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South China University of Technology

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

Review of R-K methodDecay problem

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

Harmonic oscillation

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

Foucault pendulum

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

A Driven Nonlinear Pendulum

Adding dissipation, driving force, and nonlinearity.

Chaos in the driven nonlinear pendulum

Unpredictable vs. determination

Code demonstration !

(Runge Kutta method & Euler-Cromer method)

Code

l=9.8; g=9.8; q=0.5;FD=1.465;omegaD=2/3;

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')

Everything predictable?

- 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

Chaos in the driven nonlinear pendulum

Illustration!

l=9.8; g=9.8; q=0.5;FD=0.5;omegaD=2/3;

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

codeInitialization

Calculation

Little change

Calculation

Show difference

tmp=theta;

theta(1)=0.2+0.001;

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

Lyapunov Exponent

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.

Chaos in the driven nonlinear pendulum

Illustration!

Jules Henri Poincaré (1854─1912)

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.

Code

Initialization

Calculation

Data selecting

clear

l=9.8; g=9.8; q=0.5;FD=0.5;omegaD=2/3;

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

if cos(t(ii)*omegaD)>cos(t(ii-1)*omegaD))&(cos(t(ii)*omegaD)>cos(t(ii+1)*omegaD))

re=[re

ii];

end

end

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

Routes to Chaos: Period Doubling

Only θ at t in phase of FD, i.e.

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

Here Feigenbaum constant

The Logistic Map: Why The Period Doubles

Population growth model in a collection of animals:

Illustration!

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