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.   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

[1]

P. Kaplický, Dalibor Pražák. Lyapunov exponents and the dimension of the attractor for 2d shear-thinning incompressible flow. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 961-974. doi: 10.3934/dcds.2008.20.961
[2]

L. Dieci, M. S Jolly, Ricardo Rosa, E. S. Van Vleck. Error in approximation of Lyapunov exponents on inertial manifolds: The Kuramoto-Sivashinsky equation. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 555-580. doi: 10.3934/dcdsb.2008.9.555
[3]

Matthias Rumberger. Lyapunov exponents on the orbit space. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 91-113. doi: 10.3934/dcds.2001.7.91
[4]

Edson de Faria, Pablo Guarino. Real bounds and Lyapunov exponents. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 1957-1982. doi: 10.3934/dcds.2016.36.1957
[5]

Andy Hammerlindl. Integrability and Lyapunov exponents. Journal of Modern Dynamics, 2011, 5 (1) : 107-122. doi: 10.3934/jmd.2011.5.107
[6]

Sebastian J. Schreiber. Expansion rates and Lyapunov exponents. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 433-438. doi: 10.3934/dcds.1997.3.433
[7]

Chao Liang, Wenxiang Sun, Jiagang Yang. Some results on perturbations of Lyapunov exponents. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4287-4305. doi: 10.3934/dcds.2012.32.4287
[8]

Shrihari Sridharan, Atma Ram Tiwari. The dependence of Lyapunov exponents of polynomials on their coefficients. Journal of Computational Dynamics, 2019, 6 (1) : 95-109. doi: 10.3934/jcd.2019004
[9]

Wafa Hamrouni, Ali Abdennadher. Random walk's models for fractional diffusion equation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2509-2530. doi: 10.3934/dcdsb.2016058
[10]

Hui Huang, Jian-Guo Liu. Well-posedness for the Keller-Segel equation with fractional Laplacian and the theory of propagation of chaos. Kinetic & Related Models, 2016, 9 (4) : 715-748. doi: 10.3934/krm.2016013
[11]

Zhibo Cheng, Jingli Ren. Periodic and subharmonic solutions for duffing equation with a singularity. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1557-1574. doi: 10.3934/dcds.2012.32.1557
[12]

Zhaosheng Feng, Goong Chen, Sze-Bi Hsu. A qualitative study of the damped duffing equation and applications. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1097-1112. doi: 10.3934/dcdsb.2006.6.1097
[13]

Nguyen Dinh Cong, Thai Son Doan, Stefan Siegmund. On Lyapunov exponents of difference equations with random delay. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 861-874. doi: 10.3934/dcdsb.2015.20.861
[14]

Wilhelm Schlag. Regularity and convergence rates for the Lyapunov exponents of linear cocycles. Journal of Modern Dynamics, 2013, 7 (4) : 619-637. doi: 10.3934/jmd.2013.7.619
[15]

Jianyu Chen. On essential coexistence of zero and nonzero Lyapunov exponents. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4149-4170. doi: 10.3934/dcds.2012.32.4149
[16]

Paul L. Salceanu, H. L. Smith. Lyapunov exponents and persistence in discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 187-203. doi: 10.3934/dcdsb.2009.12.187
[17]

Andrey Gogolev, Ali Tahzibi. Center Lyapunov exponents in partially hyperbolic dynamics. Journal of Modern Dynamics, 2014, 8 (3&4) : 549-576. doi: 10.3934/jmd.2014.8.549
[18]

Luis Barreira, César Silva. Lyapunov exponents for continuous transformations and dimension theory. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 469-490. doi: 10.3934/dcds.2005.13.469
[19]

Fei Yu, Kang Zuo. Weierstrass filtration on Teichmüller curves and Lyapunov exponents. Journal of Modern Dynamics, 2013, 7 (2) : 209-237. doi: 10.3934/jmd.2013.7.209
[20]

Lucas Backes, Aaron Brown, Clark Butler. Continuity of Lyapunov exponents for cocycles with invariant holonomies. Journal of Modern Dynamics, 2018, 12: 223-260. doi: 10.3934/jmd.2018009

 Impact Factor: 

Tools

Metrics

  • PDF downloads (19)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences