Thank you for suggestion. I'll visit the forum you've indicated later. I think to estimate the influence of the error in the numerical analysis is difficult particularly for chaotic system since the effect of any small error could become tremendously great after repeating many iterations. By the...
The "translation" I got is;
dx/dt = 0
dy/dt = 1
(Do you think they are too simple?) Although the solution of above differential equations is a line in two dimensional Euclidean space, the solution in the following quotient space...
The logistic equation;
x(n+1) = rx(n)(1-x(n))
shows chaotic behavior under some values of r. It is a discrete chaotic system. Is it possible to translate it to continuous chaotic system? I tried it in the following site;
http://geocities.com/tontokohirorin/mathematics/moduloid/moduloid2.htm
The discussion of a chaotic map different from the horseshoe map in the viewpoint of quotient space is newly added in the following site;
http://geocities.com/tontokohirorin/mathematics/moduloid/moduloid2.htm
Sorry for my response has been delaying. Oh, that's a nice drawing for that limit cycle! By the way I've been continuing to find any other limit cycle in this dynamical system. And I found it at last! Strange enough, the parameters for that limit cycle are exactly same as the previous ones, i.e...
I updated the webpage with implementing software for calculating the moduloid (or magma) for the cases of torus, sphere, real projective plane, and Klein Bottle as,
http://geocities.com/tontokohirorin/mathematics/moduloid/moduloid2.htm
Thank you for your comment. My website is regarding a generalization of residue arithmetic, in short. You may imagine the residue space in the linear space. I substituted the linear space by some quotient spaces such as sphere, real projective plane, Klein bottle, etc. Then I found the addition...
Magma as the mathematical object may be too big to be dealt with.
However I found the magma which is commutative and has the unit element
has some interesting properties which might be applicable to algebra
and topology. For details, please visit...
I updated the site;
http://www.geocities.com/tontokohirorin/mathematics/quadratic/complex2.htm
with some examples of hyperbolic holomorphic functions, which are shown graphically. Also I added some graphical examples of hyperbolic Moebius transformation and the simulator for it written by...
>Hello Tom. Got Mathematica?
Unfortunately I don't have it. It seems too expensive to buy for me... Anyway thank you for your analysis using Mathematica. The result of Feigenbaum plot is very interesting. Did you notice the blank around d=0.0985 in that plot? I imagined that blank might imply...
>My first thought is have you drawn a Feigenbaum plot?
No, I haven't done Feigenbaum plot for that dynamical system.
>What are the values of the parameters which yield a chaotic trajectory and what >are some values which yield a limit cycle?
As shown in my website, when setting the...
I agree with you. Generally in dynamical system the chaotic behavior will be changed to the recurrent motion converging to a simple limit cycle by tuning parameters. However if my calculation is correct, the Lyapunov exponent on the "limit cycle" appeared on the dynamical system shown in my...
The 3D dynamical system introduced in the following site;
http://www.geocities.com/tontokohirorin/mathematics/limitcycle/limitcycle2.htm
shows not only chaotic behaviour, but also converges to a strange-shaped limit cycle under a certain parameter setting!
I've found the former description of calculation for residue in my webpage was ineffective. So I modified the integral path in order to calculate the residue of f(x-yc) on the origin of Q(c) and updated the webpage;
http://www.geocities.com/tontokohirorin/mathematics/quadratic/complex2.htm