Baixe mapa quadrático e mapa do círculo e outras Exercícios em PDF para Física, somente na Docsity!
Terceiro relat´orio de Introdu¸c˜ao ao Caos
Miguel Mendes Ruiz
Instituto de F´ısica
Universidade de S˜ao Paulo
7 de dezembro de 2010
Mapa quadr´atico
A equa¸c˜ao para o mapa quadr´atico ´e dada por:
xn+1 = f (xn) = C − x
2 n
- Modifique o programa da atividade 57 (itera) para visualizar a evolu¸c˜ao do mapa quadr´atico. Confira o
comportamento no intervalo C ∈ [− 0 .25 2].
programas/iteraMapQuad.m
function iteraMapQuad ( x0 , c , N)
% f u n c t i o n i t e r a m a p l o g ( x0 , c , N)
% chama e limpa a f i g u r a f i g u r e ( 1 ) ;
c l f ; hold on ;
% c r i a a p a r a b o l a e a d i a g o n a l
diag=linspace ( −2 , 2 , 1 0 0 0 0 ) ; parab=c ∗ 1 − diag. ∗ diag ;
% s o l u ¸c ˜a o i t e r a d a
x=mapQuad( x0 , c , N ) ;
plot ( diag , diag , diag , parab ) ; %r e t a x , r e t a y , ’ k ’
t i t l e ( [ ’ e v o l u ¸c ˜a o do mapa q u a d r ´a t i c o | c = ’ num2str ( c ) ] ) ; xlabel ( ’ x {n} ’ ) ;
ylabel ( ’ x {n+1} ’ ) ;
f o r n=3:N plot ( [ x ( n−2) x ( n − 1 ) ] , [ x ( n−1) x ( n − 1 ) ] , ’ k ’ ) ;
plot ( [ x ( n−1) x ( n − 1 ) ] , [ x ( n−1) x ( n ) ] , ’ k ’ ) ; plot ( x ( n −1) , x ( n ) , ’ o r ’ ) ;
drawnow ; pause ( 0. 1 ) ;
end
c s t r = num2str ( c ) ;
c s t r = strrep ( c s t r , ’. ’ , ’− ’ ) ;
print ( ’−dpng ’ , [ ’ mapQuad c ’ c s t r ’. png ’ ] ) ;
hold o f f ;
end
Figura 1: Evolu¸c˜ao do mapa quadr´atico.
- (a) C = − 0 25 (b) C =
- (c) C = 0. 25 (d) C = 0. 50 (e) C = 0.
- (f) C = 1 (g) C = 1. 25 (h) C = 1. - (i) C = 1. 75 (j) C =
programas/bifMapQuad.m
function bifMapQuad ( cmin , cmax , N, t r a n s i e n t e )
%f u n c t i o n bifMapQuad ( cmin , cmax , N, t r a n s i e n t e )
% chama e limpa a f i g u r a f i g u r e ( 1 ) ; hold o f f ;
c l f ;
% c o n d i ¸c ˜o e s i n i c i a i s x=zeros (N, 1 ) ;
c=linspace ( cmin , cmax , N ) ;
% t r a n s i e n t e x0 = 0. 1 ;
f o r i =1: t r a n s i e n t e x0=cmin−x0 ∗ x0 ;
end ;
% g r ´a f i c o s hold on ; plot ( [ cmin cmax ] , [ − 2 2 ] , ’w ’ ) ;
xlabel ( ’ parˆametro de c o n t r o l e ( c ) ’ ) ; ylabel ( ’ x n ’ ) ;
plot ( [ cmin cmax ] , [ 0 0 ] , ’ k ’ ) ; drawnow ;
x (1)= x0 ;
f o r i =2:N
x ( i )=c ( i )−x ( i −1)∗x ( i −1); end ;
plot ( c , x , ’. r ’ , ’ M a r k e r S i z e ’ , 1 ) ; drawnow ;
hold o f f end
programas/chamaBifMapQuad.m
bifMapQuad ( − 0. 2 5 , 2 , 2 7 0 0 , 2 5 0 ) ;
print ( ’−dpng ’ , ’.. / f i g u r a s / bifurcaMapQuad. png ’ ) ;
Utilizando os c´odigos anteriores chego `a figura 2.
