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Reference numerical computation for: Feigenbaum Constant α
The Feigenbaum constant α governs the geometric scaling of the attractor in
period-doubling bifurcations. It is defined via the functional equation for
the universal function g(x) at the accumulation point of bifurcations:
g(x) = -α · g(g(-x/α))
where g(0) = 1 and g'(0) = 0 (g has a quadratic maximum at 0).
The scaling factor α = 2.502907875095892822... is universal.
"""
from mpmath import mp, mpf
# Set precision to 110 decimal places
mp.dps = 110
def compute():
"""
Return the Feigenbaum constant α.
The constant can be computed via:
1. The renormalization group fixed-point equation
2. Measuring the scaling of superstable periodic orbits
3. The width ratio of the attractor at successive period doublings
For ground truth, we use the high-precision published value computed via
renormalization group methods.
The value has been computed to 1000+ digits by Briggs (1997) and others.
"""
# Feigenbaum α computed to 100+ digits
# Source: K. Briggs (1997), D. Broadhurst (1999)
# Available here: https://oeis.org/A006891
alpha = mpf(
"2.50290787509589282228390287321821578638127137672714997733619205677923546317959020670329964974643383412959"
)
return alpha
if __name__ == "__main__":
result = compute()
print(str(result))
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