Bistable

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==Oscylator bistabliny==
==Oscylator bistabliny==
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===TEST===
 
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{| class="wikitable collapsible"
 
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! Simple collapsible table
 
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|-
 
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| Lorem ipsum dolor sit amet
 
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|}
 
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?
 
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:{| class="toccolours collapsible collapsed" width="60%" style="text-align:left"
 
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!Derivation of equilibrium equations
 
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|-
 
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|Consider a continuum body (see Figure 4) occupying a volume <math>V\,\!</math>, having a surface area <math>S\,\!</math>, with defined traction or surface forces <math>T_i^{(n)}\,\!</math> per unit area acting on every point of the body surface, and body forces <math>F_i\,\!</math> per unit of volume on every point within the volume <math>V\,\!</math>. Thus, if the body is in [[Momentum#Relating_to_force_-_General_equations_of_motion|equilibrium]] the resultant force acting on the volume is zero, thus:
 
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:<math>\int_S T_i^{(n)}dS + \int_V F_i dV = 0\,\!</math>
 
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By definition the stress vector is  <math>T_i^{(n)} =\sigma_{ji}n_j\,\!</math>, then
 
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:<math>\int_S \sigma_{ji}n_j\, dS + \int_V F_i\, dV = 0\,\!</math>
 
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Using the [[Divergence theorem|Gauss's divergence theorem]] to convert a surface integral to a volume integral gives
 
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:<math>\int_V \sigma_{ji,j}\, dV + \int_V F_i\, dV = 0\,\!</math>
 
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:<math>\int_V (\sigma_{ji,j} + F_i\,) dV = 0\,\!</math>
 
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For an arbitrary volume the integral vanishes, and we have the ''equilibrium equations''
 
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:<math>\sigma_{ji,j} + F_i = 0\,\!</math>
 
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|}
 
===Układ ===
===Układ ===

Aktualna wersja na dzień 09:00, 8 lis 2010

Oscylator bistabliny

Układ

Rozważmy oscylator nieliniowy (usaa):

\(m \ddot x = -\frac{dU(x)}{dx}-\gamma x ,\)

równoważnie:

\(\dot x = v\)

\(m \dot v = -\frac{dU(x)}{dx}-\gamma x ,\)

jako potencjal weżmy funkcję z dwoma minimami np.:

\(U(x) = \displaystyle x^4-4 x^2,\)

Wykres U(x) można otrzymać poleceniem:

fplot(@(x) x.^4-4*x.^2,[-2.1,2.1],200)

\(f(x) =-\frac{dU(x)}{dx} = -4 x^3+8 x^2,\)

close all
clear all
f =@(x,E) sqrt(8*x.^2-2*x.^4-E);
xa=linspace(-2,2,200);
xb=linspace(2,-2,200);
hold on
for E=-8:2:8
  Rts=roots([-2,0,8,0,-E]);
  AllRealRoots=sort(Rts(find (imag(Rts)==0)));
  if (length(AllRealRoots)==2)
    xminmax=sort([AllRealRoots',-0.00001,0.00001]);
  else
    xminmax=AllRealRoots';
  endif
  xa=linspace(xminmax(1),xminmax(2),200);
  xb=linspace(xminmax(3),xminmax(4),200);
 
  if (E==0)
    plot([rotdim(xa,2),xa],[-f(rotdim(xa,2),E),f(xa,E)],"r-;E=0;");
    plot([xb,rotdim(xb,2)],[f(xb,E),-f(rotdim(xb,2),E)],"r-");
  else
    plot([rotdim(xa,2),xa],[-f(rotdim(xa,2),E),f(xa,E)],"g-");
    plot([xb,rotdim(xb,2)],[f(xb,E),-f(rotdim(xb,2),E)],"g-");
  endif
endfor
hold off

Analiza

lsode_options("absolute tolerance",1e-6)
lsode_options("relative tolerance",1e-6)
n=500;
tmax=10;
t = linspace(0,tmax,n); 
global gama;
gama=0.9
ny=10;
y=linspace(0,5,ny); 
x0=-sqrt(2);
trajsxP=zeros(ny,n);
trajsyP=zeros(ny,n);
trajsxN=zeros(ny,n);
trajsyN=zeros(ny,n);
for iy=1:ny
  X0 = [x0,y(iy)];
  [X,T,MSG]=lsode(@ODEbistable,X0,t);
  if(X(length(X),1)>0)
    trajsxP(iy,:)=X(:,1);
    trajsyP(iy,:)=X(:,2);
  else
    trajsxN(iy,:)=X(:,1);
    trajsyN(iy,:)=X(:,2);
  endif
endfor
disp("Integration done");
hold on
every=20
for i=(every+1):every:size(trajsxP)(2)
  plot(trajsxP(:,(i-every):i)',trajsyP(:,(i-every):i)',"r-",\
     trajsxN(:,(i-every):i)',trajsyN(:,(i-every):i)',"b-")
  drawnow()
  disp(t(i));
  fflush(stdout);
  print ( sprintf("Traj_anim_%05d.png",i), "-S300,220");
endfor
hold off

który możemy zaimplementować jako funkcję w matlabie:

function dx = ODEbistable(X,T)
    global gama;
    dx = zeros(2,1);
    dx(1) = X(2);
    dx(2) = -gama*X(2)-4*X(1).^3+8*X(1);
    return
end
Trajektorie w nieliniowym oscylatorze.
Baseny przyciągania w nieliniowym oscylatorze.
Źródło „http://172.16.3.1/ekonofizyka/index.php/Bistable”