Characterizing chaos in a type of fractional Duffing's equation

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2015, 2015(special): 660-669. doi: 10.3934/proc.2015.0660

Characterizing chaos in a type of fractional Duffing's equation

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

Received September 2014 Revised January 2015 Published November 2015

Abstract
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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.

Keywords: strange attractor, chaos., Fractional Duffing's equation, Lyapunov exponents.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.

Citation: S. Jiménez, Pedro J. Zufiria. Characterizing chaos in a type of fractional Duffing's equation. Conference Publications, 2015, 2015 (special) : 660-669. doi: 10.3934/proc.2015.0660

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.
[2]	J. Guckenheimer and Ph. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields,, Springer-Verlag, (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), 21 (2011), 275.
[4]	X. Gao and J. Yu, Chaos in the fractional order periodically forced complex Duffing's oscillators,, Chaos, 24 (2005), 1097.
[5]	L.J. Sheu, H.K. Chen, J.H. Chen and L.M. Tam, Chaotic dynamics of the fractionally damped Duffing equation,, Chaos, 32 (2007), 1459.
[6]	A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations,, North-Holland Mathematics Studies 204, (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)*, (1997), 223.
[8]	V. Volterra, Theory of functionals and of integral and integro-differential equations, Dover Publications, (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), 194 (2005), 743.
[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), 14 (2004), 1.
[11]	J.-P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors,, Reviews of Modern Physics 57 (3) (1985), 57 (1985), 617.
[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), 13A (1976), 1921.
[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), 57A (1991), 2787.
[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), 49 (1983), 185.
[15]	H.D.I. Abarbanel, Analysis of observed Chaotic data,, Springer-Verlag, (1996).
[16]	P. Walters, A dynamical proof of the multiplicative ergodic theorem, Transactions of the American Mathematical Society 335 (1993), 335 (1993), 245.

show all references

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.
[2]	J. Guckenheimer and Ph. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields,, Springer-Verlag, (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), 21 (2011), 275.
[4]	X. Gao and J. Yu, Chaos in the fractional order periodically forced complex Duffing's oscillators,, Chaos, 24 (2005), 1097.
[5]	L.J. Sheu, H.K. Chen, J.H. Chen and L.M. Tam, Chaotic dynamics of the fractionally damped Duffing equation,, Chaos, 32 (2007), 1459.
[6]	A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations,, North-Holland Mathematics Studies 204, (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)*, (1997), 223.
[8]	V. Volterra, Theory of functionals and of integral and integro-differential equations, Dover Publications, (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), 194 (2005), 743.
[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), 14 (2004), 1.
[11]	J.-P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors,, Reviews of Modern Physics 57 (3) (1985), 57 (1985), 617.
[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), 13A (1976), 1921.
[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), 57A (1991), 2787.
[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), 49 (1983), 185.
[15]	H.D.I. Abarbanel, Analysis of observed Chaotic data,, Springer-Verlag, (1996).
[16]	P. Walters, A dynamical proof of the multiplicative ergodic theorem, Transactions of the American Mathematical Society 335 (1993), 335 (1993), 245.

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