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 #1 Oct 16, 2017

2 An introduction to Matlab Matlab by Mathworks. MAT(rix) LAB(oratory) Matlab is installed on all pc in the Department of Engineering Open source alternative: Octave (available on Ubuntu repository)

3 Matlab Command Window MATLAB is easier than general-purpose languages like C++ and Java. Indeed: It is interpreted No compiler head-aches! It is oriented to fast prototyping It is a weakly typed programming language You don t have to care about the variable type! Anyway... It is case sensitive (i.e. sin(x) = Sin(x)) The syntax is important... (i.e. 1+ does not make any sense!)

4 Define a row vector Collect entries between square brackets [ ] separated by spaces... >> x = [ ] x = >> y = [1.1, 0, 0.4]... or separated by commas, y = Suppress the output using semicolon ; >> z = [1 9-8];

5 Define a column vector >> w = [0; 7; -1] Collect entries between square brackets [ ] separated by semicolons ; w = >> v = [1, 2, 3]... or transpose a row vector using v = 1 2 3

6 Linearly spaced vectors >> x = 2:5 Row vector starting from 2, ending with 5, step 1 x = >> y = -1:2:7 Row vector starting from -1, ending with 7, step 2 y = Row vector starting from 0, ending with 1, formed by 4 elements (the step is calculated automatically) >> z = linspace(0, 1, 4) z =

7 Why do we use vectors? Typical usages for this course We want to draw the right-hand side of an ODE... ẋ = x(1 x)...or we want to draw the solution of an ODE x(t) = e t x 0 e t x x 0 In general, we want to draw some function!

8 Why do we use vectors? Consider the function f (x) = x(1 x) Define a linearly spaced vector x Choose an appropriate range (for example [ 1, 2]) Choose an appropriate step (for example 0.1)......or an appropriate number of elements (for example 100) >> x = -1:0.1:2; or >> x = linspace(-1, 2, 100); We want to define another vector f such that: x = [ ]; f = [f(-1.0) f(-0.9) f(-0.8)... f(1.8) f(1.9) f(2.0)];

9 Why do we use vectors? Consider the function f (x) = x(1 x) Define a linearly spaced vector x Choose an appropriate range (for example [ 1, 2]) Choose an appropriate step (for example 0.1)......or an appropriate number of elements (for example 100) >> x = -1:0.1:2; or >> x = linspace(-1, 2, 100); We want to define another vector f such that: x = [ ]; f = [f(-1.0) f(-0.9) f(-0.8)... f(1.8) f(1.9) f(2.0)];

10 Operations with vectors We start with a vector x: >> x = -1:0.1:2; f = x.*(1-x) f (x) = x(1 x) f = f (x) = x 1 x + 2 f (x) = x ( ) x 2 1 x+5 >> f = (x-1)./(x+2) f = >> f = x.*(x.^2-1./(x+5)) f =

11 Summary Addition operator: + Subtraction operator: - Multiplication operator:.* Don t forget the point!!! You can use only * to multiply a vector with a scalar Multiplying two vectors using only * corresponds to the standard linear algebra product (next lessons?) Division operator:./ Power operator:.ˆ Group using only round brackets ( )

12 Built-in functions We start with a vector t: >> t = linspace(0, 10, 100); e t x 0 x(t) = e t x x 0 with x 0 = 0.6 x(t) = t cos(ωt) with ω = 3 >> x0 = 0.6; >> xt = exp(t)*x0./(exp(t)*x x0); xt = >> omega = 3; >> xt = t.* cos(omega * t) xt = >> xt = t.^{2} - sin(omega * t) x(t) = t 2 sin(10t) xt =

13 Plot with Matlab Type clear all in your Matlab workspace to clear all the defined vectors Define the vector x >> x = -1:0.1:2; Define the vector f >> f = x.*(1-x);... and now, plot it! >> plot(x, f)

14 Plot with Matlab Define the vector t >> t = linspace(0, 10, 100); Define the vector xt >> x0 = 0.6; >> xt = exp(t)*x0./(exp(t)*x x0);... and now, plot it! >> plot(t, xt, r, LineWidth, 2) r means red line Try other colors! g, b, c, k, y, m LineWidth, 2 stands for a line with width 2

15 Plot with Matlab Plot the asymptote >> hold on >> asymptote = ones(1, 100); >> plot(t, asymptote, k-- ); >> ylim([0 1.5]) hold on: don t create another figure, just plot over the previous one asymptote = ones(1, 100): create a vector of 1 s with 1 row and 100 columns (same size of vector t) k-- : draw as a dashed (--) black line (k) ylim([0 1.5]): chnage the y range (from 0 to 1.5)

16 Custom Matlab functions Download plotlogistic.m from the course web site: mocenni/compdynsys-teach html function plotlogistic ( tf, Nt, x0list ) N = numel ( x0list ); % Number of in. cond. t = linspace (0, tf, Nt); % Time instants for end i =1: N x0 = x0list (i); xt = exp (t)*x0./( exp (t)*x x0); plot (t, xt, ' LineWidth ', 2); hold all asymptote = ones (1, 100) ; plot (t, asymptote, 'k-- ') end

17 Custom Matlab functions In the Matlab workspace, type: plotlogistic (10, 100, [0.1, 0.5, 1.1, 1.5])

18 Remarks N = numel ( v); % Number of elements of the vector v for end i =1: N % Do something N times a = v( i); % Read the i- th element of vector v hold all % Same as hold on, but it also automatically changes the color of the new plot