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A new topological indicator for chaos in mechanical systems

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Abstract

The main goal in this paper was to provide a novel chaos indicator based on a topological model which allows to calculate the fractal dimension of any curve. A fractal structure is a topological tool whose recursiveness becomes ideal to generalize the concept of fractal dimension. In this paper, we provide an algorithm to calculate a new fractal dimension specially designed for a parametrization of a curve or a random process, whose definition is made by means of fractal structures. As an application, we explore the use of this new concept of fractal dimension as a chaos indicator for dynamical systems, in a similar way to the classical maximal Lyapunov exponent. To illustrate it, we apply the new fractal dimension as an indicator to model the chaotic behavior of a satellite which is moving around a planet whose gravity field is approximated by the field of a point mass.

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Acknowledgments

M. Fernández-Martínez especially acknowledges the valuable support provided by Centro Universitario de la Defensa en la Academia General del Aire de San Javier (Murcia, Spain). M.A. Sánchez-Granero acknowledges the support of the Ministry of Economy and Competitiveness of Spain, Grant MTM2012-37894-C02-01. J. E. Trinidad Segovia appreciates the support of Junta de Andalucía, Grant P09-SEJ-5404. J.A. Vera-López acknowledges the support of MICINN/FEDER, Grant No. MTM2011-22587.

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Authors and Affiliations

  1. University Centre of Defence at the Spanish Air Force Academy, MDE-UPCT, Santiago de la Ribera, 30720, Murcia, Spain

    M. Fernández-Martínez & J. A. Vera-López

  2. Department of Mathematics, Universidad de Almería, 04120, Almería, Spain

    M. A. Sánchez-Granero

  3. Department of Accounting and Finance, Universidad de Almería, 04120, Almería, Spain

    J. E. Trinidad Segovia

Authors
  1. M. Fernández-Martínez
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  2. M. A. Sánchez-Granero
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  3. J. E. Trinidad Segovia
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  4. J. A. Vera-López
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Corresponding author

Correspondence to J. A. Vera-López.

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Fernández-Martínez, M., Sánchez-Granero, M.A., Trinidad Segovia, J.E. et al. A new topological indicator for chaos in mechanical systems. Nonlinear Dyn 84, 51–63 (2016). https://doi.org/10.1007/s11071-015-2207-x

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  • DOI: https://doi.org/10.1007/s11071-015-2207-x

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