Simulating, Analyzing, and Animating Dynamical Systems by Bard Ermentrout



In [1]:

    
from pyndamics import *

Section 2.1

# Linear2d.ode
#
# right hand sides
x' =a*x+b*y
y'=c*x+d*y
#
# parameters
par a=0,b=l,c=-l,d=0
#
# some initial conditions init x=l,y=0
#
# we are done
done



In [2]:

    
sim=Simulation()
sim.add("x'=a*x+b*y",1,plot=True)
sim.add("y'=c*x+d*y",0,plot=True)
sim.params(a=0,b=1,c=-1,d=0)
sim.run(0,20)









    












    












    





<matplotlib.figure.Figure at 0x11204bef0>



In [3]:

    
phase_plot(sim,'x','y')
axis('equal')









    Out[3]:





(-1.0, 1.0, -1.0, 1.0)

Section 2.7

# Fitzhugh-Nagumo equations
v'=I+v*(1-v)*(v-a) -w
w'=eps*(v-gamma*w)
par 1=0,a=.1,eps=.1,gamma=.25
@ xp=V,yp=w,xlo=-.25,xhi=l.25,ylo=-.5,yhi=l,total=100 @ maxstor=10000
done



In [4]:

    
sim=Simulation()
sim.add("v'=I+v*(1-v)*(v-a) -w",0,plot=True)
sim.add("w'=eps*(v-gamma*w)",0,plot=True)
sim.params(I=0.4,a=0.1,eps=0.1,gamma=0.25)
sim.run(100)









    












    












    





<matplotlib.figure.Figure at 0x112e812e8>



In [5]:

    
vector_field(sim,v=linspace(-.25,1.25,30),w=linspace(-.5,1,30))
phase_plot(sim,'v','w')

Section 3.2.1

# forced duffing equation
x'=v
v'=a*x*(l-x~2) -f*v+c*cos(omega*t) 
par a=l,f=.2,c=.3,omega=l
init x=.l,v=.l 
done



In [6]:

    
sim=Simulation()
sim.add("x'=v",.1)
sim.add("v'=a*x*(1-x**2)-f*v+c*cos(omega*t)",.1)
sim.params(a=1,f=.2,c=.3,omega=1)
sim.run(200,num_iterations=20000)
phase_plot(sim,'x','v')

Section 3.3.1 - User defined functions

# the wilson-cowan equations 
u'=-u+f(a*u-b*v+p) 
v'=-v+f(c*u-d*v+q) 
f(u)=!/(l+exp(-u)
par a=16,b=12,c=16,d=5,p=-l,q=-4
@ xp-u,yp=v,xlo=-.125,ylo=-.125,xhi=l,yhi=l 
done



In [7]:

    
def f(u):
    return 1.0/(1+exp(u))

sim=Simulation()
sim.add("u'=-u+f(a*u-b*v+p)",0,plot=True)
sim.add("v'=-v+f(c*u-d*v+q)",0,plot=True)
sim.functions(f)
sim.params(a=16,b=12,c=16,d=5,p=-1,q=-4)
sim.run(10)









    












    












    





<matplotlib.figure.Figure at 0x11415fd30>

Section 3.4 - Auxiliary variables

# the undamped pendulum 
theta'=v 
v'=-(g/1)*sin(theta) 
par g=9.8,1=2,ra=l
aux E=m*((l*v)~2/2+g*l*(1-cos(theta))) 
done



In [8]:

    
sim=Simulation()
sim.add("theta'=v",10,plot=True)
sim.add("v'=-(g/L)*sin(theta)",plot=True)
sim.add("E=m*((L*v)**2/2+g*L*(1-cos(theta)))",plot=True)
sim.params(g=9.8,L=2,m=1)
sim.run(10)









    












    












    












    





<matplotlib.figure.Figure at 0x114794160>

Section 3.4.1 - Fixed Variables

# standard nonlinear oscillator 
R=x"2+y~2
x'=x*(a-R)-y*(l+q*R) 
y'=y*(a-R)+x*(l+q*R)
par a=l,q=l 
done



In [9]:

    
sim=Simulation()
sim.add("R=x**2+y**2")
sim.add("x'=x*(a-R)-y*(1+q*R)",10,plot=True)
sim.add("y'=y*(a-R)+x*(1+q*R)",0,plot=True)
sim.params(a=1,q=1 )
sim.run(10)









    












    












    





<matplotlib.figure.Figure at 0x1135ba7f0>

Section 4.5

# example from Golubitsky #
x'=x+y+x~2-y~2+0.1 
y'=y-2*x*y+x~2/2+y~2
@ xp=x,yp=y
@ x l o = - 6 , x h i =6 , y l o = - 6 , y h i =6 , c l t = . 02 
done



In [10]:

    
sim=Simulation()
sim.add("x'=x+y+x**2-y**2+0.1",0.1,plot=True)
sim.add("y'=y-2*x*y+x**2/2+y**2",0,plot=True)
sim.run(1.2)









    












    












    





<matplotlib.figure.Figure at 0x1141378d0>



In [11]:

    
vector_field(sim,x=linspace(-6,6,30),y=linspace(-6,6,30))



In [ ]:



In [ ]:

￼Simulating, Analyzing, and Animating Dynamical Systems by Bard Ermentrout

Section 2.1

Section 2.7

Section 3.2.1

Section 3.3.1 - User defined functions

Section 3.4 - Auxiliary variables

Section 3.4.1 - Fixed Variables

Section 4.5

Simulating, Analyzing, and Animating Dynamical Systems by Bard Ermentrout