Lapunov
Description:
A concept of Lapunov exponents was introduced by Oseledec (1968) named for Russian mathematician A. M. Lapunov (1857-1918). Lapunov exponents are measure of sensitivity for initial conditions. Let's have two different staritng points that differ of e: x and x + e. After n iterations divergence can be describe by:
e(n) = eenl
l<0 trajectories are converged
l>0 evolution is chaotic.
Let:
xn+1 = f(xn)
[fn(x+e) - fn(x)] = eenl
ln[(fn(x+e) - fn(x))/e] = nl
l = 1/n ln[dfn/dx]
l = 1/n*[ln(f'(x1))+ln(f'(x2))+ln(f'(x3))+...+ln(f'(xn))]
For n dimentional representation we have n Lapunov exponents:
V = V0e(l1+l2+...+ln)n
for dissipative systems sum of Lapunov exponents is less then zero.
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