Re: connection of fractional derivative to Kaplan-Yorke dimension?
From: Roger L. Bagula (rlbtftn_at_netscape.net)
Date: 09/13/04
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Date: Mon, 13 Sep 2004 23:30:28 GMT
Result of today's experimentation:
ds/dt=-s+1+1/Gamma[2-s]
converges to 1.53033.
Some experimentation in Mathematica on this and the map type gives:
s[n+1]=1+1/Gamma[2-s[n]/s0]
where s0 seems to act as an integer scale
like:
s0=1/l; l=1,2,3,4,....
The result with the right starting point of
s[0]=s0+delta
for smaller delta,
gives a complex and sometimes converging chaos.
This result appears to indicate that some scaling factor exists between
the fractional differential value and the fractal dimensional roughness
for the special case I have used. It is far from a "proof",
but it is closer than before to an understandable answer.
My experience with the partial angular fractional derivative done in IFS
is that the resulting set has a lower fractal dimension for between
angles ( for example a space filling 4 part Sierpinski when a
PAD is used on it gets "hole" and doesn't completely cover a space fill
anymore).
It shouldn't be unexpected that the actual fractal roughness factor is
scaled by the fractional differential.
It does make questions:
1) Is it only true in this special case?
2) Is the actual scale some constant of nature? ( like E or Pi )
3) Can some practical use be made of this relationship of calculus
and fractals?
Roger L. Bagula wrote:
> Strange result for the equation:
> s[n]=1+1/Log[Gamma[2-s[n-1]]]:
> It goes complex and is initial point dependent, but
> the attractors are clear:
> {1, Infinity,-Infinity}
> That's one from below and one from above to give a four term sequence.
>
> To correct a mistake:
> L(s)=Sum[ Log[Abs[ d^sx[n]/dt^s]],{n,1,m}]/m
> where
> d^sx[n]/dt^s
> is the Abel-Louville fractional derivative as real number 0<s<1.
> The recognized polynomial formula is:
> d^sx^n/dx^s=Gamma[n+1]* x^(n-s)/Gamma[n-s+1]
> Which formula give for n=1 and Gamma[2]=1:
> d^sx[n]/dt^s=x(1-s)/Gamma[2-s]
> where
> Gamma[n]=(n-1)!
>
> The paper that inspired this rumination is:
> Fractional differentiability of nowhere differentiatiable functions and
> dimensions
> by Kiran M. Kolwankar and Anil D. Gangal, chaos and dynamics 21 nov 1996
> which has a marvelous bibliography of people who have addressed the
> problem of fractional differentiation and it's relationship to dimension
> starting with Mandelbrot and van Ness and including Dr. M. Zähle
> and Dr. Falconer.
>
> Roger Bagula wrote:
>
>> I have an idea of how I can connect a Lyapunov exponent
>> and Kaplan -Yorke dimension to the fractional derivative
>> using the Hölder exponent type of idea in a paper I downloaded.
>> The idea is that the "roughness" region 0<s<1 in any fractal
>> is the part that is affectred by the fractional derivative and
>> not the topological dimension and behaves like a Hölder exponent.
>> I all ready have the roughness expoment based on the 2nd derivative
>> Bezier limit :
>> r=Sum[ Log[1+Abs[ d^2x[n]/dt2]/4],{n,1,m}]/m
>> which seems to be very much like the Lyapunov exponent in detecting
>> fractal dimensions
>> in my experiements. My idea is to invent a fractional Lyapunov exponent:
>> L(s)=Sum[ Log[Abs[ d^sx[n]/dt^s]],{n,1,m}]/m
>> where: Gamma[2]=1
>> d^sx[n]/dt^s=Gamma[2]*x[n]^(1-s)/Gamma[2-s]
>> Or
>> L(s)=Sum[ Log[Abs[x[n]^(1-s)/Gamma[2-s]],{n,1,m}]/m
>> If the average of x[n] is one then or Log[1]=0:
>> L[s]=Log[Gamma[2-s]]
>> which is always negative since:
>> 0<=Gamma[2-s]<=1 on 0<=s<=1
>> which gives a Kaplan -Yorke dimension of: (d0 the topological dimension)
>> if the other exponents are one
>> dky(s)=d0+1/Log[Gamma[2-s]]
>> Which gives a roughness gap in the range:
>> s'=1+1/Log[Gamma[2-s]]
>> That would give the connection of the fractional derivative to
>> the dimension.
>> There are a lot of "if"'s involved in this "not a proof" derivation, but
>> it does show that a connection may exist.
>> A lot of people have tried to make this kind of connection in a
>> provable way
>> and failed so far as I know.
>>
>>
>>
>> Respectfully, Roger L. Bagula
>>
>> tftn@earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel:
>> 619-5610814 :
>> URL : http://home.earthlink.net/~tftn
>> URL : http://victorian.fortunecity.com/carmelita/435/
>>
>
>
-- Respectfully, Roger L. Bagula tftn@earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 : URL : http://home.earthlink.net/~tftn URL : http://victorian.fortunecity.com/carmelita/435/
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