Recently there has been increasing interest in finding and studying rare examples of simple chaotic systems such as those in which there are no equilibria or in which any existing equilibria are stable [1-7]. They belong to a new defined category of chaotic systems which is called “chaotic systems with hidden attractor” [8-9]. However they are all about chaotic flows and I haven’t seen any similar work about chaotic maps. Does anybody know about a chaotic map with no equilibria or only stable equilibria?
References
[1] Jafari, S., Sprott, J.C., Golpayegani, S.M.R.H. [2013] “Elementary quadratic chaotic flows with no equilibria,” Phys. Lett. A, 377, 699-702.
[2] Wei, Z. [2011] “Dynamical behaviors of a chaotic system with no equilibria,” Phys. Lett. A 376, 102-108.
[3] Wang, X., Chen, G. [2012] “A chaotic system with only one stable equilibrium,” Commun.Nonlinear Sci. Numer.Simulat. 17, 1264-1272.
[4] Wang, X., Chen, G. [2013] “Constructing a chaotic system with any number of equilibria,” Nonlinear Dyn. 71, 429-436.
[5] Wei, Z., Yang, Q. [2012] “Dynamical analysis of the generalized Sprott C system with only two stable equilibria,” Nonlinear Dyn. 68, 543-554.
[6] Malihe Molaie, Sajad Jafari, J. C. Sprott, S. Mohammad Reza Hashemi Golpayegani, “Simple chaotic flows with one stable equilibrium,” International Journal of Bifurcation and Chaos, In press.
[7] Sajad Jafari, J. C. Sprott, “Simple chaotic flows with a line equilibrium,” Chaos Solitons Fract, In press.
[8] Leonov, G.A., Kuznetsov, N.V., Vagaitsev, V. I., [2011] “Localization of hidden Chua’s attractors,” Phys. Lett. A 375, 2230–2233.
[9] Leonov, G.A., Kuznetsov, N.V. Vagaitsev, V.I., [2012] “Hidden attractor in smooth Chua systems,” Physica D 241, 1482–1486.
Dear Nikolay,
I suppose that your suggestion needs additional conditions, otherwise it can be constructed easily by defining continuous maps on subintervals and gluing them properly.
Regards.
Dear Sajad,
in discrete dynamical systems (of any dimensions) an invariant set is called "chaotic" only if the periodic solutions are dense in the set X and are all unstable. Then I think it is impossible to have either stable equilibria or no equilibrium.
For further details, I advise you to contact Laura Gardini (University of Urbino, Italy): she is really expert in discrete-time dynamical systems analysis!
Hope this helps.
Dear Sajad,
The question is quite open and admits contardictory answers because there are several definitions of chaos which are not equivalent and the topological space in which the dynamical system is defined can be also wide. For instance, it is possible to construct a compact subset of the unit interval [0,1] and a continuous selfmap on it such that the set of periodic points is empty and the map is chaotic in the sense of Li and Yorke. On the opposite, if you fix the interval [0,1] and consider a Devaney chaotic map, by force, there are unstable equilibria.
Best regards,
jose canovas
Dear Jose,
Can you please give me a simple example of each case?
I especially like to investigate simple 1D maps:
x(n+1) = f(x(n))
Thank you
Dear Sajad,
The logistic map 4x(1-x) is transitive and hence Devaney chaotic. The maximum of the function has a trajectory which is eventually the fixed point 0, which is unstable. Since the map has negative Schwarzian derivative, by Singer's theorem any stable periodic obit has to attract the maximum. Since it is impossible due to 0 is unstable, the map is Devaney chaotic and all the eauilibria are unstable.
The example for Li-Yorke chaos is more difficult to explain, because it is constructed on a Li-Yorke chaotic map with zero topological entropy. It is based on the existence of infinite omega-limit sets, called solenoids (recently I read also the word odometer) and it is necesary a technical symbolic dynamics to describe it. Basically, such solenoids should be non-empty and such that wandering intervals exist. I can send you references if you wish so.
Anyway, there are also exmaples of minimal homeomorphisms with positive topological entropy. By minimality there are not periodic orbits. As topological entropy is positive, it is chaotic in the sense of Li and Yorke.
As I told you before, the question is quite "open", and admits multiple answers.
Best regards,
jose canovas
is it a problem for you, if the map is discontinuous?
if not, I will give you such a map immediately :-)))
Dear Professor Avrutin,
I have discussed this topic with Professor Gardini too. As you mentioned, there can be many examples of those systems if we consider discontinuous maps. However, we can have chaotic maps with stable fixed-points in continuous maps too (just by connecting discontinuous parts of abovementioned examples in such a way that the fixed-points be stable).
Thank you very much.
Regards,
Sajad
I suggest to construct a discrete map with stable equilibria and stable periodic solution and with period doubling bifurcation (route to chaos to chaos).
Recently I read a paper on counterexample to disrete Kalman's conjecture (disrete dynamical system with the only stabe equilibrium and stable periodic solution was presented there).
P.S. Dear Victor, as you know the topic is very interesting for us. May be you will say a few words about n your presentation in April?
Dear Nikolay,
I suppose that your suggestion needs additional conditions, otherwise it can be constructed easily by defining continuous maps on subintervals and gluing them properly.
Regards.
Dear Sajad,
discrete chaotic map presented in paper "A new discrete chaotic map based on the composition of permutations" does not have fixed points.
Best regards
Article A new discrete chaotic map based on the composition of permutations
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