1 / 30

# South China University of Technology - PowerPoint PPT Presentation

South China University of Technology. Chaos. Xiao- Bao Yang Department of Physics. www.compphys.cn. Typically, there are m Eqs and m + m ( m −1) / 2 unknowns. Review of R-K method. Decay problem. for j=1:nstep; tmp =Nu(j)-Nu(j)/tau* dt /2; Nu(j+1)=Nu(j)- tmp /tau* dt ;

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about ' South China University of Technology' - whitney

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

Chaos

Xiao-BaoYang

Department of Physics

www.compphys.cn

Typically, there are mEqs and m + m(m −1)/2 unknowns.

Review of R-K method

for j=1:nstep;

tmp=Nu(j)-Nu(j)/tau*dt/2;

Nu(j+1)=Nu(j)-tmp/tau*dt;

t(j+1)=t(j)+dt;

end

for j=1:1000;

E(j+1)=.5*m*l^2*(omega(j)^2+g/l*theta(j)^2);

d1 = theta(j)+0.5*dt * omega(j);

k1 = omega(j)+0.5*dt * (-g/l*theta(j));

theta(j+1)=theta(j)+k1*dt;

omega(j+1)=omega(j)+dt*(-g/l*d1);

t(j+1)=t(j)+dt;

end

for ii=1:7000

vx(ii+1)=vx(ii)+(-x(ii)*g/l+2*omega*vy(ii)*sin(fi))*dt;

vy(ii+1)=vy(ii)+(-y(ii)*g/l-2*omega*vx(ii)*sin(fi))*dt;

x(ii+1)=x(ii)+vx(ii+1)*dt;

y(ii+1)=y(ii)+vy(ii+1)*dt;

t(ii+1)=t(ii)+dt;

end

Adding dissipation, driving force, and nonlinearity.

Unpredictable vs. determination

Code demonstration !

(Runge Kutta method & Euler-Cromer method)

t(1)=0; theta(1)=0.2; omega(1)=0;

dt = 0.04;

Initialization

for i=1:2500;

omega(i+1)=omega(i)+dt * (-g/l*(sin(theta(i))) - q * omega(i) + FD*sin(omegaD*t(i)));

theta(i+1)=theta(i)+omega(i+1)*dt;

t(i+1) = t(i) + dt;

%angle transitons

end

while(theta(i+1)>pi)

theta(i+1)=theta(i+1)-2*pi;

end

while(theta(i+1)<-pi)

theta(i+1)=theta(i+1)+2*pi;

end

plot(t,theta,'k')

plot(t,mod(theta+pi,2*pi)-pi,'k')

• 洛倫茨……結束了笛卡爾宇宙觀統治的時代，繼相對論和量子力學之後，開啟了20世紀第三次科學革命。”——麻省理工學院大氣學教授伊曼紐爾

•     “他的混沌理論繼牛頓之後，為人類自然觀帶來了最為戲劇性的改變。”

• ——1991年“京都獎”評委會致辭

Illustration!

t(1)=0; theta(1)=0.2; omega(1)=0;

dt = 0.04;

for i=1:1500;

omega(i+1)=omega(i)+dt * (-g/l*(sin(theta(i))) - q * omega(i) + FD*sin(omegaD*t(i)));

theta(i+1)=theta(i)+omega(i+1)*dt;

t(i+1) = t(i) + dt;

end

code

Initialization

Calculation

Little change

Calculation

Show difference

tmp=theta;

theta(1)=0.2+0.001;

plot(t,log(abs(tmp-theta))/log(10))

In mathematics, the Lyapunov exponent or Lyapunov

characteristic exponent of a dynamical system is a quantity

that characterizes the rate of separation

of infinitesimally close trajectories.

Lyapunov (1857-1918) is known for his development of the stability

theory of a dynamical system, as well as for his many

contributions to mathematical physics and probability theory.

Illustration!

In mathematics, particularly in dynamical systems,

a first recurrence map or Poincaré map, named after Henri Poincaré,

is the intersection of a periodic orbit in the state space of a continuous

dynamical system with a certain lower dimensional subspace,

called the Poincaré section, transversal to the flow of the system.

In mathematics, the Poincaré recurrence theorem states that certain systems will,

after a sufficiently long time, return to a state very close to the initial state.

The Poincaré recurrence time is the length of time elapsed until the recurrence.

The result applies to physical systems in which energy is conserved. The theorem

is commonly discussed in the context of ergodic theory, dynamical systems and

statistical mechanics. The theorem is named after Henri Poincaré, published in 1890.

Initialization

Calculation

Data selecting

clear

t(1)=0; theta(1)=0.2; omega(1)=0;

dt = 0.04;

for i=1:20000;

omega(i+1)=omega(i)+dt * (-g/l*(sin(theta(i))) - q * omega(i) + FD*sin(omegaD*t(i)));

theta(i+1)=theta(i)+omega(i+1)*dt;

t(i+1) = t(i) + dt;

end

re=[];

for ii=500:length(t)-1

re=[re

ii];

end

end

theta=mod(theta+pi,2*pi)-pi; plot(theta(re),omega(re),'.')

Only θ at t in phase of FD, i.e.

(observe θ at a particular time in the drive cycle)

Here Feigenbaum constant

Population growth model in a collection of animals:

Illustration!

Lorentz model (Simplified Navier-Stokes equations)

Numerical solution

Root finding

Newton-Raphson method

secant method

• Exercise:

• 3.9, 3.12, 3.23

• Both Results and source codes are required.