(Sharkovsky Forcing Theorem ). If m is a period for f and m⊲ l , then l is also a period for f . I have the following question:
Let f be a such map having a period three, So f is chaotic. But for some initial conditions f tending to infinity. How I can guarantee the existence of other all periods for f in this case?
The Sharkovsky theorem in part guarantees that the existence, for a continuous mapping from R to R, of a three-periodic point entails the existence of all other periods, in particular, of periods 2ⁿ for any integer n>0, closely related with the chaotic motion, yet not ensuing it a priori, and nothing more. The mapping to be chaotic still additionally should staisfy the transitivity property, which happens rarely enough.
Regards!
The fact that a map has a cycle of period 3 and also trajectories which are divergent is quite usual. The standard example is the logistic map f(x)=ax(1-x) for a>4: a cycle of period 3 exists as well all the cycles of any period (all unstable), belonging to an invariant chaotic Cantor set in [0,1].
To: Laura Gardini
Please, could you cite some reference on a proof that logistic map f(x)=ax(1-x) for a>4, and what about a
Dear Anatolij K. Prykarpatski
This is a very old result, reported in almost all the books, see for example in Devaney, Introduction to chaotic dynamical systems, 1989,
in pag.35, see Fig,5.4: taking off all the preimages of the interval A0 (of divergent trajectories) what is left in [0,1] is an invariant chaotic Cantor set.
In the same book as well as in C. Mira, Chaotic Dynamics, World Scientific, Singapore, 1987 you can see all what occurs in the logistic map also for a
The question reminds me of Milnor attractors (to which I was once introduced by Chico Doria).
a Milnor attractor is an invariant sets not attracting in the topological sense (i.e. without an attracting neighbourhood) with a stable set of positive measure, for example this always occurs at a fold bifurcation of a cycle in the logistic map
but this is different from the case of divergent trajectories (what is left is a true repelling Cantor set, not a Milnor attractor)
With Georg C.Hartmann, I published a few papers on attractors with very large finite points included, derived as perturbations. They were called "flare attractors." But probably they are trivial in your context?
Does anyone know how Chico is doing?
To @Laura Gardini
Dear Laura, thanks for the info and refs. As I understand, the logistic map has sense only for 0
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