American Institute of Mathematical Sciences

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

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, Spain, Spain

Received  September 2014 Revised  January 2015 Published  November 2015

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), 1350089-1-1350089-13. Google Scholar

[2]

J. Guckenheimer and Ph. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Springer-Verlag, New York, 1986. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[8]

V. Volterra, Theory of functionals and of integral and integro-differential equations Dover Publications, Inc., USA, 1959. Google Scholar

[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. Google Scholar

[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. Google Scholar

[11]

J.-P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors, Reviews of Modern Physics 57 (3) (1985), 617-656. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[15]

H.D.I. Abarbanel, Analysis of observed Chaotic data, Springer-Verlag, New York, 1996. Google Scholar

[16]

P. Walters, A dynamical proof of the multiplicative ergodic theorem Transactions of the American Mathematical Society 335 (1993), 245-257. Google Scholar

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), 1350089-1-1350089-13. Google Scholar

[2]

J. Guckenheimer and Ph. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Springer-Verlag, New York, 1986. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[8]

V. Volterra, Theory of functionals and of integral and integro-differential equations Dover Publications, Inc., USA, 1959. Google Scholar

[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. Google Scholar

[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. Google Scholar

[11]

J.-P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors, Reviews of Modern Physics 57 (3) (1985), 617-656. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[15]

H.D.I. Abarbanel, Analysis of observed Chaotic data, Springer-Verlag, New York, 1996. Google Scholar

[16]

P. Walters, A dynamical proof of the multiplicative ergodic theorem Transactions of the American Mathematical Society 335 (1993), 245-257. Google Scholar

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