Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2013, Article ID 851970, 11 pages
http://dx.doi.org/10.1155/2013/851970
Research Article

Determining the Lyapunov Spectrum of Continuous-Time 1D and 2D Multiscroll Chaotic Oscillators via the Solution of -PWL Variational Equations

Facultad de Ciencias de la Electrónica, Benemérita Universidad Autónoma de Puebla, 72570 Puebla, PUE, Mexico

Received 25 July 2013; Accepted 21 October 2013

Academic Editor: Andrei Korobeinikov

Copyright © 2013 Jesus Manuel Munoz-Pacheco et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. S. H. Strogatz, Nonlinear Dynamics and Chaos with Applications to Physics, Biology, Chemistry, and Engineering, Westview Press, New York, NY, USA, 2001.
  2. H. Yu, G. Cai, and Y. Li, “Dynamic analysis and control of a new hyperchaotic finance system,” Nonlinear Dynamics, vol. 67, no. 3, pp. 2171–2182, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. M. EtehadTavakol, E. Y. K. Ng, C. Lucas, S. Sadri, and M. Ataei, “Nonlinear analysis using Lyapunov exponents in breast thermograms to identify abnormal lesions,” Infrared Physics and Technology, vol. 55, no. 4, pp. 345–352, 2012. View at Publisher · View at Google Scholar · View at Scopus
  4. T. S. Parker and L. O. Chua, Practical Numerical Algorithms for Chaotic Systems, Springer, New York, NY, USA, 1992. View at Publisher · View at Google Scholar · View at MathSciNet
  5. A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano, “Determining Lyapunov exponents from a time series,” Physica D, vol. 16, no. 3, pp. 285–317, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. J. Lü and G. Chen, “Generating multiscroll chaotic attractors: theories, methods and applications,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 16, no. 4, pp. 775–858, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. J. Lü, F. Han, X. Yu, and G. Chen, “Generating 3-D multi-scroll chaotic attractors: a hysteresis series switching method,” Automatica, vol. 40, no. 10, pp. 1677–1687, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  8. C. H. Wang, H. Xu, and F. Yu, “A novel approach for constructing high-order Chua's circuit with multi-directional multi-scroll chaotic attractors,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 23, no. 2, Article ID 1350022, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. J. M. Munoz-Pacheco and E. Tlelo-Cuautle, Electronic Design Automation of Multi-Scroll Chaos Generators, Bentham Sciences Publishers, Dubai, UAE, 2010.
  10. F. Han, J. Lü, X. Yu, G. Chen, and Y. Feng, “Generating multi-scroll chaotic attractors via a linear second-order hysteresis system,” Dynamics of Continuous, Discrete & Impulsive Systems B, vol. 12, no. 1, pp. 95–110, 2005. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. Z. Zhang and G. Chen, “Liquid mixing enhancement by chaotic perturbations in stirred tanks,” Chaos, Solitons and Fractals, vol. 36, no. 1, pp. 144–149, 2008. View at Publisher · View at Google Scholar · View at Scopus
  12. C. K. Volos, I. M. Kyprianidis, and I. N. Stouboulos, “A chaotic path planning generator for autonomous mobile robots,” Robotics and Autonomous Systems, vol. 60, no. 4, pp. 651–656, 2012. View at Publisher · View at Google Scholar · View at Scopus
  13. J. M. Muñoz-Pacheco, E. Zambrano-Serrano, O. Félix-Belt\'ran, L. C. Gómez-Pavón, and A. Luis-Ramos, “Synchronization of PWL function-based 2D and 3D multi-scroll chaotic systems,” Nonlinear Dynamics, vol. 70, no. 2, pp. 1633–1643, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  14. Y.-J. Sun, “DII-based linear feedback control design for practical synchronization of chaotic systems with uncertain input nonlinearity and application to secure communication,” Abstract and Applied Analysis, vol. 2012, Article ID 369267, 14 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. D. Chen, Z. Sun, X. Ma, and L. Chen, “Circuit implementation and model of a new multi-scroll chaotic system,” International Journal of Circuit Theory and Applications, 2012. View at Publisher · View at Google Scholar
  16. J. M. Munoz-Pacheco, W. Campos-López, E. Tlelo-Cuautle, and C. Sánchez-López, “OpAmp-, CFOA- and OTA-based configurations to design multi-scroll chaotic oscillators,” Trends in Applied Sciences Research, vol. 7, no. 2, pp. 168–174, 2012. View at Publisher · View at Google Scholar
  17. J. Jung, J. Lee, and H. Song, “Implementation of an integrated op-amp based chaotic neuron model and observation of its chaotic dynamics,” Chaos, vol. 21, no. 1, Article ID 013105, 2011. View at Publisher · View at Google Scholar · View at Scopus
  18. K. Gopakumar, B. Premlet, and K. G. Gopchandran, “Chua's oscillator in integrated circuit form with inbuilt control option,” Journal of Circuits, Systems and Computers, vol. 20, no. 8, pp. 1591–1604, 2011. View at Publisher · View at Google Scholar · View at Scopus
  19. R. Trejo-Guerra, E. Tlelo-Cuautle, J. M. Jiménez-Fuentes et al., “Integrated circuit generating 3- and 5-scroll attractors,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 11, pp. 4328–4335, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  20. E. Tlelo-Cuautle, J. M. Muñoz-Pacheco, and J. Martínez-Carballido, “Frequency scaling simulation of Chua's circuit by automatic determination and control of step-size,” Applied Mathematics and Computation, vol. 194, no. 2, pp. 486–491, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. I. Zelinka, G. Chen, and S. Celikovsky, “Chaos synthesis by means of evolutionary algorithms,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 18, no. 4, pp. 911–942, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. L. G. de la Fraga, E. Tlelo-Cuautle, V. H. Carbajal-Gómez, and J. M. Munoz-Pacheco, “On maximizing positive Lyapunov exponents in a chaotic oscillator with heuristics,” Revista Mexicana de Física, vol. 58, no. 3, pp. 274–281, 2012. View at Google Scholar
  23. K. Ramasubramanian and M. S. Sriram, “A comparative study of computation of Lyapunov spectra with different algorithms,” Physica D, vol. 139, no. 1-2, pp. 72–86, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. L. Dieci and E. S. Van Vleck, “On the error in computing Lyapunov exponents by QR methods,” Numerische Mathematik, vol. 101, no. 4, pp. 619–642, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. Z.-M. Chen, K. Djidjeli, and W. G. Price, “Computing Lyapunov exponents based on the solution expression of the variational system,” Applied Mathematics and Computation, vol. 174, no. 2, pp. 982–996, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. R. Trejo-Guerra, E. Tlelo-Cuautle, J. M. Muñoz-Pacheco, C. Sánchez-López, and C. Cruz-Hernández, “On the relation between the number of scrolls and the lyapunov exponents in PWL-functions-based n-scroll chaotic oscillators,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 11, no. 11, pp. 903–910, 2010. View at Google Scholar · View at Scopus
  27. T. Stachowiak and M. Szydlowski, “A differential algorithm for the Lyapunov spectrum,” Physica D, vol. 240, no. 16, pp. 1221–1227, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. J. Razjouyan, S. Gharibzadeh, A. Fallah et al., “A neuro-fuzzy based model for accurate estimation of the Lyapunov exponents of an unknown dynamical system,” International Journal of Bifurcation and Chaos, vol. 22, no. 3, Article ID 1250043, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  29. B. Cannas and F. Pisano, “A fast algorithm for Lyapunov exponents calculation in piecewise linear systems,” in Proceedings of the AIP Conference, vol. 1389, pp. 1844–1847.
  30. J. M. Munoz-Pacheco and E. Tlelo-Cuautle, “Automatic synthesis of 2D-n-scrolls chaotic systems by behavioral modeling,” Journal of Applied Research and Technology, vol. 7, no. 1, pp. 5–14, 2009. View at Google Scholar