Modeling a chaotic system
The pendulum is a classical physics system that we’re all aware of. Be it a grandfather clock or a baby on a swing, we have now seen the common, periodic movement of the pendulum. A single pendulum is nicely outlined in classical physics, however the double pendulum (a pendulum hooked up to the top of one other pendulum) is literal chaos. On this article, we’re going to construct on our intuitive understanding of pendulums and mannequin the chaos of the double pendulum. The physics is fascinating and the numerical strategies wanted are an important instrument in anybody’s arsenal.
On this article we’ll:
- Study harmonic movement and mannequin the conduct of a single pendulum
- Be taught the basics of chaos principle
- Mannequin the chaotic conduct of a double pendulum numerically
Easy Harmonic Movement
We describe the periodic oscillating motion of a pendulum as harmonic movement. Harmonic movement happens when there’s motion in a system that’s balanced out by a proportional restoring power in the other way of mentioned motion. We see an instance of this in determine 2 the place a mass on a spring is being pulled down as a result of gravity, however this places power into the spring which then recoils and pulls the mass again up. Subsequent to the spring system, we see the peak of the mass going round in a circle referred to as a phasor diagram which additional illustrates the common movement of the system.
Harmonic movement may be damped (lowering in amplitude as a result of drag forces) or pushed (growing in amplitude as a result of outdoors power being added), however we’ll begin with the only case of indefinite harmonic movement with no outdoors forces appearing on it (undamped movement). That is sort of movement is an efficient approximation for modeling a single pendulum that swings at a small angle/low amplitude. On this case we are able to mannequin the movement with equation 1 beneath.
We will simply put this operate into code and simulate a easy pendulum over time.
def simple_pendulum(theta_0, omega, t, phi):
theta = theta_0*np.cos(omega*t + phi)
return theta#parameters of our system
theta_0 = np.radians(15) #levels to radians
g = 9.8 #m/s^2
l = 1.0 #m
omega = np.sqrt(g/l)
phi = 0 #for small angle
time_span = np.linspace(0,20,300) #simulate for 20s cut up into 300 time intervals
theta = []
for t in time_span:
theta.append(simple_pendulum(theta_0, omega, t, phi))
#Convert again to cartesian coordinates
x = l*np.sin(theta)
y = -l*np.cos(theta) #adverse to ensure the pendulum is dealing with down
Full Pendulum Movement with Lagrangian Mechanics
A easy small angle pendulum is an efficient begin, however we wish to transcend this and mannequin the movement of a full pendulum. Since we are able to not use small angle approximations it’s best to mannequin the pendulum utilizing Lagrangian mechanics. That is an important instrument in physics that switches us from wanting on the forces in a system to wanting on the power in a system. We’re switching our body of reference from driving power vs restoring power to kinetic vs potential power.
The Lagrangain is the distinction between kinetic and potential power given in equation 2.
Substituting within the Kinetic and Potential of a pendulum given in equation 3 yields the Lagrangain for a pendulum seen is equation 4
With the Lagrangian for a pendulum we now describe the power of our system. There’s one final math step to undergo to remodel this into one thing that we are able to construct a simulation on. We have to bridge again to the dynamic/power oriented reference from the power reference utilizing the Euler-Lagrange equation. Utilizing this equation we are able to use the Lagrangian to get the angular acceleration of our pendulum.
After going by the maths, we have now angular acceleration which we are able to use to get angular velocity and angle itself. This may require some numerical integration that shall be specified by our full pendulum simulation. Even for a single pendulum, the non-linear dynamics means there isn’t a analytical answer for fixing for theta, thus the necessity for a numerical answer. The mixing is kind of easy (however highly effective), we use angular acceleration to replace angular velocity and angular velocity to replace theta by including the previous amount to the latter and multiplying this by a while step. This will get us an approximation for the realm below the acceleration/velocity curve. The smaller the time step, the extra correct the approximation.
def full_pendulum(g,l,theta,theta_velocity, time_step):
#Numerical Integration
theta_acceleration = -(g/l)*np.sin(theta) #Get acceleration
theta_velocity += time_step*theta_acceleration #Replace velocity with acceleration
theta += time_step*theta_velocity #Replace angle with angular velocity
return theta, theta_velocityg = 9.8 #m/s^2
l = 1.0 #m
theta = [np.radians(90)] #theta_0
theta_velocity = 0 #Begin with 0 velocity
time_step = 20/300 #Outline a time step
time_span = np.linspace(0,20,300) #simulate for 20s cut up into 300 time intervals
for t in time_span:
theta_new, theta_velocity = full_pendulum(g,l,theta[-1], theta_velocity, time_step)
theta.append(theta_new)
#Convert again to cartesian coordinates
x = l*np.sin(theta)
y = -l*np.cos(theta)
We’ve got simulated a full pendulum, however that is nonetheless a nicely outlined system. It’s now time to step into the chaos of the double pendulum.
