Introduction to Dynamical Systems. Tuesday, September 9, 14

Transcription

1 Introduction to Dynamical Systems 1

2 Dynamical Systems A dynamical system is a set of related phenomena that change over time in a deterministic way. The future states of the system can be predicted from past states during lifetime of system Specification of a dynamical system: definition of the state of the system rule for change (the dynamic) Successfully characterize systems on a wide variety of scales (physics, biology, psychology, ecology...) 2

3 Population Change State of system: N of organisms (P n ) Rule for change: P n+1 = r. P n r = 1 + b(irth rate) - d(eath rate) System can be modeled by iterating rule for change Initial conditions must be specified 3

4 Example Growth Parameter Values: b =.05, d =.01!! r = 1.04 Initial Conditions: P 0 = 100 State of system changes over time, but system is fixed! P n+1 = r. P n 4

5 Simple Simulation code % pop.m p(1) = 100; r =.95; niter = 100 for i = 2:niter p(i) = r*p(i-1); end plot (p, 'o-'); xlabel ('Generation', 'FontSize', 18); ylabel ('Population', 'FontSize', 18); shg 5

6 System Parameters and Qualitative behavior r = 1.04 r =.95 6

7 Attractors When r < 1, system has an attractor at 0. All initial conditions result in P = 0. Even if the system is perturbed (new seagulls join community), end state is the same. When r > 1, no attractor System grows indefinitely Effect of initial conditions (or perturbations) is always there. When r = 1 Fixed point Neutral stability Dependence on initial conditions and perturbations. 7

8 Iterative Simulation vs. Analytical Solutions Some systems have analytical solutions Function giving state of system at any time n 8

9 System Activation and Change System with fixed set of parameter values is valid for some interval of time, during which we can say that the system is active. parameters of system remain fixed state of system evolves over time Parameter change may occur e.g. change in b and d due to environment change may reflect state of higher-order system Different from perturbation to a system e.g., changing state from outside the system BUT: Nonlinear system can have multiple attractors and perturbations may push system into a different attractor. 9

10 Spring Dynamical System I spring: I Displace a spring from its resting position (stretch or compress) and it returns smoothly. I Sti ness of spring determines how quickly it returns: slinky vs. your skin I Rule for change: Change in x = kx I k is related to the sti ness of the spring. I I Rule for change: Change in x = kx I The higher the value of k, the faster the goal is attained. Money in Bank Time

11 I I mass: I I Mass k is related to the sti ness of the spring. When displaced, it doesn t return Set it into motion and it keeps moving.

12 Mass + Spring I What happens if you combine a mass and a spring? I Pull the object at the end of the spring, and it will return to its rest position, but because the mass is in motion, it wants to stay in motion. (That is what masses do). I Motion causes spring to compress, then the spring wants to return to its rest position again I Result is oscillation around rest position.

13 Spring: Change in x = 1 2 x Water in bathtub Time Position Time Mass + Spring

14 Mass+Spring: Harmonic motion time A x 0 + A + A Block resting on air hockey table, with a spring attaching it to the side of the table. Such systems are effectively frictionless. Block will oscillate about neutral position What is rule for change for the block,? such that we can find the change in state (e.g., position) from the current state. `

15 Kinematics Kinematics are state descriptors of change (not a dynamical system) in a continuous, multi-dimensional space over time. Position Xn Velocity Vn = Δ(Xn) Acceleration An = Δ(Δ(Xn)) Discrete kinematics: states defined at sequence of discrete instants in time (samples) 15

16 Kinematic Equations for Velocity and Acceleration 16

17 Time Functions Ball rolls to wall, bounces off, rolls back 17

18 Relevant physical laws Newton s 2nd Law (masses): Sum of Forces Applied to an Object F = m a = Object Mass x Object Acceleration Inertial Force Hooke s Law (springs): F = k(x x 0 ) k(stiffness) x Inertial Force is proportional to displacement from x0

19 Deriving the Rule for Change x=f(t) What is the function? F = m a F = k(x x 0 ) ma = k(x x 0 ) a = k m (x x 0) ẍ = k m (x x 0) x = x x 0 ẍ = ( k m )x Differential Equation (DE): Second derivative of x is proportional to x with a constant = -k/m This is a feedback system! The position causes change in position. To solve analytically find x=f(t) such that it satisfies the DE.

