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On global dynamics in a multi-dimensional discrete map
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 |
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. Google Scholar |
[2] |
J. Guckenheimer and Ph. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields,, Springer-Verlag, (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), 21 (2011), 275. Google Scholar |
[4] |
X. Gao and J. Yu, Chaos in the fractional order periodically forced complex Duffing's oscillators,, Chaos, 24 (2005), 1097. 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, 32 (2007), 1459. 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, (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), (1997), 223. Google Scholar |
[8] |
V. Volterra, Theory of functionals and of integral and integro-differential equations, Dover Publications, (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), 194 (2005), 743. 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), 14 (2004), 1. Google Scholar |
[11] |
J.-P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors,, Reviews of Modern Physics 57 (3) (1985), 57 (1985), 617. 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), 13A (1976), 1921. 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), 57A (1991), 2787. 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), 49 (1983), 185. Google Scholar |
[15] |
H.D.I. Abarbanel, Analysis of observed Chaotic data,, Springer-Verlag, (1996). Google Scholar |
[16] |
P. Walters, A dynamical proof of the multiplicative ergodic theorem, Transactions of the American Mathematical Society 335 (1993), 335 (1993), 245. 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. Google Scholar |
[2] |
J. Guckenheimer and Ph. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields,, Springer-Verlag, (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), 21 (2011), 275. Google Scholar |
[4] |
X. Gao and J. Yu, Chaos in the fractional order periodically forced complex Duffing's oscillators,, Chaos, 24 (2005), 1097. 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, 32 (2007), 1459. 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, (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), (1997), 223. Google Scholar |
[8] |
V. Volterra, Theory of functionals and of integral and integro-differential equations, Dover Publications, (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), 194 (2005), 743. 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), 14 (2004), 1. Google Scholar |
[11] |
J.-P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors,, Reviews of Modern Physics 57 (3) (1985), 57 (1985), 617. 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), 13A (1976), 1921. 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), 57A (1991), 2787. 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), 49 (1983), 185. Google Scholar |
[15] |
H.D.I. Abarbanel, Analysis of observed Chaotic data,, Springer-Verlag, (1996). Google Scholar |
[16] |
P. Walters, A dynamical proof of the multiplicative ergodic theorem, Transactions of the American Mathematical Society 335 (1993), 335 (1993), 245. Google Scholar |
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