Chaos, within the mathematical sense, refers to methods which are extremely delicate to their preliminary circumstances. Even slight modifications within the system’s begin will result in vastly completely different behaviors because the system evolves. This completely describes the movement of the double pendulum. In contrast to the one pendulum, it’s not a nicely behaved system and can evolve in a vastly completely different method with even slight modifications in beginning angle.
To mannequin the movement of the double pendulum, we’ll use the identical Lagrangian strategy as earlier than (see full derivation).
We may even be utilizing the identical numerical integration scheme as earlier than when implementing this equation into code and discovering theta.
#Get theta1 acceleration
def theta1_acceleration(m1,m2,l1,l2,theta1,theta2,theta1_velocity,theta2_velocity,g):
mass1 = -g*(2*m1 + m2)*np.sin(theta1)
mass2 = -m2*g*np.sin(theta1 - 2*theta2)
interplay = -2*np.sin(theta1 - theta2)*m2*np.cos(theta2_velocity**2*l2 + theta1_velocity**2*l1*np.cos(theta1 - theta2))
normalization = l1*(2*m1 + m2 - m2*np.cos(2*theta1 - 2*theta2))theta1_ddot = (mass1 + mass2 + interplay)/normalization
return theta1_ddot
#Get theta2 acceleration
def theta2_acceleration(m1,m2,l1,l2,theta1,theta2,theta1_velocity,theta2_velocity,g):
system = 2*np.sin(theta1 - theta2)*(theta1_velocity**2*l1*(m1 + m2) + g*(m1 + m2)*np.cos(theta1) + theta2_velocity**2*l2*m2*np.cos(theta1 - theta2))
normalization = l1*(2*m1 + m2 - m2*np.cos(2*theta1 - 2*theta2))
theta2_ddot = system/normalization
return theta2_ddot
#Replace theta1
def theta1_update(m1,m2,l1,l2,theta1,theta2,theta1_velocity,theta2_velocity,g,time_step):
#Numerical Integration
theta1_velocity += time_step*theta1_acceleration(m1,m2,l1,l2,theta1,theta2,theta1_velocity,theta2_velocity,g)
theta1 += time_step*theta1_velocity
return theta1, theta1_velocity
#Replace theta2
def theta2_update(m1,m2,l1,l2,theta1,theta2,theta1_velocity,theta2_velocity,g,time_step):
#Numerical Integration
theta2_velocity += time_step*theta2_acceleration(m1,m2,l1,l2,theta1,theta2,theta1_velocity,theta2_velocity,g)
theta2 += time_step*theta2_velocity
return theta2, theta2_velocity
#Run full double pendulum
def double_pendulum(m1,m2,l1,l2,theta1,theta2,theta1_velocity,theta2_velocity,g,time_step,time_span):
theta1_list = [theta1]
theta2_list = [theta2]
for t in time_span:
theta1, theta1_velocity = theta1_update(m1,m2,l1,l2,theta1,theta2,theta1_velocity,theta2_velocity,g,time_step)
theta2, theta2_velocity = theta2_update(m1,m2,l1,l2,theta1,theta2,theta1_velocity,theta2_velocity,g,time_step)
theta1_list.append(theta1)
theta2_list.append(theta2)
x1 = l1*np.sin(theta1_list) #Pendulum 1 x
y1 = -l1*np.cos(theta1_list) #Pendulum 1 y
x2 = l1*np.sin(theta1_list) + l2*np.sin(theta2_list) #Pendulum 2 x
y2 = -l1*np.cos(theta1_list) - l2*np.cos(theta2_list) #Pendulum 2 y
return x1,y1,x2,y2
#Outline system parameters
g = 9.8 #m/s^2m1 = 1 #kg
m2 = 1 #kg
l1 = 1 #m
l2 = 1 #m
theta1 = np.radians(90)
theta2 = np.radians(45)
theta1_velocity = 0 #m/s
theta2_velocity = 0 #m/s
theta1_list = [theta1]
theta2_list = [theta2]
time_step = 20/300
time_span = np.linspace(0,20,300)
x1,y1,x2,y2 = double_pendulum(m1,m2,l1,l2,theta1,theta2,theta1_velocity,theta2_velocity,g,time_step,time_span)
We’ve lastly completed it! We’ve got efficiently modeled a double pendulum, however now it’s time to watch some chaos. Our remaining simulation shall be of two double pendulums with barely completely different beginning situation. We’ll set one pendulum to have a theta 1 of 90 levels and the opposite to have a theta 1 of 91 levels. Let’s see what occurs.
