def dfdx(self):
"""
Jacobian of x + dt*dxdt.
"""
nU,J,h,b,c = self.nU,self.J,self.h,self.b,self.c
assert is1d(x)
F = np.zeros((ndim(),ndim()))
# X
md = lambda i: np.mod(i,nU)
for i in range(nU):
# wrt. X
F[i,i] = - dt + 1
F[i,md(i-2)] = - dt * x[md(i-1)]
F[i,md(i+1)] = + dt * x[md(i-1)]
F[i,md(i-1)] = dt *(x[md(i+1)]-x[md(i-2)])
# wrt. Y
F[i,nU+self.iiY[i]] = dt * -h*c/b
# Y
md = lambda i: nU + np.mod(i-nU,nU*J)
for i in range(nU,(J+1)*nU):
# wrt. Y
F[i,i] = -dt*c + 1
F[i,md(i-1)] = +dt*c*b * x[md(i+1)]
F[i,md(i+1)] = -dt*c*b * (x[md(i+2)]-x[md(i-1)])
F[i,md(i+2)] = -dt*c*b * x[md(i+1)]
# wrt. X
F[i,self.iiX[i-nU]] = dt * h*c/b
return F
def TLM(x):
"""Tangent linear model"""
assert is1d(x)
x,y,z = x
TLM=np.array(
[[-sig , sig , 0],
[rho-z , -1 , -x],
[y , x , -beta]])
return TLM
def TLM(x):
"""Tangent linear model"""
assert is1d(x)
x,y,z = x
TLM=np.array(
[[-a , -2*y , -2*z],
[y-b*z , x-1 , -b*x],
[b*y+z , b*x , x-1]])
return TLM
def TLM(x):
"""Tangent linear model"""
assert is1d(x)
m = len(x)
TLM = np.zeros((m,m))
md = lambda i: np.mod(i,m)
for i in range(m):
TLM[i,i] = -1.0
TLM[i, i-2 ] = -x[i-1]
TLM[i,md(i+1)] = +x[i-1]
TLM[i, i-1 ] = x[md(i+1)]-x[i-2]
return TLM