If you are a researcher working with chaos and chaotic systems, which unsolved problem or question you have in mind
Here I have collected some open problems and questions related to chaos and chaotic systems for anyone interested. Most of them are borrowed from Professor J. C. Sprott.
Do you have anything more to add?
Best Regards,
Sajad
1. Can we mathematically prove that the system Sprott A (see file 01) is conservative? It has many coexisting nested tori and one chaotic sea. Some hints can be found in file 02.
2. See file 03. Can we have a better model that results in more similarity between figure 5 and figure 4?
3. Find the algebraically simplest example of an attracting 2-torus in a three-dimensional autonomous system of ordinary differential equations.
4. A common problem is to find a mathematical model that mimics the apparently chaotic dynamics of an experimental system. Models that give good short-term predictability tend to give very inaccurate long-term behavior, even to the point of having unbounded or non-chaotic solutions. Is it possible to find models of data that give the right topology of their strange attractor at the expense of short-term predictability?
5. Under some conditions (such as for the Hénon map) the boundary of the basin of attraction is smooth, and under other conditions (such as for the Mandelbrot set) it is fractal. What conditions determine the shape and size of the basin of attraction? Is there a correlation of its fractal dimension with the dimension of the attractor or other quantity? What role do the Cauchy-Reimann equations play, if any? Can two-dimensional maps that satisfy the Cauchy-Reimann equations have chaos on a set of nonzero measure in their parameter space?
6. Power spectrum analysis is not very useful for distinguishing chaos from noise since it appears possible to construct a chaotic system that produces an arbitrary power spectrum. For that purpose, people rely on the correlation dimension (Grassberger and Pracaccia, Phys. Rev. Lett. 50, 346-349 (1983)). However, Osborne & Provenzale (Physica D 35, 357, 1989) have shown that colored noise can give a spuriously low correlation dimension. Can it be shown analytically or numerically that an appropriately chosen noise spectrum can produce the same correlation integral as an arbitrary chaotic system?
7. Some earlier work indicates that the probability that a polynomial map with arbitrarily chosen coefficients is chaotic decreases with the dimension of the map. This result is counterintuitive and contradicts results for polynomial flows and for discrete-time neural networks. What is the reason for the different behavior?
I believe problems 1 and 3 have been solved, at least to my satisfaction:
1. Through a rescaling of time, the Sprott-A system, more properly called the Nose-Hoover system, can be written as the Nose system which is Hamiltonian with a slave variable. See, for example, Dettmann and Morriss, Phys. Rev. E 55, 3693 (1997).
3. I believe the simplest 3-D autonomous system with polynomial nonlinearities whose solution is an attracting torus is the one given by Mehrabbeik, et al., Phys. Lett. A 451, 128427 (2022).
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