We will see that each pendulums begin off with related trajectories however shortly diverge. That is what we imply after we say chaos, even a 1 diploma distinction in angle cascades into vastly completely different finish conduct.
On this article we discovered about pendulum movement and easy methods to mannequin it. We began from the only harmonic movement mannequin and constructed as much as the advanced and chaotic double pendulum. Alongside the way in which we discovered in regards to the Lagrangian, chaos, and numerical integration.
The double pendulum is the only instance of a chaotic system. These methods exist in all places in our world from inhabitants dynamics, local weather, and even billiards. We will take the teachings we have now discovered from the double pendulum and apply them each time we encounter a chaotic methods.
Key Take Aways
- Chaotic methods are very delicate to preliminary circumstances and can evolve in vastly alternative ways with even slight modifications to their begin.
- When coping with a system, particularly a chaotic system, is there one other body of reference to have a look at it that makes it simpler to work with? (Just like the power reference body to the power reference body)
- When methods get too sophisticated we have to implement numerical options to resolve them. These options are easy however highly effective and provide good approximations to the precise conduct.
All figures used on this article have been both created by the writer or are from Math Photos and full below the GNU Free Documentation License 1.2
Classical Mechanics, John Taylor https://neuroself.information.wordpress.com/2020/09/taylor-2005-classical-mechanics.pdf
Easy Pendulum
def makeGif(x,y,title):
!mkdir framescounter=0
photographs = []
for i in vary(0,len(x)):
plt.determine(figsize = (6,6))
plt.plot([0,x[i]],[0,y[i]], "o-", shade = "b", markersize = 7, linewidth=.7 )
plt.title("Pendulum")
plt.xlim(-1.1,1.1)
plt.ylim(-1.1,1.1)
plt.savefig("frames/" + str(counter)+ ".png")
photographs.append(imageio.imread("frames/" + str(counter)+ ".png"))
counter += 1
plt.shut()
imageio.mimsave(title, photographs)
!rm -r frames
def simple_pendulum(theta_0, omega, t, phi):
theta = theta_0*np.cos(omega*t + phi)
return theta
#parameters of our system
theta_0 = np.radians(15) #levels to radians
g = 9.8 #m/s^2
l = 1.0 #m
omega = np.sqrt(g/l)
phi = 0 #for small angle
time_span = np.linspace(0,20,300) #simulate for 20s cut up into 300 time intervals
theta = []
for t in time_span:
theta.append(simple_pendulum(theta_0, omega, t, phi))
x = l*np.sin(theta)
y = -l*np.cos(theta) #adverse to ensure the pendulum is dealing with down
Pendulum
def full_pendulum(g,l,theta,theta_velocity, time_step):
theta_acceleration = -(g/l)*np.sin(theta)
theta_velocity += time_step*theta_acceleration
theta += time_step*theta_velocity
return theta, theta_velocityg = 9.8 #m/s^2
l = 1.0 #m
theta = [np.radians(90)] #theta_0
theta_velocity = 0
time_step = 20/300
time_span = np.linspace(0,20,300) #simulate for 20s cut up into 300 time intervals
for t in time_span:
theta_new, theta_velocity = full_pendulum(g,l,theta[-1], theta_velocity, time_step)
theta.append(theta_new)
#Convert again to cartesian coordinates
x = l*np.sin(theta)
y = -l*np.cos(theta)
#Use similar operate from easy pendulum
