[SciPy-user] Volterra system identification

Georg Holzmann grh@mur...
Mon Apr 28 05:47:58 CDT 2008


Hallo!

Thanks for the answer - but I don't understand it ... ;)
I mean volterra series for nonlinear system identification.

LG
Georg

>>From a previous post:
> 
> import numpy as n
> import pylab as p
> import scipy.integrate as integrate
> 
> """
> If you look closely to the second graph, you can see that the trajectory
> crosses some arrows of the direction field. I had this problem too,
> before forcing matplotlib to use equal axis.
> """
> 
> alpha, delta = 1, .25
> beta, gamma = .2, .05
> 
> def dr(r, f): return alpha*r - beta*r*f    
>     
> def df(r, f): return gamma*r*f - delta*f
> 
> def derivs(state, t):
>     """ Map the state variable [rabbits, foxes] to the derivitives
> [deltar, deltaf] at time t """
>     #print t, state
>     r, f = state  # rabbits and foxes
>     deltar = dr(r, f)  # change in rabbits
>     deltaf = df(r, f) # change in foxes
>     return deltar, deltaf
> 
> # the initial population of rabbits and foxes
> r0 = 20
> f0 = 10
> 
> t = n.arange(0.0, 100, 0.1)
> 
> y0 = [r0, f0]  # the initial [rabbits, foxes] state vector
> y = integrate.odeint(derivs, y0, t)
> r = y[:,0]  # extract the rabbits vector
> f = y[:,1]  # extract the foxes vector
> 
> p.figure()
> p.plot(t, r, label='rabbits')
> p.plot(t, f, label='foxes')
> p.xlabel('time (years)')
> p.ylabel('population')
> p.title('population trajectories')
> p.grid()
> p.legend()
> #p.savefig('lotka_volterra.png', dpi=150)
> #p.savefig('lotka_volterra.eps')
> 
> p.figure()
> p.plot(r, f)
> p.xlabel('rabbits')
> p.ylabel('foxes')
> p.title('phase plane')
> 
> 
> # make a direction field plot with quiver
> rmax = 1.1 * r.max()
> fmax = 1.1 * f.max()
> R, F = n.meshgrid(n.arange(-1, rmax), n.arange(-1, fmax))
> dR = dr(R, F)
> dF = df(R, F)
> p.quiver(R, F, dR, dF)
> 
> 
> R, F = n.meshgrid(n.arange(-1, rmax, .1), n.arange(-1, fmax, .1))
> dR = dr(R, F)
> dF = df(R, F)
> 
> p.contour(R, F, dR, levels=[0], linewidths=3, colors='black')
> p.contour(R, F, dF, levels=[0], linewidths=3, colors='black')
> p.ylabel('foxes')
> p.title('trajectory, direction field and null clines')
> 
> #p.savefig('lotka_volterra_pplane.png', dpi=150)
> #p.savefig('lotka_volterra_pplane.eps')
> 
> 
> p.show()
> 
>> Thanks for any hint,
>> LG
>> Georg
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