Abstract
We consider a family of simple flows in tori that display chaotic behavior in a wide sense. But these flows do not have homoclinic nor heteroclinic orbits. They have only a fixed point which is of parabolic type. However, the dynamics returns infinitely many times near the fixed point due to quasi-periodicity. A preliminary example is given for maps introduced in a paper containing many examples of strange attractors in [6]. Recently, a family of maps similar to the flows considered here was studied in [9]. In the present paper we consider the case of 2D tori and the extension to tori of arbitrary finite dimension. Some other facts about exceptional frequencies and behavior around parabolic fixed points are also included.
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Acknowledgments
Thanks to J. Timoneda for maintaining the computing facilities of the Dynamical Systems Group of the Universitat de Barcelona, which have been largely used in this work to make many simulations.
Funding
This work has been supported by grants MTM2016-80117-P (Spain) and 2017-SGR-1374 (Catalonia).
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Dedicated to Prof. Valery Vasilievich Kozlov on the occasion of his 70th birthday
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Simó, C. Simple Flows on Tori with Uncommon Chaos. Regul. Chaot. Dyn. 25, 199–214 (2020). https://doi.org/10.1134/S1560354720020057
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DOI: https://doi.org/10.1134/S1560354720020057
Keywords
- chaos without homoclinic/heteroclinic points
- chaotic flows on tori
- the returning role of quasi-periodicity
- zero maximal Lyapunov exponents
- the role of parabolic points
- exceptional frequencies