makeGif(x,y,"pendulum.gif")
Double Pendulum
def theta1_acceleration(m1,m2,l1,l2,theta1,theta2,theta1_velocity,theta2_velocity,g):
mass1 = -g*(2*m1 + m2)*np.sin(theta1)
mass2 = -m2*g*np.sin(theta1 - 2*theta2)
interplay = -2*np.sin(theta1 - theta2)*m2*np.cos(theta2_velocity**2*l2 + theta1_velocity**2*l1*np.cos(theta1 - theta2))
normalization = l1*(2*m1 + m2 - m2*np.cos(2*theta1 - 2*theta2))theta1_ddot = (mass1 + mass2 + interplay)/normalization
return theta1_ddot
def theta2_acceleration(m1,m2,l1,l2,theta1,theta2,theta1_velocity,theta2_velocity,g):
system = 2*np.sin(theta1 - theta2)*(theta1_velocity**2*l1*(m1 + m2) + g*(m1 + m2)*np.cos(theta1) + theta2_velocity**2*l2*m2*np.cos(theta1 - theta2))
normalization = l1*(2*m1 + m2 - m2*np.cos(2*theta1 - 2*theta2))
theta2_ddot = system/normalization
return theta2_ddot
def theta1_update(m1,m2,l1,l2,theta1,theta2,theta1_velocity,theta2_velocity,g,time_step):
theta1_velocity += time_step*theta1_acceleration(m1,m2,l1,l2,theta1,theta2,theta1_velocity,theta2_velocity,g)
theta1 += time_step*theta1_velocity
return theta1, theta1_velocity
def theta2_update(m1,m2,l1,l2,theta1,theta2,theta1_velocity,theta2_velocity,g,time_step):
theta2_velocity += time_step*theta2_acceleration(m1,m2,l1,l2,theta1,theta2,theta1_velocity,theta2_velocity,g)
theta2 += time_step*theta2_velocity
return theta2, theta2_velocity
def double_pendulum(m1,m2,l1,l2,theta1,theta2,theta1_velocity,theta2_velocity,g,time_step,time_span):
theta1_list = [theta1]
theta2_list = [theta2]
for t in time_span:
theta1, theta1_velocity = theta1_update(m1,m2,l1,l2,theta1,theta2,theta1_velocity,theta2_velocity,g,time_step)
theta2, theta2_velocity = theta2_update(m1,m2,l1,l2,theta1,theta2,theta1_velocity,theta2_velocity,g,time_step)
theta1_list.append(theta1)
theta2_list.append(theta2)
x1 = l1*np.sin(theta1_list)
y1 = -l1*np.cos(theta1_list)
x2 = l1*np.sin(theta1_list) + l2*np.sin(theta2_list)
y2 = -l1*np.cos(theta1_list) - l2*np.cos(theta2_list)
return x1,y1,x2,y2
#Outline system parameters, run double pendulum
g = 9.8 #m/s^2m1 = 1 #kg
m2 = 1 #kg
l1 = 1 #m
l2 = 1 #m
theta1 = np.radians(90)
theta2 = np.radians(45)
theta1_velocity = 0 #m/s
theta2_velocity = 0 #m/s
theta1_list = [theta1]
theta2_list = [theta2]
time_step = 20/300
time_span = np.linspace(0,20,300)
for t in time_span:
theta1, theta1_velocity = theta1_update(m1,m2,l1,l2,theta1,theta2,theta1_velocity,theta2_velocity,g,time_step)
theta2, theta2_velocity = theta2_update(m1,m2,l1,l2,theta1,theta2,theta1_velocity,theta2_velocity,g,time_step)
theta1_list.append(theta1)
theta2_list.append(theta2)
x1 = l1*np.sin(theta1_list)
y1 = -l1*np.cos(theta1_list)
x2 = l1*np.sin(theta1_list) + l2*np.sin(theta2_list)
y2 = -l1*np.cos(theta1_list) - l2*np.cos(theta2_list)
#Make Gif
!mkdir framescounter=0
photographs = []
for i in vary(0,len(x1)):
plt.determine(figsize = (6,6))
plt.determine(figsize = (6,6))
plt.plot([0,x1[i]],[0,y1[i]], "o-", shade = "b", markersize = 7, linewidth=.7 )
plt.plot([x1[i],x2[i]],[y1[i],y2[i]], "o-", shade = "b", markersize = 7, linewidth=.7 )
plt.title("Double Pendulum")
plt.xlim(-2.1,2.1)
plt.ylim(-2.1,2.1)
plt.savefig("frames/" + str(counter)+ ".png")
photographs.append(imageio.imread("frames/" + str(counter)+ ".png"))
counter += 1
plt.shut()
imageio.mimsave("double_pendulum.gif", photographs)
!rm -r frames