20 Mass-spring: iteration %mass_spring.m x(1)=10; %initial position v(1)=0; %initial velocity a(1)=0; %initial acceleration k=1; m=5; dt = 1; niter = 100; for i=2:niter a(i) = -(k/m)*x(i-1); v(i) = v(i-1)+(a(i)*dt); x(i) = x(i-1)+(v(i)*dt); end subplot (3,1,1), plot(x); ylabel (gca,' Position', 'FontSize', 18); title (['k=' num2str(k) ' ; m= ' num2str(m)],'fontsize', 18 ) subplot (3,1,2), plot(v); ylabel (gca,' Velocity', 'FontSize', 18); subplot (3,1,3), plot(a); ylabel (gca,' Acceleration', 'FontSize', 18); xlabel (gca,' Time(s)', 'FontSize', 18); shg

21 Effect of decreasing m

22 Effect of increasing k

23 Analytic Solution? x=cos(ωt) d sin( t) = cos( t) dt d cos( t) = sin( t) dt d 2 dt 2 cos(!t) = d dt ( d dt cos(!t)) d 2 dt 2 cos(!t) = d dt (!sin(!t)) d 2 dt 2 cos( t) = 2 cos( t) Curvature of cos(ωt) is proportional to it its value

24 Slope Curvature Time (s)

25 Solutions of mass-spring differential equation What function for x will satisfy d 2 x this equation? dt 2 = k m x x = cos( t) d 2 dt 2 cos( t) = 2 cos( t) FOR 2 = k m So the solution is an oscillator, whose frequency is: k = m ω will be proportional to stiffness of the spring, and inversely proportional to mass.

26 Are frequencies correct?

27 Control Parameter of system k/m Increasing stiffness of spring is equivalent to decreasing mass. Frequency of oscillation depends on k/m ω 2 = k/m Attractor?? 27

28 Real-life Systems: Damping A simple mass-spring system neither loses nor gains energy and is called undamped Real-life systems are not frictionless and display both energy losses and gains Thus, more realistic models of skilled activity must include some sort of friction component: Friction Force + Spring Force = Inertial Force (resists motion) 28

29 Damped Mass-Spring b x k(x x ) = m x 0 ẍ = b mẋ k m (x x 0) 29

30 30

31 Damped Mass-spring: iteration %mass_spring_damped.m x(1)=10; %initial position v(1)=0; %initial velocity a(1)=0; %initial acceleration k=1; m=5; b=1; dt = 1; niter = 100; for i=2:niter a(i) = (-(k/m)*x(i-1)) - (b/m)*v(i-1); v(i) = v(i-1)+(a(i)*dt); x(i) = x(i-1)+(v(i)*dt); end subplot (3,1,1), plot(x); xlabel (gca,' Position', 'FontSize', 18); title (['k=' num2str(k) '; m= ' num2str(m) '; b= ' num2str(b)],'fontsize', 18 ) subplot (3,1,2), plot(v); xlabel (gca,' Velocity', 'FontSize', 18); subplot (3,1,3), plot(a); xlabel (gca,' Acceleration', 'FontSize', 18); shg

32 Damped Mass-Spring

33 Analytic Solution of Damped Mass- Spring bw = b m = bw 2 e bw 2 t 33

34 Critical Damping Approaches rest position, but does not oscillate. Note that there is still a natural frequency (ω), even though there is no oscillation. = k m b =2 p mk 34

35 Damping Ratio If ratio = 1, damping is critical. = b 2 p mk 35

36 Damped Mass-Spring System: Time & Phase TIME SERIES PHASE PLANE TRAJECTORY 36

37 Damped Mass-Spring System: > velocity > > > > System reaches target regardless of initial conditions position Ex. Reaching for a cup from any starting position Equifinality > 37

38 Point Attractors The behavior of a damped mass-spring system is attracted to a single point in the phase plane called an equilibrium point. This type of system is governed by point attractor dynamics. Stability: Having reached its equilibrium point, a point attractor system will remain there if there are no disturbing forces. If perturbed away from its equilibrium point, the system will return to it. 38