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# Fireflies Synchronization
''' @Author: Sanyukta Suman
Purpose: To design a model of firefly synchronization in 2 dimension
For now my purpose is to just design a model.
also analyze firefly response to the stimulus.
Steps to design a model:
1. Define equation
2. Find the equilibrium point
3. Fixed point
4. Stability
5. Ploting a graph
6. trajectories and phases
'''
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# Step 1: Defining Equations
''' I am using a simplified model found in the chapter on firefly
synchronization in Strogatz.
He examines a simple model of how firefly flashing rythm responds
to flashing stimuli.
Let the Phase of the firefly flashing be given by theta(t) where theta=0 corresponds to the instant
when a flash is emitted.
Assume this phase is 2pi periodic so that theta=2pi*n, where n= any integer
Then theta_dot=w -->represets its natural frequency
Now let there be a stimulus with 2pi periodic phase uptheta,
uptheta=0 corresponds to the instant the stimulus flashes
frequency----> uptheta=omega
Therefore, parameters will be w, omega for now.
We know that,
if the stimulus flashes before the firefly---> flirefly will speed up in attempt to synchronize.
if the stimulus flashes after the firefly----> firefly slows down.
The simple model will be
uptheta_dot=w+Asin(uptheta-theta)
where A is the resetting strength of the firefly.
This is a measure of the firefly's ability to change its instataenous frequency in response to a stimulus.
which also implies A>0.
From this equation, if 0<uptheta-theta<pi----> stimulus is ahead of the firefly
then upteta_dot>w --->firefly speeds up
Similarly, if -pi<uptheta-theta<0----> stimulus is behind the firefly
then optheta_dot<w---> firefly slows down
Let phi=uptheta-theta---> denotes phase difference between the stimulus and firefly.
Here theta, uptheta and phi are three are in 2pi periodic.
'''
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#Entertainment
'''Entertainemnt is a term to describe the situation in which the firefly is able to
match its instatntaneous frequency to that of the stimulus.
if the entertainemnet occurs, the phase difference (phi), approaches a fixed point.
If this constant is 0, we say firefly is syncronized.
if the constant is not equal to zero, we say firefly is phase-locked to the stimulus.
we can plot phi vs time in order to classiyf the behaviour
'''
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import numpy as np
from math import cos, exp, pi
from scipy.integrate import odeint
import matplotlib.pyplot as plt
import warnings
import random
warnings.filterwarnings('ignore')
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#Graph for Phase of firefly(theta_dot) and phase of stimulus(uptheta_dot)
#first define initial conditions and parameters
w0=12.5664
omega0=16.5664
c0=0
n=2
#Define time parameter 0 to 2 pi
t=np.linspace(0,2*np.pi,1000)
#Fuction for natural frequency of firefly
def theta_dot(w,t):
w0=w
theta_dot=n*w
return theta_dot
#Function for frequency for a stimulus
def uptheta_dot(omega,t):
omega0=omega
uptheta_dot=n*omega
return uptheta_dot
#applying in both the functions
y2=odeint(theta_dot,w0,t)
y3=odeint(uptheta_dot,omega0,t)
plt.plot(t,y2,label='Phase of firefly')
plt.plot(t,y3,label='Phase of stimulus')
#plt.plot(t,y4,label='Phase difference')
plt.ylabel('Phase')
plt.xlabel('t')
plt.legend()
plt . savefig ('Phase.png')
plt.show()
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#Function for Phase difference (uptheta-theta)
t1=np.linspace(0,2*np.pi,1000)
def p(omega,t1,w,c):
dpdt=omega-w
return dpdt
y4=odeint(p,omega0,t1,args=(w0,c0))
plt.plot(t1,y4,label='Phase difference')
plt.ylabel('Angular velocity')
plt.xlabel('t')
plt.legend()
plt.show()
plt . savefig ('ExpGrowth.png')
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#define fucntion for dpdt
A=1
w0=12.5664
omega0=14.6121
omega0=14.6121
phi0=1.5708
#y0=[phi0,w0,omega0]
t=np.linspace(0,2*np.pi,1000)
def phi_dot(phi,t,w0,omega0):
count=0
phi0=phi
phi_dot=omega0-w0-(A* np.sin(phi))
phi1=phi
return phi_dot
sol4=odeint(phi_dot,phi0,t,args=(w0,omega0))
plt.plot(t,sol4,label='w=11.5564, omega=18.5564, A=3')
plt.title('Phase Drift')
plt.ylabel('Phi')
plt.xlabel('t')
plt.legend()
plt.grid(True)
#plt . savefig ('phaselock.png')
plt.show()
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#fixed point analysis and stability
#Plotting graph phi vs phi_dot
phi=np.linspace(-4,4,100)
phi_dot=omega0-w0-(A* np.sin(phi))
plt.plot(phi,phi_dot,label='w=12.5564 Hz, omega=14.6121 Hz, A=1')
plt.autoscale(enable=True)
plt.grid(True)
plt.title('Analysis of resetting strength of firefly (A)')
plt.xlabel('Phi')
plt.ylabel('phi_dot')
plt.legend()
#plt.set_ylim(-.5, 1.5)
plt.ylim(-4, 4)
plt . savefig ('flock.png')
plt.show()
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print (phi)
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#Simulation
#Define Intial parameters
w0=12.5664 #natural frequency of firefly
omega10=14.5664 #frequency of stimulus 1
omega20=15.5664 #frequency of stimulus 2
A=3 #resetting strength of firefly - its ability to change its natural frequency to match that of stimulus
t=np.linspace(0,2*np.pi,1000)
phi10=0
phi20=0
y0=[phi10,phi20]
def dydt(y,t):
phi1,phi2=y
dydt1=omega10-w0-A*np.sin(phi1)
dydt2=omega20-w0-A*np.sin(phi2)
test=[dydt1,dydt2]
return test
sols=odeint(dydt,y0,t)
fig,(ax1, ax2, ax3)=plt.subplots(nrows=3,ncols=1,figsize=(15,10))
ax1 . plot (t,sols[:,0],label='Stimulus 1')
ax1.set_ylabel('phi1')
ax1.set_ylim(-3*np.pi, np.pi)
ax1.grid(True)
ax2 . plot (t,sols[:,1],label='Stimulus 2')
ax2.set_ylabel('phi2')
ax2.grid(True)
ax3 . plot (t,sols[:,0],label='Stimulus 1')
ax3 . plot (t,sols[:,1],label='Stimulus 2')
ax2.set_ylabel('phi2')
ax2.grid(True)
#plt.ylim(-3*np.pi, np.pi)
plt.xlabel('t')
plt.legend()
plt . savefig ('twostimuli.png')
plt.show()
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In [158]:
plt.scatter(y,z)
plt.show()
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import matplotlib.cm as cm
import matplotlib.colors as colors
norm = colors.Normalize(vmin=-14, vmax=86)
f2rgb = cm.ScalarMappable(norm=norm, cmap=cm.get_cmap('RdYlGn'))
def f2hex(f2rgb, f):
rgb = f2rgb.to_rgba(f)[:3]
return '#%02x%02x%02x' % tuple([255*fc for fc in rgb])