Complex Dynamic Systems

Transcription

1 Complex Dynamic Systems Department of Information Engineering and Mathematics University of Siena (Italy) (mocenni at dii.unisi.it) (madeo at dii.unisi.it) (roberto.zingone at unisi.it) Lab Session #5 Dec 4, 2017

2 A discrete time system A generic discrete time system (first order) is described by an equation in the form: Consider the Logistic equation: x n+1 = f (x n, r). x n+1 = rx n (1 x n ), where r is a positive real parameter. Steady states - Find x such that x n+1 = x n = x. For the logistic equation, we have two steady states: x1 = 0 x2 = r 1 r

3 Stability analysis (1/12) To study the stability of the steady states, we need the derivative of f wrt to x: f = r 2rx. if f (x ) < 1, then the steady state x is asymptotically stable. if f (x ) > 1, then the steady state x is unstable. if f (x ) = 1, we can t say nothing about the stability of x.

4 Stability analysis (2/12) Stability of x 1 : Then f (x 1 ) = r. if 0 < r < 1, x1 is asymptotically stable. if 1 < r 4, x1 is unstable. r = 1 could be a bifurcation point. Notice that: f (x 1 ) r=1 = 1. Then here we can have a saddle-node bifurcation, or a transcritical bifurcation, or a pitchfork bifurcation.

5 Stability analysis (3/12) Stability of x2 : f (x2 ) = 2 r. Then if 1 < r < 3, x2 is asymptotically stable. if 0 r < 1, x1 is unstable. if 3 < r 4, x1 is unstable. r = 1 could be a bifurcation point. Notice that: f (x 2 ) r=1 = 1. Then for r = 1 we can have a saddle-node bifurcation, or a transcritical bifurcation, or a pitchfork bifurcation. Moreover, notice that: f (x 2 ) r=3 = 1. Then for r = 3 we can have a flip bifurcation.

6 Stability analysis (4/12) Summarizing: For r = 1, we have a transcritical bifurcation (exchange of stability between two points) For r = 3, we may have a flip bifurcation (this must verified via simulations)

7 Stability analysis (5/12) Download DTanalysis.m from the course web site: mocenni/compdynsys-teach html function DTanalysis () %% System parameter ( varying ) rrange = 0:0.05:4; %% Time steps NT = 100; %% Initial condition x0 = 0.3; %% Set up the dimension of the images xmin = 0; xmax = 1;

8 Stability analysis (6/12) %% If tpause = 0, then the program waits for a key stroke in the %% Command windows before going on. %% If tpause > 0, then you will obtain a movie. The delay between %% each frame is equal to tpause ( in seconds ). tpause = 0.; %% Technical objects laststeps = 10; npoints = 100; xplotrange = linspace ( xmin, xmax, npoints );

9 Stability analysis (7/12) %% Run the simulations for each value of r and plot figure % Normal ( from smallest to biggest r) for i =1: numel ( rrange ) % % Reverse ( from biggest to smallest r) % for i= numel ( rrange ) : -1:1 %% Fix parameter r r = rrange (i); %% Steady states xss (1) = 0; xss (2) = (r -1) /r; %% Simulate x = zeros (NT +1, 1); x (1) = x0; for n =1: NT x(n +1) = f(x(n), r);

10 Stability analysis (8/12) %% Plot in the x_{n}/ x_{n +1} plane subplot (1,2,1) %% Plot the bisectrix plot ( xplotrange, xplotrange, 'k '); hold on %% Plot the right - hand function f ( x_{ n +1} = f( x_{n}) fplot = zeros ( npoints, 1); for j =1: npoints fplot (j) = f( xplotrange (j), r); plot ( xplotrange, fplot, 'g ');

11 Stability analysis (9/12) %% Plot the steady states for j =1: numel ( xss ) if ( isreal ( xss (j))) scatter ( xss (j), f( xss (j), r), 120,, ' MarkerFaceColor ', 'c '); 'c ' %% Plot the trajectory on f ftraj = zeros (NT +1, 1); for j =1: NT +1 ftraj (j) = f(x(j), r); plot (x, ftraj, '.b ', ' MarkerSize ', 20) ; %% Plot the last laststeps points of the trajectory on f plot (x( - laststeps : ), ftraj ( - laststeps : ), '.r ', ' MarkerSize ', 20) ;

