Characterizing chaos in a type of fractional Duffing's equation

S.  Jiménez; Pedro J.  Zufiria

doi:10.3934/proc.2015.0660

Article Contents

2015, Volume 2015: 660-669. Doi: 10.3934/proc.2015.0660 open access

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Characterizing chaos in a type of fractional Duffing's equation

S. Jiménez^1, and
Pedro J. Zufiria^1,

1.
Dept. Matemática Aplicada a las TIC, E.T.S.I. Telecomunicación, Universidad Politécnica de Madrid, Av. Complutense 30, 28040-Madrid

Received: August 31, 2014

Revised: December 31, 2014

Published: October 2015

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Abstract

We characterize the chaos in a fractional Duffing's equation computing the Lyapunov exponents and the dimension of the strange attractor in the effective phase space of the system. We develop a specific analytical method to estimate all Lyapunov exponents and check the results with the fiduciary orbit technique and a time series estimation method.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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References

[1]	S. Jimé3nez, J. A. González and L. Vázquez, Fractional Duffing's equation and geometrical resonance, International Journal of Bifurcation and Chaos, 23 (2013), 1350089-1-1350089-13.
[2]	J. Guckenheimer and Ph. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Springer-Verlag, New York, 1986.
[3]	S. Jeyakumari, V. Chinnathambi, S. Rajasekar and M.A.F. Sanjuán, Vibrational resonance in an asymmetric Duffing oscillator, International Journal of Bifurcation and Chaos 21 (2011), 275-286.
[4]	X. Gao and J. Yu, Chaos in the fractional order periodically forced complex Duffing's oscillators, Chaos, Solitons and Fractals 24 (2005), 1097-1104.
[5]	L.J. Sheu, H.K. Chen, J.H. Chen and L.M. Tam, Chaotic dynamics of the fractionally damped Duffing equation, Chaos, Solitons and Fractals 32 (2007), 1459-1468.
[6]	A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies 204, Elsevier, The Netherlands, 2006.
[7]	R. Gorenflo, F. Mainardi, Fractional Calculus: Integral and Differential Equations of Fractional Order, in Fractals and Fractional Calculus in Continuum Mechanics (eds. A. Carpinteri and F. Mainardi), Springer Verlag,(1997), 223-276.
[8]	V. Volterra, Theory of functionals and of integral and integro-differential equations Dover Publications, Inc., USA, 1959.
[9]	K. Diethelm, N.J. Ford, A.D. Freed and Yu. Luchko, Algorithms for the fractional calculus: A selection of numerical methods, Computer Methods in Applied Mechanics and Engineering 194 (2005), 743-773.
[10]	S. Jiménez, P. Pascual, C. Aguirre and L. Vázquez, A Panoramic View of Some Perturbed Nonlinear Wave Equations, International Journal of Bifurcation and Chaos 14 (2004), 1-40.
[11]	J.-P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors, Reviews of Modern Physics 57 (3) (1985), 617-656.
[12]	M. Casartelli, E. Diana, L. Galgani and A. Scott, Numerical computations on a stochastic parameter related to the Kolmogorov entropy, Physical Review 13A (5) (1976), 1921-1925.
[13]	R. Brown, P. Bryant and H.D.I. Abarbanel, Computing the Lyapunov spectrum of a dynamical sustem from an observed time series, Physical Review 57A (6) (1991), 2787-2806.
[14]	P. Frederickson, J.L. Kaplan, E.D. Yorke And J.A. Yorke, The Liapunov Dimension of Strange Attractors, Journal of Differential Equations 49 (1983), 185-207.
[15]	H.D.I. Abarbanel, Analysis of observed Chaotic data, Springer-Verlag, New York, 1996.
[16]	P. Walters, A dynamical proof of the multiplicative ergodic theorem Transactions of the American Mathematical Society 335 (1993), 245-257.