×
Fernández-Martínez, M.; Sánchez-Granero, M. A.; Trinidad Segovia, J. E.; Vera-López, J. A.

A new topological indicator for chaos in mechanical systems. (English) Zbl 1354.70044

Nonlinear Dyn. 84, No. 1, 51-63 (2016).
Summary: 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.

Cited in 2 Documents

MSC:

70K55 Transition to stochasticity (chaotic behavior) for nonlinear problems in mechanics
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
28A80 Fractals
37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
37C45 Dimension theory of smooth dynamical systems

Keywords:

fractal structure; fractal dimension; DIM3 algorithm; chaos indicator; maximal Lyapunov exponent; Beletsky’s equation

Software:

APL
Cite Review PDF
Full Text: DOI

References:

[1] Arenas, F.G., Sánchez-Granero, M.A.: A characterization of non-archimedeanly quasimetrizable spaces. Rend. Istit. Mat. Univ. Trieste Suppl XXX, 21-30 (1999) · Zbl 0944.54019
[2] Beletsky, V.V.: Motion of an Artificial Satellite About a Center of Mass. Israel Program for Scientific Translations, Jerusalem (1966)
[3] Benettin, G., Galgani, L., Giorgilli, A., Strelcyn, J.M.: Lyapunov characteristic exponents for smooth dynamical systems and for hamiltonian systems; a method for computing all of them; part 1: theory. Meccanica 15(1), 9-20 (1980) · Zbl 0488.70015 · doi:10.1007/BF02128236
[4] Benettin, G., Galgani, L., Giorgilli, A., Strelcyn, J.M.: Lyapunov characteristic exponents for smooth dynamical systems and for hamiltonian systems; a method for computing all of them. Part 2: numerical application. Meccanica 15(1), 21-30 (1980) · doi:10.1007/BF02128237
[5] Falconer, K.: Fractal Geometry: Mathematical Foundations and Applications. Wiley, Chichester (1990) · Zbl 0689.28003
[6] Feder, J.: Fractals. Plenum Press, New York (1988) · Zbl 0648.28006 · doi:10.1007/978-1-4899-2124-6
[7] Fernández-Martínez, M., Sánchez-Granero, M.A.: Fractal dimension for fractal structures: a Hausdorff approach. Topol. Appl. 159(7), 1825-1837 (2012) · Zbl 1250.28005 · doi:10.1016/j.topol.2011.04.023
[8] Fernández-Martínez, M., Sánchez-Granero, M.A.: Fractal dimension for fractal structures. Topol. Appl. 163, 93-111 (2014) · Zbl 1296.37038 · doi:10.1016/j.topol.2013.10.010
[9] Fernández-Martínez, M., Sánchez-Granero, M.A.: Fractal dimension for fractal structures: a Hausdorff approach revisited. J. Math. Anal. Appl. 409(1), 321-330 (2014) · Zbl 1309.28003 · doi:10.1016/j.jmaa.2013.07.011
[10] Fernández-Martínez, M., Sánchez-Granero, M.A.: Trinidad Segovia, J.E.: Fractal dimension for fractal structures: applications to the domain of words. Appl. Math. Comput. 219(3), 1193-1199 (2012) · Zbl 1291.68306
[11] Fernández-Martínez, M., Sánchez-Granero, M.A., Trinidad Segovia, J.E.: Fractal dimensions for fractal structures and their applications to financial markets, Aracne editrice S.r.l., Roma (2013) · Zbl 1314.91006
[12] Fernández-Martínez, M., Sánchez-Granero, M.A., Trinidad Segovia, J.E.: Román-Sánchez, I.M.: An accurate algorithm to calculate the Hurst exponent of self-similar processes. Phys. Lett. A 378(32-33), 2355-2362 (2014) · Zbl 1303.60029
[13] Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer, Berlin (1983) · Zbl 0515.34001 · doi:10.1007/978-1-4612-1140-2
[14] Iverson, K.E.: A Programming Language. Wiley, New York (1962) · Zbl 0101.34503 · doi:10.1145/1460833.1460872
[15] Jeanblanc, M., Yor, M., Chesney, M.: Mathematical Methods for Financial Markets. Springer Finance, London (2009) · Zbl 1205.91003 · doi:10.1007/978-1-84628-737-4
[16] Lamperti, J.W.: Semi-stable stochastic processes. Trans. Am. Math. Soc. 104, 62-78 (1962) · Zbl 0286.60017 · doi:10.1090/S0002-9947-1962-0138128-7
[17] Mandelbrot, B.B.: Fractals: Form, Chance and Dimension. W.H. Freeman & Company, San Francisco (1977) · Zbl 0376.28020
[18] Mandelbrot, B.B.: The Fractal Geometry of Nature. W.H. Freeman & Company, New York (1982) · Zbl 0504.28001
[19] Mandelbrot, B.B.: Gaussian Self-Affinity and Fractals. Springer, New York (2002) · Zbl 1007.01020
[20] Robinson, C.: Dynamical Systems: Stability, Symbolic Dynamics and Chaos. CRC Press Inc., Boca Raton (1995) · Zbl 0853.58001
[21] Samorodnitsky, G., Taqqu, M.S.: Stable Non-gaussian Random Processes: Stochastic Models with Infinite Variance. Chapman & Hall, London (1994) · Zbl 0925.60027
[22] Sánchez-Granero, M.A.: Fractal structures, in: asymmetric topology and its applications. In: Quaderni di Matematica, vol. 26, Aracne, pp. 211-245 (2012) · Zbl 1315.54003
[23] Sánchez-Granero, M.A., Fernández-Martínez, M., Trinidad Segovia, J.E.: Introducing fractal dimension algorithms to calculate the Hurst exponent of financial time series. Eur. Phys. J. B 85(3), 1-13 (2012)
[24] Tijera, M., Maqueda, G., Cano, J.L., Yagüe, C.: Analysis offractal dimension of the wind speed and its relationships withturbulent and stability parameters, chapter 2 of [Sid-Ali Ouadfeul(ed.), Fractal Analysis and Chaos in Geosciences], Intech (2012)
[25] Trinidad Segovia, J.E., Fernández-Martánez, M., Sánchez-Granero, M.A.: A note on geometric method-based procedures to calculate the Hurst exponent. Phys. A 391(6),2209-2214 (2012)
[26] 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) · doi:10.1007/s11071-012-0504-1
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.