- Obtenha os pontos fixos resolvendo a equa¸c˜ao x ∗ = f (x ∗ ), e plote-os em fun¸c˜ao de C.
x
∗ = f (x
∗ ) ⇔ x
∗^2
∗ − C = 0
⇔ x
∗ ±
1 + 4C
x
∗
1 2
1 + 4C
x ∗ −
1 2
1 + 4C
O diagrama dos pontos fixos em fun¸c˜ao de C est´a apresentado na figura 3.
programas/chamaBifFixMapQuad.m
bifFixMapQuad ( −2 , 2 , 2 7 0 , 2 5 0 ) ; print ( ’−dpng ’ , ’.. / f i g u r a s / bifurcaFixMapQuad. png ’ ) ;
Figura 3: Diagrama de bifurca¸c˜oes dos pontos fixos do mapa quadr´atico.
programas/bifFixMapQuad.m
function bifFixMapQuad ( cmin , cmax , N, t r a n s i e n t e )
%f u n c t i o n bifFixMapQuad ( cmin , cmax , N, t r a n s i e n t e )
% chama e limpa a f i g u r a f i g u r e ( 1 ) ;
hold o f f ; c l f ;
% guardando mem´oria
x f 1=zeros (N, 1 ) ; x f 2=zeros (N, 1 ) ;
c=linspace ( cmin , cmax , N ) ;
% g r ´a f i c o s hold on ;
plot ( [ cmin cmax ] , [ −1.5 0. 5 ] , ’w ’ ) ; xlabel ( ’ parˆametro de c o n t r o l e ( c ) ’ ) ; ylabel ( ’ ponto f i x o ( x ˆ { ∗ } ) ’ ) ;
plot ( [ cmin cmax ] , [ 0 0 ] , ’ k ’ ) ; drawnow ;
f o r i =1:N
x f 1 ( i )=(−1 + sqrt ( 1 + 4 ∗ c ( i ) ) ) / 2 ; x f 2 ( i )=(− 1 − sqrt ( 1 + 4 ∗ c ( i ) ) ) / 2 ;
end ; plot ( c , xf1 , ’−r ; x ˆ{∗} +; ’ , c , xf2 , ’ ∗b ; x ˆ{∗} −; ’ , ’ M a r k e r S i z e ’ , 1. 5 ) ;
drawnow ;
hold o f f end
Figura 4: Diagrama de bifurca¸c˜oes com os pontos fixos inst´aveis e expoente de lyapunov do mapa quadr´atico.
programas/bifLyapMapQuad.m
function bifLyapMapQuad ( cmin , cmax , N, t r a n s i e n t e )
%f u n c t i o n bifLyapMapQuad ( cmin , cmax , N, t r a n s i e n t e )
% chama e limpa a f i g u r a f i g u r e ( 1 ) ;
hold o f f ; c l f ;
% c o n d i ¸c ˜o e s i n i c i a i s
x=zeros (N, 1 ) ; c=linspace ( cmin , cmax , N ) ;
% t r a n s i e n t e
x0 = 0. 1 ; f o r i =1: t r a n s i e n t e
x0=cmin−x0 ∗ x0 ; end ;
% g r ´a f i c o s hold on ;
plot ( [ cmin cmax ] , [ − 2 2 ] , ’w ’ ) ; xlabel ( ’ parˆametro de c o n t r o l e ( c ) ’ ) ;
ylabel ( ’ x n ’ ) ; drawnow ;
x (1)= x0 ;
f o r i =2:N
x ( i )=c ( i )−x ( i −1)∗x ( i −1); end ;
f o r i =1:N x f 2 ( i )=(− 1 − sqrt ( 1 + 4 ∗ c ( i ) ) ) / 2 ;
end ;
lam=lyapMapQuad ( cmin , cmax , N, N, x0 ) ;
plot ( c , x , ’. b ’ , c , lam , ’ r ’ , c , xf2 , ’ g ’ , ’ M a r k e r S i z e ’ , 1 ) ;
plot ( [ cmin cmax ] , [ 0 0 ] , ’ k ’ ) ; drawnow ;
hold o f f end
- Verifique numericamente que a bacia de atra¸c˜ao ´e dada pelo valor absoluto do ponto fixo inst´avel.
