Sharedcloud-examples / sage / 2016-10-31-131219 3D phase space.sagewsOpen in CoCalc
Author: Gerardo Emilio García Almeida
Views : 48
Description: Example of 3D phase space plot
# based on code from oddrobot, http://sagenb.org/home/pub/1532/ from sage.calculus.desolvers import desolve_system_rk4 x,y,t=var('x y t') class DESolution: def __init__(self,system,time_range,initial,stepsize=0.05,initial_points=20): self.tvar=time_range[0] self._times=srange(time_range[1],time_range[2],stepsize) self.vars=[v for v,_ in initial] self.dim=len(self.vars) self._system=system # check to see if we need one solution or multiple solutions if isinstance(initial[0][1],(list,tuple)): # multiple initial values, from the first value of each variable to the last value of each variable start = vector([a for _,(a,b) in initial]) line = vector([b for _,(a,b) in initial])-start self._soln = [desolve_odeint(system, ics=list(start+t*line), times=self._times, dvars=self.vars, ivar=self.tvar) for t in srange(0,1,step=1/(initial_points-1),include_endpoint=True)] else: self._soln = [desolve_odeint(system, ics=[v for _,v in initial],times=self._times, dvars=self.vars, ivar=self.tvar)] def phase_plot(self,vars=None,color='blue',**kwargs): # find which indices the specified variables are if vars is not None: vars_index=[self.vars.index(v) for v in vars] elif self.dim<=3: vars_index=range(self.dim) else: vars_index=range(2) p = Graphics() for s in self._soln: p+=line(s[:,vars_index],color=color,**kwargs) # add an arrow head showing which way we are going around the phase line half=int(s.shape[0]/2) p+=arrow(s[half,vars_index], s[half+1,vars_index],color='red') if len(vars_index)==2: p.axes_labels([str(self.vars[v]) for v in vars_index]) return p def coordinates(self,colors=None,**kwargs): if colors is None: colors=rainbow(len(self.vars)) p=Graphics() legend=True for s in self._soln: # only want legends the first time p+=sum(line(zip(self._times,s[:,i]), color=colors[i], legend_label=str(self.vars[i]) if legend else None,**kwargs) for i in range(self.dim)) legend=False return p #ADA 14 Equipo 2. var('x,y,z') M = matrix([[-1,2,2],[2,2,2],[-3,-6,-6]]) show(M) show(M.eigenvectors_right()) show(M.characteristic_polynomial()) show(M.characteristic_polynomial().roots()) show(M.eigenvalues()) F=[-x+2*y+2*z,2*x+2*y+2*z,-3*x-6*y-6*z] solution=DESolution(F,[t,0,2],[[x,(2,-1)],(y,[2,4]),[z,[2,10]]]) solution.coordinates(['red','blue','green']) solution.phase_plot()
(x, y, z)
(122222366)\displaystyle \left(\begin{array}{rrr} -1 & 2 & 2 \\ 2 & 2 & 2 \\ -3 & -6 & -6 \end{array}\right)
[(0\displaystyle 0, [(0,1,1)\displaystyle \left(0,\,1,\,-1\right)], 1\displaystyle 1), (2\displaystyle -2, [(1,12,0)\displaystyle \left(1,\,-\frac{1}{2},\,0\right)], 1\displaystyle 1), (3\displaystyle -3, [(1,0,1)\displaystyle \left(1,\,0,\,-1\right)], 1\displaystyle 1)]
x3+5x2+6x\displaystyle x^{3} + 5x^{2} + 6x
[(0\displaystyle 0, 1\displaystyle 1), (2\displaystyle -2, 1\displaystyle 1), (3\displaystyle -3, 1\displaystyle 1)]
[0\displaystyle 0, 2\displaystyle -2, 3\displaystyle -3]
3D rendering not yet implemented
x,y,z=var('x,y,z') # Next we define the parameters sigma=10 rho=28 beta=8/3 # The Lorenz equations lorenz=[sigma*(y-x),x*(rho-z)-y,x*y-beta*z] # Time and initial conditions N=250000 tmax=250 h=tmax/N t=srange(0,tmax+h,h) ics=[0,1,1] sol=desolve_odeint(lorenz,ics,t,[x,y,z],rtol=1e-13,atol=1e-14) X=sol[:,0] Y=sol[:,1] Z=sol[:,2] # Plot the result from mpl_toolkits.mplot3d import axes3d from matplotlib import pyplot as plt # Call the plot function if you want to plot the data def plot(): fig = plt.figure(1) ax = fig.add_subplot(111, projection='3d') ax.plot_wireframe(X, Y, Z, rstride=10, cstride=10) ax.set_xlabel('X(t)') ax.set_ylabel('Y(t)') ax.set_zlabel('Z(t)') plt.show() plot()
var('a,x,y,z,w') initial=[[x,[y,1]],[y,[z,2]],[z,[w,3]]] p=[[x,y,z],[1,2,3]] [v for v,_ in initial] [a for _,(a,b) in initial] [b for _,[_,b] in initial] [c for _,c,_ in p] initial[0][1]
(a, x, y, z, w) [x, y, z] [y, z, w] [1, 2, 3] [y, 2] [y, 1]