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# Example : Lorenz System #
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# Liu Condition for experiment04: [[a^2*b + a*b^2 - a^2*r + a^2 + 2*a*b + b^2 - a*r + a + b], [-a*b*r + a*b], [2*a*b*crv1 + b^2*crv1 - 2*a*r*crv1 + a^2*crv2 + 2*a*b*crv2 - a^2*crv3 + 2*a*crv1 + 2*b*crv1 - r*crv1 + 2*a*crv2 + 2*b*crv2 - a*crv3 + crv1 + crv2]]
# consider the case a+1 = 0
# where a^2*b+a*b^2-a^2*r+a^2+2*a*b+b^2-a*r+a+b = (a*b-a*r+b^2+a+b)*(a+1)

# Parameters and Variables
a,b,r = var("a b r")                   # Parameters
PR.<a,b,r> = PolynomialRing(QQ)
PF  = Frac(PR);  P = PF.gens();
x,y,z = var("x y z");                  # Variables
VR.<x,y,z> = PolynomialRing(PF); V = VR.gens(); 

# a vector field H 
H1 = [[a*(y-x)], [r*x-y], [-b*z]]     # Linear Polynomials
H2 = [[0],       [-x*z],  [x*y]]   # Quadratic Polynomials
ZB = [[0], [0], [0]]; # 3, 4, 5-th Homogeneous Polynomials
H  = [H1, H2, ZB, ZB, ZB]
P  = [[a+1], [a*b-a*r+b^2+a+b], [1]]

# Hopf bifurcation of Multiplicity 2
attach("main.sage"); hopf(2, V, P, H, PR, PF, VR, pol_opt = P)

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