Hello everyone,
Strange Nonchaotic Attractors are known as a rather new class of attractors that, despite having fractional dimensions, do not illustrate chaos in the conventional sense (exponential divergence of infinitesimally close trajectories and positive Lyapunov exponents). I have some questions about this class of attractors and their properties.
1. Is the response of the SNAs without any period? (so if we look at them, we cannot predict where EXACTLY the trajectory will be in time T, where T is a positive value).
2. Are SNAs the same as weakly chaotic systems? because I've encountered systems that illustrate non-exponential divergence but due to a lack of periodicity, are called weakly chaotic. If they are different, what is the distinction?
3. Chaotic attractors are known to have a positive, a negative, and a zero Lyapunov exponent. What are the Lyapunov exponents of SNAs?
4. If one uses the idea of Poincare maps to analyze SNAs, how the results will differ from the results of chaotic attractors?
The qualifier `` strange" means the attractor has fractal dimension. The qualifier``non-chaotic" means what it says: The system isn’t chaotic, therefore it’s not ``weakly chaotic" either.
Fractal shapes aren’t necessarily chaotic.
Chaos means exponential growth of errors. Many systems have errors that grow and eventually become unpredictable, such as two unsynchronized clocks whose error typically grows linearly. They may or may not be periodic depending on the source of the error. They will return to the same reading once a day but may not do so with a regular period. You will not be able to predict what time such a clock will read a hundred years from now.
In a weakly chaotic system, the error grows exponentially, but slowly. The Solar System is weakly chaotic, but "weak" is an imprecise term that is only meaningful in comparison with something else.
An SNA will have its largest LE exactly equal to zero even though it is typically aperiodic and unpredictable. Other exponents might be zero, negative, or undefined. Its Poincare section will be a fractal, else it would not be "strange."
The definition of Stam is clear. But I still wonder what are the areas of applications of SNAs ?
The following is non-rigorous, incomprehensive, speculative and explorative in nature :
Strangeness of the attractor typically refers to the dimension (hausdorff etc) of invariant subspaces in phase / state space while chaoticity refers to the lyapunov exponent (usually positive for chaotic systems) of "typical / most" trajectories on the system. Usually, , for ergodic systems , both viewpoints (phase space / temporal or trajectory based yield the same result). I am not sure what happens in case of non ergodic systems. Also, in certain systems, Hausdorff dimensions could be complex (Complex scaling exponents) . i am not sure whether lyapunov exponents could be complex and its implications. Quasi-periodic systems, especially those driven at frequencies involving the golden mean (and other "badly rationally approximable" (by polynomials with rational coefficients) but algebraic numbers ) seem to be stable to small nonlinear perturbations thus yielding in islands of stability in a sea of chaos (links to KAM theorem etc ??) . The quasi-periodic systems seem to be break into Cantor sets , the most "central" of which seems to be Irrational Cantor Sets involving the Golden Mean possibly resulting from the final breakdown of the above islands
The hypothesis is that the interwoven nature of the quasi-periodic trajectories could possibly lead to strange attractors with fractal dimension
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