12 Stability analysis (10/12) %% Plot the initial condition of the trajectory on f plot (x (1), ftraj (1), '.m ', ' MarkerSize ', 20) ; %% Plot the tangents... for j =1: numel ( xss ) if ( isreal ( xss (j))) plot ( xplotrange, f( xss (j), r) + fprime ( xss (j),r)*( xplotrange - xss (j)), 'c-- ', ' LineWidth ', 2); title ( sprintf ( 'r = xlabel ( 'x_n ') ylabel ( 'x_{n +1} ') xlim ([ xmin xmax ]) ylim ([ xmin xmax ]) hold off %.2 f ', r));

13 Stability analysis (11/12) %% Plot over time subplot (1,2,2) %% Plot the trajectory over time plot (0: NT, x, 'b.- ', ' MarkerSize ', 20, 'Color ', 'k ', ' MarkerEdgeColor ', 'b ') hold on %% Plot the last laststeps points of the trajectory over time in red plot (NT - laststeps :NT, x(nt - laststeps +1: NT +1), r. ', ' MarkerSize ', 20) ' %% Plot the initial condition in magenta plot (0, x (1), 'm. ', ' MarkerSize ', 20) plot (0, x (2), 'm. ', ' MarkerSize ', 20) xlabel ( 'time (n) ') ylabel ( 'f( x_n ) ') xlim ([0 NT ]) ylim ([ xmin xmax ]) hold off

14 Stability analysis (12/12) %% Pause if ( tpause == 0) pause else pause ( tpause ) %% Customizable function ( right hand part of the DT system, x_{n +1} = f(x_{n})) function xnext = f(x, r) xnext = r*x*(1 -x); %% Derivative of the customizable function (f '( x)) function fp = fprime (x, r) fp = r - 2* r* x;

15 Now, run the script! Running the script, you get the following images: For r = 0.5, x1 is stable, x 2 is unstable.

16 Now, run the script! Running the script, you get the following images: For r = 1.5, x1 is unstable, x 2 is stable.

17 Now, run the script! Running the script, you get the following images: For r = 3, we observe a flip bifurcation for x 2.

18 Now, run the script! Running the script, you get the following images: Here, we observe another flip bifurcation for x 2.

19 Now, run the script! Running the script, you get the following images: For bigger value of r, we observe chaotic behavior!

20 Quadratic map (1/2) Let s study another discrete time system: x n+1 = x 2 n + r, for r [ 2, 1] The steady states are: x1 = r 2 x2 = 1 1 4r 2 NOTE: these steady states exist for r 1 4. Matlab code: %% Steady states xss (1) = (1 + sqrt (1-4*r)) /2; xss (2) = (1 - sqrt (1-4*r)) /2;

21 Quadratic map (2/2) Stability analysis: For r = 1 4, we have a saddle-node bifurcation For r = 3 4, we may have a flip bifurcation (this must verified via simulations) Do you observe chaos? For which values of r? Tips Try to run DTanalysis.m in reverse mode (from the biggest to the smallest r value) Adjust xmin and xmax

22 Two-steps dynamics of the logistic equation (1/3) Download DTanalysis f2.m from the course web site: mocenni/compdynsys-teach html. The file DTanalysis f2.m has the same structure of DTanalysis.m, but it is divided into two parts: 1 Analysis of x n+1 = f (x n ) 2 Analysis of x n+2 = f (f (x n )) (two-steps dynamics) Hereafter, we explain some important parts.

23 Two-steps dynamics of the logistic equation (2/3) %% SECOND PART : two steps dynamics (x_{n +2} = f(f( x_n ))) %% Steady states xss_f2 (1) = 0; xss_f2 (2) = (r -1) /r; xss_f2 (3) = (r + sqrt ((r + 1) *(r - 3)) + 1) /(2* r); xss_f2 (4) = (r - sqrt ((r + 1) *(r - 3)) + 1) /(2* r); Here you must specify the steady states of f (f (x)).

24 Two-steps dynamics of the logistic equation (3/3) %% f(f(x)) function xnext = f2(x, r) xnext = f(f(x,r), r); %% Derivative of f( f( x)) %% Evaluated using the chain rule %% d(f(f(x)))/dx = f '(x) f '(f(x)) function fp = f2prime (x, r) fp = fprime (x,r)* fprime (f(x,r), r); These two functions evaluates f (f (x)) and f (f (x)) dx is evaluated using the chain rule: f (f (x)) dx = f (x)f (f (x)) f (f (x)). Notice that dx

25 Now, run the script! Running the script, you get the following images: Here we are just before the flip bifurcation for the original system.

