Example Dynamical Systems

Transcription

1 Example Dynamical Systems Reading for this lecture: NDAC, Sec , , , &

2 1D Flows: Fixed Points model of static equilibrium 1D Flow: x R ẋ = F (x) Fixed Points: x R such that ẋ x =0 or F (x )=0 2

3 1D Flows: Fixed Points... Stability: What is linearized system at? Investigate evolution of perturbations: x x 0 = x + x Local Flow: ẋ = df dx x(t) x Local Linear System: ẋ = x Solution: x(t) / e t x(0) 3

4 1D Flows... Stability Classification of Fixed Points: Slope of F (x) at : 1. Stable: < 0 x ẋ 2. Unstable: > 0 Attractor Repellor Attractor x 3. Neutral: =0 4

5 2D Flows: Fixed Points model of static equilibrium 2D Flow: x R 2 ẋ = F (x) Fixed Points: (x,y ) such that ~x (x,y ) =(0, 0) or x =(x, y) F =(f,g) ẋ = f(x, y) ẏ = g(x, y) or 0=f(x,y ) 0=g(x,y ) 5

6 2D Flows: Fixed Points... Stability: What is linearized system at x? Investigate evolution of perturbations x: ~x 0 = ~x + ~x ẋ = F (x) Local Flow: δ x = F x δ x x (t) Initial conditions: x(0) x(0) 6

7 2D Flows: Fixed Points... Local Linear System: ẋ = A x Jacobian: A = F x = ( f f x y g g x y ) Solution: ~x(t) / e At ~x(0) 7

8 2D Flows: Fixed Points (an aside)... Solve linear ODEs: Find ~x (0) ~x = A~x ~x (t) given Eigenvalues and eigenvectors: j and ~v j : A~v j = j ~v j, j =1, 2 Solution: ~x (t) = 2X j=1 j j e jt ~v j where calculate : ~x (0) = 1 ~v ~v 2 8

9 2D Flows... Stability Classification of Fixed Points: Eigenvalues of Jacobian A at x : λ 1 & λ 2 C (Review: NDAC, Chapter 5) Stable fixed point (aka sink, attractor): R(λ 1 ), R(λ 2 ) < 0 9

10 2D Flows... Stability Classification of Fixed Points... Eigenvalues of Jacobian A at x : λ 1 & λ 2 C Unstable fixed point (aka source, repellor): R(λ 1 ), R(λ 2 ) > 0 10

11 2D Flows... Stability Classification of Fixed Points: Eigenvalues of Jacobian at x : λ 1 & λ 2 C Saddle fixed point (mixed stability): R(λ 1 ) > 0&R(λ 2 ) < 0 11

12 2D Flows... Stability Classification of Fixed Points: Eigenvalues of Jacobian at x : λ 1 & λ 2 C Center: R(λ 1 )=R(λ 2 )=0 12

13 2D Flows... Stability Classification of Fixed Points... Magnitude of (in)stability: Det(A) =λ 1 λ 2 Det(A) < 0:λ 1,λ 2 R, λ 1 > 0 λ 2 < 0 Saddles Det(A) > 0: Stable: Tr(A) < 0 Tr(A) =λ 1 + λ 2 Unstable: Tr(A) > 0 Marginal: Tr(A) =0 13

14 2D Flows... Stability Classification of Fixed Points... Tr(A) Unstable Tr 2 (A) 4Det(A) =0 Saddles Unstable Spirals Det(A) Saddles Stable Spirals Stable 14

15 2D Flows... Stability Classification of Fixed Points... Tr(A) Unstable Tr 2 (A) 4Det(A) =0 Saddles Unstable Spirals Det(A) Saddles Stable Spirals Stable 14

16 2D Flows... Stability Classification of Fixed Points... Tr(A) Unstable Tr 2 (A) 4Det(A) =0 Saddles Unstable Spirals Det(A) Saddles Stable Spirals Stable 14

17 2D Flows... Stability Classification of Fixed Points... Tr(A) Unstable Tr 2 (A) 4Det(A) =0 Saddles Unstable Spirals Det(A) Saddles Stable Spirals Stable 14

18 2D Flows... Stability Classification of Fixed Points... Tr(A) Unstable Tr 2 (A) 4Det(A) =0 Saddles Unstable Spirals Det(A) Saddles Stable Spirals Stable 14

19 2D Flows... Stability Classification of Fixed Points... Tr(A) Unstable Tr 2 (A) 4Det(A) =0 Saddles Unstable Spirals Det(A) Saddles Stable Spirals Stable 14

20 2D Flows... Stability Classification of Fixed Points... Tr(A) Unstable Tr 2 (A) 4Det(A) =0 Saddles Unstable Spirals Det(A) Saddles Stable Spirals Stable 14

21 2D Flows... Stability Classification of Fixed Points... W s W u Hyperbolic intersection of and : Robust, if R(λ i ) 0, i W s W s Fixed point persists under perturbation W u W u 15

22 2D Flows... Stability Classification of Fixed Points... Non-hyperbolic intersection of and : W s W u Fragile W s W u Fixed point changes structure under perturbation 16

23 2D Flows: Limit Cycles isolated, closed trajectory: a periodic orbit: model of stable oscillation this is a new behavior type not possible in 1D flows ~x(t) =~x(t + p), for all t ( p is the period) Stable limit cycle 17

