In [2]:
%pylab inline
from numpy import *
from scipy import integrate
from ipywidgets import interact
In [3]:
def driven_pendulum(t,theta,A,w,Q):
return [
theta[1],
A*sin(w*t) - theta[1]/Q - sin(theta[0])
]
In [4]:
def chaotic(init_angle=0,init_vel=0,driv_ampl=1.5,driv_freq=2./3,qual=0.5):
initial_conditions,t0 = [init_angle, init_vel],0
time_list = []
sol = []
time_step = 0.01
end_time = 20
system = integrate.ode(driven_pendulum)
system.set_f_params(driv_ampl,driv_freq,qual)
system.set_initial_value(initial_conditions,t0)
while system.successful and system.t < end_time:
time_list.append(system.t)
sol.append(system.integrate(system.t+time_step))
sol =asarray(sol)
fig = figure(figsize=(20,5))
ax1 = fig.add_subplot(1,2,1)
ax1.set_ylim(-2,2)
ax1.set_xlabel("time").set_color('white')
ax1.plot(time_list,sol)
ax2 = fig.add_subplot(1,2,2)
# ax2.set_xlim(-2,2)
# ax2.set_ylim(-2,2)
ax2.set_xlabel("position").set_color('white')
ax2.set_ylabel("velocity").set_color('white')
ax2.set_title("Phase space plot").set_color('white')
ax2.plot(sol.T[0],sol.T[1])
interact(
chaotic,
init_angle=(-1.0,1.0,0.1),
init_vel=(-2.0,2.0,0.1),
driv_ampl=(0.0,10.0,0.5),
driv_freq=(0.0,2.0,0.1),
qual=(0.1,10.0,0.01)
)
Out[4]:
In [5]:
def resonance(qual=0.5):
initial_conditions,t0 = [0,0],0
time_step = 0.01
end_time = 20
freq_list = linspace(0,5,10**2)
p2p = []
for driv_freq in freq_list:
sol = []
system = integrate.ode(driven_pendulum)
system.set_f_params(1,driv_freq,qual)
system.set_initial_value(initial_conditions,t0)
while system.successful and system.t < end_time:
# time_list.append(system.t)
sol.append(system.integrate(system.t+time_step))
sol=asarray(sol)
# Peak to Peak current
p2p.append(sol.T[1].max() - sol.T[1].min())
fig = figure(figsize=(20,5))
ax1 = fig.add_subplot(1,1,1)
ax1.set_ylim(0,10)
ax1.set_xlabel("frequency").set_color('white')
ax1.set_ylabel("response current").set_color('white')
ax1.plot(freq_list,p2p)
interact(
resonance,
qual=(0.1,10.0,0.01)
)
Out[5]:
So this resembles nothing like the beautiful resonance curves we see. Chaos has no resonance.