26 Now, run the script! Running the script, you get the following images: For r = 3, we observe a pitchfork bifurcation for the two-steps dynamical system.

27 Now, run the script! Running the script, you get the following images: The second flip for the original system corresponds to a flip for the two-steps dynamical system.

28 Sensitivity to initial conditions (1/7) Download DTsensitivity.m from the course web site: mocenni/compdynsys-teach html function DTsensitivity () %% Fix the parameter r = 0.5; %% Time steps NT = 30; %% Initial conditions x0_1 = 0.5; x0_2 = 0.501; %% Set up the dimension of the images xmin = 0; xmax = 1;

29 Sensitivity to initial conditions (2/7) %% If tpause = 0, then the program waits %% for a key stroke in the %% Command windows before going on. %% If tpause > 0, then you will obtain a movie. %% The delay between each frame is %% equal to tpause ( in seconds ). tpause = 0.1; %% Technical objects npoints = 100; xplotrange = linspace ( xmin, xmax, npoints );

30 Sensitivity to initial conditions (3/7) %% Set up the x_{n}/ x_{n +1} plane figure subplot (1,2,1) %% Plot the bisectrix plot ( xplotrange, xplotrange, 'k '); hold on %% Plot the right - hand function f (x_{n +1} = f(x_{n}) fplot = zeros ( npoints, 1); for j =1: npoints fplot (j) = f( xplotrange (j), r); plot ( xplotrange, fplot, 'g '); title ( sprintf ( 'r = xlabel ( 'x_n ') ylabel ( 'x_{n +1} ') %.2 f ', r));

31 Sensitivity to initial conditions (4/7) %% Setup the plot over time subplot (1,2,2) %% Draw the initial conditions scatter (0, x0_1, 120, 'b ', ' MarkerFaceColor ', 'b '); hold on scatter (0, x0_2, 120, 'r ', ' MarkerFaceColor ', 'r '); title ( sprintf ( ' Trajectories over time - r = %.2 f ', r)); xlabel ( 'time (n) ') ylabel ( 'f( x_n ) ')

32 Sensitivity to initial conditions (5/7) %% Run the simulations x1 = x0_1 ; x2 = x0_2 ; for n =0: NT -1 x1next x2next = f(x1, r); = f(x2, r); %% Plot the cobwebs subplot (1,2,1) plot ([ x1 x1], [x1 x1next ], 'b '); plot ([ x1 x1next ], [ x1next, x1next ], 'b ') plot ([ x2 x2], [x2 x2next ], 'r '); plot ([ x2 x2next ], [ x2next, x2next ], 'r ') xlim ([ xmin xmax ]) ylim ([ xmin xmax ])

33 Sensitivity to initial conditions (6/7) %% Plot the trajectories over time subplot (1,2,2) plot (n:n+1, [x1 x1next ], 'b.-- ',... ' MarkerSize ', 20, 'Color ', 'b ', ' MarkerEdgeColor ', 'b ') plot (n:n+1, [x2 x2next ], 'r.-- ',... ' MarkerSize ', 20, 'Color ', 'r ', ' MarkerEdgeColor ', 'r ') xlim ([0 NT ]) ylim ([ xmin xmax ])

34 Sensitivity to initial conditions (7/7) %% Pause if ( tpause == 0) pause else pause ( tpause ) %% Copy x1 and x2 to x1next and x2next x1 = x1next ; x2 = x2next ; %% Customizable function ( right hand part %% of the DT system, x_{n +1} = f(x_{n})) function xnext = f(x, r) xnext = r*x*(1 -x);

35 Now, run the script! Running the script, you get the following images: For r = 0.5, the trajectories are (almost) indistinguishable.

36 Now, run the script! Set r = 1.5, x 1 (0) = 0.8 and x 2 (0) = Running the script, you get the following images: Again, the trajectories are (almost) indistinguishable.

37 Now, run the script! Set r = 3.9 (chaotic regime). Running the script, you get the following images: Chaotic behavior is characterized by a strong sensitivity to initial conditions. Indeed, after same time, the trajectories are very different, although bounded.