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The Fast Norm Vector Indicator (FNVI) method: a new dynamical parameter for detecting order and chaos in Hamiltonian systems

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Abstract

In the present article, we introduce and also deploy a new, simple, very fast, and efficient method, the Fast Norm Vector Indicator (FNVI) in order to distinguish rapidly and with certainty between ordered and chaotic motion in Hamiltonian systems. This distinction is based on the different behavior of the FNVI for the two cases: the indicator after a very short transient period of fluctuation displays a nearly constant value for regular orbits, while it continues to fluctuate significantly for chaotic orbits. In order to quantify the results obtained by the FNVI method, we establish the dFNVI, which is the quantified numerical version of the FNVI. A thorough study of the method’s ability to achieve an early and clear detection of an orbit’s behavior is presented both in two and three degrees of freedom (2D and 3D) Hamiltonians. Exploiting the advantages of the dFNVI method, we demonstrate how one can rapidly identify even tiny regions of order or chaos in the phase space of Hamiltonian systems. The new method can also be applied in order to follow the time evolution of sticky orbits. A detailed comparison between the new FNVI method and some other well-known dynamical methods of chaos detection reveals the great efficiency and the leading role of this new dynamical indicator.

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Acknowledgements

I would like to express my warmest thanks to Dr. D.D. Carpintero for all the illuminating and creative discussions during this research. The author would also like to thank the two anonymous referees for the careful reading of the manuscript and for their very useful and aptly suggestions and comments which allowed us to improve significantly the quality and the clarity of the present article.

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  1. Department of Physics, Section of Astrophysics, Astronomy and Mechanics, Aristotle University of Thessaloniki, 541 24, Thessaloniki, Greece

    Euaggelos E. Zotos

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Zotos, E.E. The Fast Norm Vector Indicator (FNVI) method: a new dynamical parameter for detecting order and chaos in Hamiltonian systems. Nonlinear Dyn 70, 951–978 (2012). https://doi.org/10.1007/s11071-012-0504-1

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