 7 Patrick McSh Linear motion in one dimension Circular motion Continuous description of linear motion in one dimension e.g. continuous compound interest Equation of motion: ẋ = a Solution: (t) = ep(at) Growth if a > and deca if a < Determinism implies that a trajector cannot intersect itself Poincaré-Bendison theorem: no chaos in two dimensions Require a three-dimensional flow and nonlinearit for chaos Linear motion in two dimensions e.g. Planet orbiting the sun Solution is given b: ẋ = ẏ = (t) = r cos t (t) = r sin t + = r, motion on a circle with constant radius r Sstem has one equilibrium point at (, ) = (, ) (, ) is known as a centre A centre is neutrall stable, it doesn t attract or repel! Nonlinear dnamics and chaos c 7 Patrick McSharr p.7 Nonlinear dnamics and chaos c 7 Patrick McSh
Mass and spring (Hooke s law) Simple Pendulum Newton s Law: F = m v where v = ẋ Hooke s Law: F = k, k = spring constant Equation of motion: m v = k Mass undergoes harmonic motion Simplif b setting m = k = : ẋ = v v = Similar to circular motion, but in (, v) state-space Note that F = mv = mẍ is a second order ODE requiring a two-dimensional state space Energ is conserved: E = mv / + k / Again, no attractor or repeller, just a centre An eample of a simple two-dimensional dnamical sstem From Newtons s second law, knowledge of the forces, position and velocit are sufficien determine future motion Pendulum (constrained to move in the plane) Dnamics full specified b the displacement angle θ(t) and the angular velocit θ(t) State vector given b (t) = [θ(t), θ(t)] Let m be the mass of the pendulum g is the acceleration due to gravit l is the length of the pendulum Tangential restoring force due to gravit: mg sin θ Tangential force due to angular acceleration: ml θ In the absence of friction, dnamics are governed b Nonlinear dnamics and chaos c 7 Patrick McSharr p.9 d dt θ = θ d θ dt = g l sin θ Nonlinear dnamics and chaos c 7 Patrick McSha Pendulum dnamics: moment of inertia Pendulum dnamics: Lagrangian formalism Moment of inertia I is defined as: I = ml where m is the mass of the bob and l is the length of the pendulum The tangential restoring force is F = mg sin θ Kinetic energ is Gravitational potential energ is T = ml θ V = mgl cos θ Equations of motion can be derived from the torque τ = Fl: τ = mgl sin θ = Iα = I θ where α is the angular acceleration and g is the acceleration due to gravit Finall, θ = g l sin θ (where the zero potential energ is defined b V = at θ = π The Lagrangian is Lagrangian formula, gives L(θ, θ) = T V = ml θ + mgl cos θ d dt L θ «= L θ θ = g l sin θ Nonlinear dnamics and chaos c 7 Patrick McSharr p. Nonlinear dnamics and chaos c 7 Patrick McSha
Pendulum dnamics: Hamiltonian formalism Pendulum dnamics (small oscillations) Momentum is The Hamiltonian is given b p = L θ = ml θ H(p, θ) = p θ L(θ, θ) = ml θ mg cos θ For θ, use a Talor series epansion for sin θ: sin θ = θ θ3 3! + θ! +... Assume sin θ = θ giving approimate equation of motion: θ = g l θ The Hamiltonian equations of motion are = p mgl cos θ ml This corresponds to harmonic motion with solution θ(t) = θ ma cos(ω t + φ) θ = H p = p ml () ṗ = H θ = mgl sin θ () Nonlinear dnamics and chaos c 7 Patrick McSharr p.3 where and the natural period is r g ω = l s l T = π g Nonlinear dnamics and chaos c 7 Patrick McSha Pendulum energ (small oscillations) Pendulum energ (large oscillations) Total energ is given b E = T + V = ml θ mgl cos θ For θ, use a Talor series epansion for cos θ: Total energ is given b Eamine specific case of E = mgl E = ml θ mgl cos θ cos θ = θ + θ4 4!... E = ml θ mgl cos θ = mgl Assume cos θ = θ giving approimate equation of the total energ: Dividing across b mgl gives Defining E = E/mgl + mgl, we obtain E = ml θ + mgl θ mgl " # E = θ + θ For small energ (small oscillations), the lines of constant energ are ellipses ω Nonlinear dnamics and chaos c 7 Patrick McSharr p. Lines of constant energ: θ ω = + cos θ = + cos ( θ ) = cos θ θ = ±ω cos θ Nonlinear dnamics and chaos c 7 Patrick McSha
