2D Systems



In [1]:
%run Steppers.ipynb #Import stepper functions (Euler, RK4, etc)
%run ODEs.ipynb     #Import ODEs (Pendulum etc)
%run Tools.ipynb
import matplotlib
matplotlib.use('nbagg')
import matplotlib.pyplot as plt

In [2]:
ncycles = 5
t0 = 0
tf = 5*(2*np.pi)
h  = 0.05
t  = np.linspace(t0,tf,int((tf-t0)/h))  #Generate timesteps

x0=(0,1)
omega=1
b=0

vectorPoints=np.array([(0,1),(0,-1),(1,0),(2,0),(-2,0),(-1,0),(1,1),(-1,-1)])
vectors=vectorPoints.copy()

for i, x in enumerate(vectorPoints):
    vectors[i]=(SimplePendulum(x,t,b,omega))

sol = RK4(SimplePendulum, x0, t, args=(b,omega,))
x, v = np.hsplit(sol, len(sol[0]))
plt.plot(x,v, label=r"$x_0,y_0={}$".format(x0))
plt.xlabel("x")
plt.ylabel("v")

x,y=np.hsplit(vectorPoints,2)
u,v=np.hsplit(vectors,2)

plt.quiver(x,y,u,v,)
plt.axis("scaled")
plt.title("Phase Portrait")
plt.legend()
plt.show()



In [3]:
ncycles = 5
t0 = 0
tf = 5*(2*np.pi)
h  = 0.05
t  = np.linspace(t0,tf,int((tf-t0)/h))  #Generate timesteps

x0=(0,2)
omega=1
b=0.5

vectorPoints=np.array([(0,1),(0,-1),(1,0),(2,0),(-2,0),(-1,0),(1,1),(-1,-1)])
vectors=vectorPoints.copy()


for i, x in enumerate(vectorPoints):
    vectors[i]=SimplePendulum(x,t,b,omega)

sol = RK4(SimplePendulum, x0, t, args=(b,omega,))
x, v = np.hsplit(sol, len(sol[0]))
plt.plot(x,v, label=r"$x_0,y_0={}$".format(x0))
plt.xlabel("x")
plt.ylabel("v")

x,y=np.hsplit(vectorPoints,2)
u,v=np.hsplit(vectors,2)

plt.quiver(x,y,u,v)
plt.axis("scaled")
plt.title(r"Phase Portrait with damping $b={}$".format(b))
plt.legend()
plt.show()



In [4]:
t = np.arange(0,30,0.01)

f0=(-1,-2)

sol=RK4(Nonlinear4, f0, t)
x,y=np.hsplit(sol,len(f0))

lim=10
y[x>lim]=np.inf
x[x>lim]=np.inf
y[x<-lim]=-np.inf
x[x<-lim]=-np.inf

plt.plot(t,x)
plt.plot(t,y)
plt.show()

for f0 in ((1.99,1.99),(1,1),(-3,-5),(-4,-4),(-4,5)):
    sol=RK4(Nonlinear4, f0, t)
    x,y=np.hsplit(sol,len(f0))
    plt.plot(x,y, color="C0")
plt.axis("scaled")
plt.xlim((-10,10))
plt.ylim((-10,10))
plt.show()


/home/adam/.local/lib/python3.6/site-packages/ipykernel_launcher.py:4: RuntimeWarning: overflow encountered in double_scalars
  after removing the cwd from sys.path.

In [5]:
t = np.arange(0,30,0.01)
xLims=(-6,5)
yLims=(-6,5)
f0=(-1,-2)

sol=RK4(Nonlinear5, f0, t)
x,y=np.hsplit(sol,len(f0))

lim=10
y[x>lim]=np.inf
x[x>lim]=np.inf
y[x<-lim]=-np.inf
x[x<-lim]=-np.inf

plt.plot(t,x)
plt.plot(t,y)
plt.show()

for f0 in ((1.99,1.99),(1,1),(-3,-5),(-4,-4),(-4,5)):
    sol=RK4(Nonlinear5, f0, t)
    x,y=np.hsplit(sol,len(f0))
    plt.plot(x,y, color="C0")
plt.axis("scaled")
PlotQuiver(Nonlinear5,xLims=(xLims[0],xLims[1]),yLims=(yLims[0],yLims[1]),spacing=0.3)
plt.xlim(xLims)
plt.ylim(yLims)
plt.show()


/home/adam/.local/lib/python3.6/site-packages/ipykernel_launcher.py:4: RuntimeWarning: overflow encountered in double_scalars
  after removing the cwd from sys.path.
/home/adam/.local/lib/python3.6/site-packages/ipykernel_launcher.py:3: RuntimeWarning: invalid value encountered in double_scalars
  This is separate from the ipykernel package so we can avoid doing imports until

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