+- Tetration Forum (https://tetrationforum.org)
+-- Forum: Tetration and Related Topics (https://tetrationforum.org/forumdisplay.php?fid=1)
+--- Forum: Mathematical and General Discussion (https://tetrationforum.org/forumdisplay.php?fid=3)
+--- Thread: Second derivitive of a dynamical system (/showthread.php?tid=36)
Second derivitive of a dynamical system - Daniel - 08/23/2007
Without lose of generality, set the fixed point of \( f(x) \) at zero. It is well known that \( D f^t(0)=f'(0)^t \). Consider that
\( D^2 f(g(x)) = f''(g(x)) g'(x)^2 + f'(g(x)) g''(x) \), then
\( D^2 f(f^{t-1}(x)) = f''(f^{t-1}(x)) (D f^{t-1}(x))^2 + f'(f^{t-1}(x)) D^2 f^{t-1}(x) \). But this gives a functional equation for a geometrical progression solved at \( x=0 \) by
\( D^2 f^t(0) = f''(0) \sum_{k=0}^{t-1} f'(0)^{2t-k-2} \).
RE: Second derivitive of a dynamical system - bo198214 - 08/23/2007
Daniel Wrote:It is well known that \( D f^t(0)=f'(0)^t \).
This is not well-known but one requirement for uniqueness.
RE: Second derivitive of a dynamical system - Daniel - 08/24/2007
The idea here is that the Taylor series for an arbitrary dynamical system \( f^t(x) \) can be derived. The second derivitive's dependancy on a geometrical progression results in the different types of fixed points. The higher derivitives also are built up from nested geometrical progressions. The regular formula for geometrical progressions breaks down when the roots of unity are involved which gives an explaination for why therre are different types of fixed points.
RE: Second derivitive of a dynamical system - bo198214 - 08/24/2007
Daniel Wrote:The regular formula for geometrical progressionsWhat do you mean by this?
RE: Second derivitive of a dynamical system - Daniel - 08/24/2007
bo198214 Wrote:Daniel Wrote:The regular formula for geometrical progressionsWhat do you mean by this?
That \( \sum_{k=0}^n x^k = \frac{1-x^{n+1}}{1-x} \) unless \( x^m=1 \) for some integer \( m \).
For \( x=1 \) we have \( \sum_{k=0}^n x^k = n+1 \).
Here \( x \) is the Lyponouv characteristic - the first derivitive at a fixed point. Note that the regular formula for geometrical progressions blows up for \( x=1 \) which happens at \( a=e^{\frac{1}{e}} \).