Pendulum state space with constant energ contours Damped harmonic oscillator... dθ/dt.. Obtained b adding friction to the harmonic oscillator Assume that friction is proportional to velocit: F = βv Equations of motion: Stable equilibrium at (, v) = (, ) Effect of friction is to lose energ Dissipative sstem ẋ = v v = βv Possible equilibria for damped oscillator: Spiral point (focus) if β < (underdamped) Radial point (node) if β > (overdamped). π π π π θ Nonlinear dnamics and chaos c 7 Patrick McSharr p.7 Nonlinear dnamics and chaos c 7 Patrick McSha van Der Pol oscillator van Der Pol series The van Der Pol Oscillator was the first relaation oscillator and provided a model of the human heartbeat [The Heartbeat considered as a Relaation oscillation, and an Electrical Model of the Heart, Balth. van der Pol and J van der Mark, Phil. Mag. Suppl. 6:763 77 (98)] 3 In D: ẍ + + ɛ( )ẋ = 4 6 8 3 ẋ = ẏ = ɛ( ) Unstable equilibrium point (for ɛ > ) at (, ) = (, ). Growth for small values of and Deca for large values of and Sstem has a stable attracting limit ccle All points in the (, )-plane lie in the basis of attraction 4 6 8 Top: ɛ =., Bottom: ɛ = Nonlinear dnamics and chaos c 7 Patrick McSharr p.9 Nonlinear dnamics and chaos c 7 Patrick McSha
Rössler sstem Rössler series The Rössler equations are ẋ = z, ẏ = + a, ż = b + z( c), Chaotic behaviour for a =., b =., and c = Proposed as a simple eample of chaos in a three-dimensional flow 3 3 4 4 z 3 3 4 4 3 z Nonlinear dnamics and chaos c 7 Patrick McSharr p. Nonlinear dnamics and chaos c 7 Patrick McSha Moore-Spiegel Moore-Spiegel series The three-dimensional Moore-Spiegel flow: ẋ =, ẏ = z, ż = z (a b + b ) a, 3 Values giving chaotic behaviour are a = 6 and b = This provides a model for a parcel of ionised gas in the atmosphere of a star z is the height of the parcel is the velocit and z is the acceleration 3 Nonlinear dnamics and chaos c 7 Patrick McSharr p.3 Nonlinear dnamics and chaos c 7 Patrick McSha
The Lorenz model Lorenz series The Lorenz equations are ẋ = σ + σ, ẏ = z + r, ż = bz, Chaotic behaviour for σ =, b = 8/3, and r = 8 Raleigh-Bernard convection: flow of fluid between two rigid horizontal plates subject to gravit, with temperature gradient T between them, the top tpicall being cooler The fluid near the lower plate epands, and buoanc causes the fluid to rise, while the cooler more dense fluid near the top plate falls For some ranges of temperature gradient between the plates, T, stead convective cellular flow occurs As T increases, the flow becomes chaotic The variable is proportional to the circulator fluid flow velocit, characterises the temperature difference between rising and falling fluid regions, and z characterises vertical temperature variation: σ and r are proportional to the Prandtl number and Raleigh number respectivel. 3 3 4 z 4 3 Nonlinear dnamics and chaos c 7 Patrick McSharr p. Nonlinear dnamics and chaos c 7 Patrick McSha Lorenz return map Stabilit analsis 48 46 44 4 Lorenz equations of motion: Fied points at ẋ = σ + σ ẏ = z + r ż = bz (,, ) 4 z ma (i+) 38 36 34 Jacobian is given b (± p b(r ), ± p b(r ), r ) 6 J() = 4 σ σ r b 3 7 3 3 3 3 34 36 38 4 4 44 46 48 z ma (i) Nonlinear dnamics and chaos c 7 Patrick McSharr p.7 Nonlinear dnamics and chaos c 7 Patrick McSha
Stabilit analsis of (,,) Stabilit analsis for pair of fied points Stabilit matri at (,, ) Eigenvalues are (,, ) is stable for r < 6 4 λ, = σ + (,, ) becomes a saddle point for r > σ σ r b ± 3 7 q (σ + ) + 4(r )σ λ 3 = b Stabilit matri for other fied points 6 4 Eigenvalues are roots of σ σ ± p b(r ) ± p b(r ) ± p b(r ) b P(λ) = λ 3 + (σ + b + )λ + b(σ + r)λ + bσ(r ) = r = λ = λ = b λ 3 = (σ + ) Define Hopf boundar: r H = σ σ + b + 3 σ b For < r < r H, smmetric pair of fied points are stable For r > r H real parts of λ, > and this pair becomes unstable 3 7 Nonlinear dnamics and chaos c 7 Patrick McSharr p.9 Nonlinear dnamics and chaos c 7 Patrick McSha Lorenz stabilit graph Power spectra 4 4 CHAOS 3 3 r STEADY 3 3 4 b+ σ Nonlinear dnamics and chaos c 7 Patrick McSharr p.3 Nonlinear dnamics and chaos c 7 Patrick McSha