programas/quad.m
function q=quad( t t , x , C)
%q=quad ( t t , x , C)
q = C − x ∗ x ; end
programas/chamaQuad.m
c l e a r ;
t i c ;
% −−−−−−−−−− problema
p a s s o = 0. 0 1 ;
t 0 = 0 ;
t e m p o I n t e g r a =500;
xx0 = linspace ( − 2. 5 , 2. 5 , 1 0 ) ;
CC=[ −0.2 0. 2 1 1. 5 ] ;
f o r i =1: length (CC) C=CC( i ) ;
f o r j =1: length ( xx0 )
x0 = xx0 ( j ) ;
q=RK4MMR10( @quad , passo , x0 , t0 , tempoIntegra , C ) ;
t=q ( : , 1 ) ; Np=length ( t ) ;
t=q ( f i x (Np / 2 ) : Np , 1 ) ; x=q ( f i x (Np / 2 ) : Np , 2 ) ;
%% imprime
s t r = [ ”C = ” num2str (C) ” \ t& x0 = ” num2str ( x0 ) ” \ t& x \ r i g h t a r r o w ” num2str ( x ( end ) ) ] ; dlmwrite ( ’ r e s u l t s Q u a d. t x t ’ , s t r , ’ append ’ , ’ on ’ , ’ d e l i m i t e r ’ , ’ ’ , ’ n e w l i n e ’ , [ ” \\\\ \n” ] ) ;
end ;
end ; tempoPassado=toc ;
save ( ’− a s c i i ’ , ’ demoraQuad. t x t ’ , ’ tempoPassado ’ ) ;
Mapa do c´ırculo
O mapa do c´ırculo bidimensional ´e dado por:
xn+1 = xn + Ω −
K
2 π
sin(2πxn) + byn (mod1)
yn+1 = byn −
K
2 π
sin(2πxn)
onde Ω ´e a rela¸c˜ao entre as freq¨uˆencias dos dois osciladores quando desacoplados, b ´e um fator de dissipa¸c˜ao
ou de amortecimento, e K ´e a intensidade do acoplamento, e ser´a usado como parˆametro de controle, e
(mod 1) significa que devemos pegar s´o a parte fracion´aria de x e y. Para b = 0.1 :
- Escreva um programa para obter os atratores do mapa do c´ırculo, plotando yn vs. xn.
- Construa os mapas de primeiro retorno yn+1 vs. yn e xn+1 vs. xn para Ω obtido da seguinte maneira:
U=´ultimo algarismo do seu n´umero USP e nn = 0. 1 , 0. 6 e 0. 9 e Ω = U + nn. (exemplo: se o seu n´umero
USP for 5, U = 5 teremos Ω = 5. 1 , 5. 6 e 5. 9 )
- Obtenha o diagrama de bifurca¸c˜oes yn vs. K, com K variando entre 0 e 5 , mantendo Ω fixo para os
valores 1. 0 , 1. 4 , 1. 8 e 2. 4.
programas/bifMapCirc.m
function b i f MapCirc ( kmin , kmax , omega , N, t r a n s i e n t e )
%f u n c t i o n b i fMapCirc ( kmin , kmax , omega , N, t r a n s i e n t e )
% chama e limpa a f i g u r a
f i g u r e ( 1 ) ; hold o f f ;
c l f ;
% c o n d i ¸c ˜o e s i n i c i a i s x = zeros (N, 1 ) ; y = zeros (N, 1 ) ;
k = linspace ( kmin , kmax , N ) ; b = 0. 1 ;
% t r a n s i e n t e
x0 = 0. 1 ; y0 = 0. 1 ;
f o r i = 1 : t r a n s i e n t e x0 = mod( x0 + omega − ( kmin / ( 2 ∗ pi ) ) ∗ sin ( 2 ∗ pi ∗ x0)+b∗y0 , 1 ) ;
y0 = b∗ y0 − ( kmin / ( 2 ∗ pi ) ) ∗ sin ( 2 ∗ pi ∗ x0 ) ; end ;
% g r ´a f i c o s hold on ;
plot ( [ kmin kmax ] , [ 0 1 ] , ’w ’ ) ; xlabel ( ’ parˆametro de c o n t r o l e (K) ’ ) ;
ylabel ( ’ x n ’ ) ; plot ( [ kmin kmax ] , [ 0 0 ] , ’ k ’ ) ;
drawnow ;
x (1)= x0 ;
f o r i =2:N x ( i ) = mod( x ( i −1) + omega − ( k ( i ) / ( 2 ∗ pi ) ) ∗ sin ( 2 ∗ pi ∗x ( i −1))+b∗y ( i − 1 ) , 1 ) ;
y ( i ) = b∗y ( i −1) − ( k ( i ) / ( 2 ∗ pi ) ) ∗ sin ( 2 ∗ pi ∗x ( i − 1 ) ) ; end ;
plot ( k , x , ’. r ’ , ’ M a r k e r S i z e ’ , 1 ) ; drawnow ;
hold o f f
end
programas/chamaBifMapCirc.m
omega = [ 1. 0 1. 4 1. 8 2. 4 ] ;
f o r i = 1 : length ( omega ) b i f MapCirc ( 0 , 5 , omega ( i ) , 2 5 0 0 0 , 2 5 0 0 0 ) ; omegastr = num2str ( omega ( i ) ) ;
omegastr = strrep ( omegastr , ’. ’ , ’− ’ ) ; print ( ’−dpng ’ , [ ’.. / f i g u r a s / bifurcaMapCircOmega ’ omegastr ’. png ’ ] ) ;
end ;
(a) Ω = 1. 0 (b) Ω = 1. 4
(c) Ω = 1. 8 (d) Ω = 2. 4
Figura 5: Diagrama de bifurca¸c˜oes do mapa do c´ırculo.
Referˆencias
[1] Jos´e Carlos Sartorelli. Introdu¸c˜ao ao Caos. IFUSP, Agosto de 2009.