24 2D Flows: Limit Cycles... Unstable cycle 18

25 2D Flows: Limit Cycles... Saddle cycle 19

26 2D Flows... Limit Cycle Examples Easy in polar coordinates: ṙ = r(1 r 2 ) θ = 1 20

27 2D Flows... Limit Cycle Examples... 2 y (a, µ) =(0.1, 10.0) Van der Pol Equations: ẍ + µ(x 2 a)ẋ + x =0 or -2 2 x ẋ = y ẏ = x + µy(a x 2 ) Nonlinear damping changes sign: Small oscillation: growth Large oscillation: damped -2 21

28 2D Flows... Limit cycle existence (requires real work to show!) Systems that can t have stable oscillations: 1. Simple harmonic oscillator 2. Gradient systems: ẋ = V (x) 3. Lyapunov systems 22

29 2D Flows... Limit cycle existence (requires real work to show!) How to find limit cycles? Poincaré-Bendixson Theorem: (a) trajectory confined to trapping region (b) no fixed points R then have limit cycle C somewhere inside R. C 23

30 3D Flows: Fixed points Limit cycles and...? 24

31 3D Flows: Quasiperiodicity product of two limit cycles: two irrational frequencies 1 = 2 ω 1 ω 2 25

32 3D Flows: Quasiperiodicity... a new kind of behavior not possible in 1D or 2D Torus attractor ω 2 ω 1 1 = 2 26

33 3D Flows: Chaos recurrent instability one way to do this: Orbit reinjection near unstable fixed point not possible in lower D flows a new behavior type 27

34 3D Flows: Chaos... A topological construction: saddle fixed point at origin: 1D unstable manifold: dim(w u (0)) = 1 2D stable manifold: dim(w s (0)) = 2 two fixed points: C + & C 0 C + C 0 Orbits Cannot Cross: Need 3D! Does any ODE implement this flow design? 28

35 3D Flows: Chaos... Does any ODE implement this design? Yes, the Lorenz equations: ẋ = σ(y x) ẏ = rx y xz ż = xy bz Parameters: σ, r, b > 0 Exercise: Show fixed point at the origin can be a saddle, with 2 stable and 1 unstable directions Exercise: Show there is a symmetry (x, y) ( x, y) 29

36 3D Flows: Chaos... Lorenz ODE properties: Trajectories stay in a bounded region near origin No stable fixed points or stable limit cycles inside Volume shrinks to zero (everywhere inside): Z V = dv r ~F (~x) region r ~F (~x) = Tr(A) = 1 b V = (σ +1+b)V V (t) =e (σ+1+b)t Region volume shrinks exponentially fast! What does the invariant set look like? 30

37 3D Flows: Chaos... Lorenz simulation demo: fixed point: limit cycle: chaotic attractor: (σ, r, b) =(10, 8/3, 28) 31

38 3D Flows: Chaos... Lorenz attractor structure Branched manifold 32

39 From Continuous-Time Flows to Discrete-Time Maps: 50 Series of z-maxima: bz 1, bz 2, bz 3,... z What happens if you plot bz n+1 versus bz n? 0-45 x + y 45 33

40 From Continuous-Time Flows to Discrete-Time Maps: Max-z Return Map: z n+1 = f(z n ) 50 Z Max Return Map Z n Zn 50 34

41 From Continuous-Time Flows to Discrete-Time Maps: Time of Return Function: T (z n ) Return Time Map: T n+1 = h(t n ) 1.5 Return Time Map 1.5 Time of Return T(Z) T n Z Tn

42 3D Flows... Lorenz reduces to a cusp 1D map: normalize to z n [0, 1] z n+1 = a(1 1 2z n b ) 1 Parameters: height: a>0 peak sharpness: 0 <b<1 z n z n 1 36

43 3D Flows... Rössler equations ẋ = y z ẏ = x + ay ż = b + z(x c) Parameters: a, b, c > 0 37

44 3D Flows... Rössler chaotic attractor Parameters: (a, b, c) =(0.2, 0.2, 5.7) 38

45 3D Flows... Rössler branched manifold 39

46 3D Flows... Rössler maximum-x return map: x n+1 = f(x n ) 14 x n x n 14 40

47 3D Flows... When normalized to x n [0, 1] get the Logistic Map: 1 x n+1 = rx n (1 x n ) x n+1 Parameter (height): r [0, 4] 0 0 x n 1 41

48 Classification of Possible Behaviors Dimension Attractor 1 Fixed point 2 Fixed point, Limit cycle 3 Fixed Point, Limit Cycle, Torus, Chaotic 4 Above + Hyperchaos 5 Above +? 42

49 Lorenz: Rössler: ẋ = σ(y x) ẏ = rx y xz ż = xy bz ẋ = y z ẏ = x + ay ż = b + z(x c) σ, r, b > 0 Cusp Map: z n [0, 1] a>0, 0 <b<1 Logistic map: z n+1 = a(1 1 2z n b ) x n+1 = rx n (1 x n ) x n [0, 1] r [0, 4] Play with these! 43

50 The Big Picture Global view of the state space structures: The attractor-basin portrait 44

51 The Learning Channel You Are Here 1 A 1 B 0 0 C α γ 0 β δ 0 System Instrument Process Modeller 45

52 Reading for next lecture: NDAC, Chapter 3. 46