We are working on a chaotic map.
The map completely continues and differentiable map with the following structure:
X_n+1 = f(X_n,Y_n)
Y_n+1 = g(X_n,Y_n)
When we plot the bifurcation diagram, instead of usual period doubling, we encounter a kind of strange period adding: 1-->3-->10-->25-->…
You can find the bifurcation diagram in the attached picture. Can anyone tell us what phenomenon is this? Does anyone know about something similar?
Or, to see explicitly how period-adding structures emerge:
Shrimp-shaped domain in the parameter space of an experimental dynamical system. [Ruedi Stoop, Philipp Benner, and Yoko Uwate, Phys. Rev. Lett. 105, 074102 (2010)]
(I am sorry for the self-promotion, but regarding your question, this seems to do a good job )
Which mapping is this ? If the map is continous it should obey the Sarkovskii's theorem:
http://en.wikipedia.org/wiki/Sharkovskii%27s_theorem
I don't know if this is helpful.
Bruno
Dear Sajad,
It depends on the map you are working on. For a differentiable interval maps the above diagram is not possible, so I imagine that you are working either on higher dimnesional spaces or a non-autonomous system. Without more information it is impossible to say more.
Regards,
jose canovas
The map is a completely continues and differentiable map with the following structure:
X_n+1 = f(X_n,Y_n)
Y_n+1 = g(X_n,Y_n)
Dear Sajad,
Take a look at this paper:
Miguel A. F. Sanjuįn. Symmetry-restoring crises, period-adding and chaotic transitions in the cubic Van der Pol oscillator. J. of Sound and Vibration, 193(4):863-875, 1996
Or, to see explicitly how period-adding structures emerge:
Shrimp-shaped domain in the parameter space of an experimental dynamical system. [Ruedi Stoop, Philipp Benner, and Yoko Uwate, Phys. Rev. Lett. 105, 074102 (2010)]
(I am sorry for the self-promotion, but regarding your question, this seems to do a good job )
Another continuous-time 3D system showing period adding is the Hindmarsh-Rose neuron model (I have a couple of works on it, if useful). About 2D maps, you can have a look to works coauthored by Iryna Sushko and Laura Gardini and also to works by Soumitro Banerjee and co-workers.
Of course this behaviour is possible in two dim maps, you can construct this depending on the parameters chosen and the projections you choose to get your, so called, "bifurcation diagrams". But you will never get an answer if you do not show the actual map... You do not even show the number of parameters the types and classification of the singularities and what you change and what you show in the pictures... The question is not serious, sorry to say...
Dear Pepe Canovas, you can have a degenerate period doubling with zero Schwarzian derivative with a big jump in the periods even in dim one families with the parameters necessary to do the unfolding, see for instance the book by Kuznetsov on bifurcation (chapter 9 if I remember), you get also that in high degeneracy singularities for periodic points with positive 1 derivative... In the swallowtail you can jump from zero to 4 periodic points, and so on... In the picture you do not know if that is the same orbit or a multi-orbit diagram... Basically the picture is a pile of nothing without the necessary Intel.
Dear Sajad,
just from the personal experience: 1D bifurcation diagrams are nice to present but frequently they do not help to understand the regularities of the system.
To me it was many times much more helpful to look on 2D parameter
planes. Of course, if your system has many parameters, the question is,
which two of them have to be varied... If you have found a plane in which you
can understand the structure, the job is done. Good luck!
And by the way, I would not recommend to use the name "period adding" here.
In fact, this name is used for many totally different phenomena (I know three of them,
different from yours and not related to each other, and maybe there are more), so if this name appears for example in a title of a paper, it is totally unpredictable what the readers will think :-)
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