The Big Picture. Python environment? Discuss Examples of unpredictability. Chaos, Scientific American (1986)

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1 The Big Picture Python environment? Discuss Examples of unpredictability homework to me: Chaos, Scientific American (1986) Odds, Stanislaw Lem, The New Yorker (1974) 1

2 Nonlinear Physics: Modeling Chaos and Complexity Jim Crutchfield chaos Spring WWW: chaos/courses/nlp/ Questionnaire 1. Name: 2. Graduate or Undergraduate (circle one) address: 4. Major/Field: 5. What programming language(s) have you used? (circle all appropriate) C or C++ or Java or Fortran or Python or Perl or Other What level of programming experience do you have? (circle one) Little or Moderate or Extensive Are you familiar with Unix? Yes or No (circle one) 8. Do you have a laptop? Yes or No (circle one) 9. Which OS(es) does it run? (circle all appropriate) Windows or OS X or Linux Do you have a desktop machine? Yes or No (circle one) 11. Which OS(es) does it run? (circle all appropriate) Windows or OS X or Linux Auditors 2 JavaScript 1 Lisp 1 Ruby 1 Math a 1 PHY 3 MAT 5 CSE 1 2

3 The Pendulum 3

4 Qualitative Dynamics (Reading: NDAC, Chapters 1 and 2) What is it? Analyze nonlinear systems without solving the equations. Why is it needed? In general, nonlinear systems cannot be solved in closed form. Three tools: Statistics Computation: e.g., simulation Mathematics: Dynamical Systems Theory Why each is good. Why each fails in some way. 4

5 Dynamical System: {X, T} State Space: X State: x X x X Dynamic: T : X X x, x X x x X X 5

6 Dynamical System... {X, T} Initial Condition: x 0 X Behavior: x 0,x 1,x 2,x 2, x 0 x 1 x 2 x 3 X 6

7 Dynamical System... For example, discrete time... Map: x t+1 = F ( x t ) t =0, 1, 2,... State: x t R n State Space x = (x 1,x 2,...,x n ) Dimension: n Initial condition: x 0 Dynamic: F : R n R n F = (f 1,f 2,...,f n ) Solution : x 0, x 1, x 2, x 3, x 4,... 7

8 Dynamical System... For example, continuous time... Ordinary differential equation (ODE): x = F ( x ) = d dt State: x (t) R n x = (x 1,x 2,...,x n ) Dimension: n Initial condition: x (0) Dynamic: F : R n R n F = (f 1,f 2,...,f n ) 8

9 Flow field for an ODE (aka Phase Portrait) Geometric view of an ODE: dx dt = F (x) dx dt x t = x x t x = x + t F (x) Each state x has a vector attached F (x) that says to what next state to go: x = x + t F (x). 9

10 Flow field for an ODE (aka Phase Portrait) Geometric view of an ODE... X = R 2 x = (x 1, x 2 ) F = (f 1 (x), f 2 (x)) x = x + t F ( x ) x 2 = tf 2 (x) x x 1 = tf 1 (x) 10

11 Geometric view of an ODE... Vector field (aka Phase Portrait): A set of rules: Each state has a vector attached That says to what next state to go X = R 2 11

12 Solving the ODE: Integrate the differential equation! x = F ( x ) x(t )=x(0) + x(t )=x(0) + T 0 T 0 dt x(t) dt F (x(t)) Time-T Flow: φ T The solution of the ODE, starting from a given IC x (T )=φ T ( x (0)) φ T : X X 12

13 Trajectory or Orbit: the solution, starting from some IC simply follow the arrows x (0) x (T ) 13

14 Time-T Flow: x (T )=φ T ( x (0)) x (0) x (T ) φ T Point: ODE is only instantaneous, Time-T Flow gives state for any time t 14

15 Time-T Flow: x (T )=φ T ( x (0)) x (0) x (T ) φ T Point: ODE is only instantaneous, behavior is the integrated, long-term result. 15

16 Example: Simple Harmonic Oscillator: ẍ = x v As two coupled, first-order DEs: ẋ = v v = x with initial condition: (x 0,v 0 ) Vector field: x Time-T flow (aka The Solution): φ T (x 0,v 0 )=(A cos(t + ω 0 ),Asin(T + ω 0 )) A = x v2 0 ω 0 = tan 1 v 0 x 0 16

17 Invariant set: Λ X Set mapped into itself by the flow: Λ = φ T (Λ) 17

18 Invariant set: Λ X Set mapped into itself by the flow: Λ = φ T (Λ) Example: Invariant point Fixed Point 18

19 Invariant set: Λ X Set mapped into itself by the flow: Λ = φ T (Λ) For example: Invariant circles Λ: Any circle Entire plane Pure Rotation (Simple Harmonic Oscillator) 19

20 Attractor: Λ X Where the flow goes at long times (1) An invariant set (2) A stable set: Perturbations off the set return to it For example: Equilibrium Λ Stable Fixed Point 20

21 Attractor: Λ X For example: Stable oscillation Λ Limit Cycle Note: Cycles in SHO, not stable in this sense; not attractors. 21

22 Preceding: A semi-local view... invariant sets and attractors in some region of the state space Next: A slightly Bigger Picture... the full roadmap for the behavior of a dynamical system 22

23 Basin of Attraction: B(Λ) The set of states that leads to an attractor Λ B(Λ) = {x X : lim t φ t (x) Λ} B(Λ 0 ) Λ 0 Λ 1 B(Λ 1 ) 23

24 Separatrix (aka Basin Boundary): B = X i B(Λ i ) The set of states that do not go to an attractor The set of states in no basin The set of states dividing multiple basins B 24

25 The Attractor-Basin Portrait: Λ 0, Λ 1, B(Λ 0 ), B(Λ 1 ), B(Λ 0 ) The collection of attractors, basins, and separatrices B(Λ 0 ) B Λ 0 Λ 1 B(Λ 1 ) 25

26 Back to the local, again Submanifolds: Split the state space into subspaces that track stability Stable manifolds of an invariant set: Points that go to the set W s (Λ) W s (Λ) = {x X : lim t φ t (x) Λ} Λ 26

27 Submanifolds... Unstable manifold: Points that go to invariant set in reverse time W u (Λ) Λ W u (Λ) = {x X : lim t φ t (x) Λ} 27

28 Example: 1D flows State Space: R State: x R Dynamic: F : R R ẋ = F (x) Flow: x(t )=φ T (x(0)) = x(0) + ẋ T 0 dt F (x(t)) x 28

29 Example: 1D flows... Invariant sets: Λ={x,x,x } ẋ Fixed points: ẋ =0 Attractors: x,x = ±1 Repellor: x =0 Attractor x x Repellor Attractor x x 29

30 Example: 1D flows... Basins: B(x )=[, 0) B(x )=(0, ] Separatrix: B = {x } x x x B(x ) B(x ) Attractor-Basin Portrait: x,x,x, B(x ), B(x ), B(x ) 30

31 Example: 1D flows... Stable manifolds: W s (x ) = [, x ) W s (x )=(x, ] Unstable manifold: W u (x ) = (x, x ) W u (x ) x x W s (x ) x W s (x ) 31

32 Example: 1D flows... Hey! Most of these dynamical systems are solvable! For example, when the dynamic is polynomial you can do the integral for the flow for all times. What s the point of all this abstraction? 32

33 Reading for next dynamics lecture: NDAC, Sec , , & Reading for next programming lecture: Python, Part II (Chapters 4 and 7-9) and Part III (Chapters 11-13). 33