The Big Picture. Discuss Examples of unpredictability. Odds, Stanisław Lem, The New Yorker (1974) Chaos, Scientific American (1986)

Transcription

1 The Big Picture Discuss Examples of unpredictability Odds, Stanisław Lem, The New Yorker (1974) Chaos, Scientific American (1986) Lecture 2: Natural Computation & Self-Organization, Physics 256A (Winter 2018); Jim Crutchfield

2 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 Mathematical: Dynamical Systems Theory Why each is good. Why each fails in some way.

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

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

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

6 Flow field for an ODE (aka Phase Portrait) x = F ( x ) ( = d dt ) Each state x =(x 1,x 2 ) has a vector attached x = x + t F ( x ) x = F ( x ) That says to what next state x to go x 2 = f 2 ( x ) x x 1 = f 1 ( x )

7 Flow field for an ODE (aka Phase Portrait) Vector field: X = R 2 A set of rules

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

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

10 Invariant set: Λ X Mapped into itself by the flow = T ( ) Example: Invariant point Fixed Point

11 Invariant set: Λ X Mapped into itself by the flow: = T ( ) For example: Invariant circles Pure Rotation (Simple Harmonic Oscillator)

12 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

13 Attractor: Λ X For example: Stable oscillation Λ Limit Cycle

14 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

15 Basin of Attraction: B(Λ) The set of states that lead to an attractor B( ) = {x 2 X : lim t!1 t(x) 2 } B(Λ 0 ) Λ 0 Λ 1 B(Λ 1 ) X

16 Separatrix (aka Basin = X [ i B( 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 X

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

18 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 ( ) = {x 2 X : lim t!1 t(x) 2 } W s (Λ) Λ

19 Submanifolds... Unstable manifold: Points that go to an invariant set in reverse time W u (Λ) Λ W u ( ) = {x 2 X : lim t!1 t(x) 2 }

20 Example: 1D flows State Space: R State: x R Dynamic: F : R R ẋ = F (x) Flow: x(t )= T (x(0)) = x(0) + Graph F (x) ẋ Z T 0 dtf (x(t)) x Vector field

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

22 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

23 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 )

24 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?

25 Reading for next lecture: NDAC, Sec , , &