Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2014 (2014), Article ID 956318, 17 pages
http://dx.doi.org/10.1155/2014/956318
Research Article

Approximate Solutions by Truncated Taylor Series Expansions of Nonlinear Differential Equations and Related Shadowing Property with Applications

1Institute of Research and Development of Processes, University of the Basque Country, Campus of Leioa (Bizkaia), P.O. Box 644, Bilbao, Barrio Sarriena, 48940 Leioa, Spain
2Department of Telecommunications and Systems Engineering, Universitat Autònoma de Barcelona, UAB, 08193 Barcelona, Spain

Received 5 December 2013; Accepted 11 June 2014; Published 7 July 2014

Academic Editor: Samir Saker

Copyright © 2014 M. De la Sen 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. M. de la Sen, “Application of the nonperiodic sampling to the identifiability and model matching problems in dynamic systems,” International Journal of Systems Science, vol. 14, no. 4, pp. 367–383, 1983. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  2. M. de la Sen, “Stability of switched feedback time-varying dynamic systems based on the properties of the gap metric for operators,” Abstract and Applied Analysis, vol. 2012, Article ID 612198, 17 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  3. M. de la Sen, “About the stabilization of a nonlinear perturbed difference equation,” Discrete Dynamics in Nature and Society, vol. 2012, Article ID 320302, 19 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  4. M. De la Sen, J. C. Soto, and A. Ibeas, “Stability and limit oscillations of a control event-based sampling criterion,” Journal of Applied Mathematics, vol. 2012, Article ID 684292, 25 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  5. R. Gu, “The average-shadowing property and topological ergodicity,” Journal of Computational and Applied Mathematics, vol. 206, no. 2, pp. 796–800, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. R. Gu, “On ergodicity of systems with the asymptotic average shadowing property,” Computers & Mathematics with Applications, vol. 55, no. 6, pp. 1137–1141, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. R. Gu, “Recurrence and the asymptotic pseudo-orbit tracing property,” Nonlinear Analysis: Theory, Methods & Applications, vol. 66, no. 8, pp. 1698–1706, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. K. Lee and K. Sakai, “Various shadowing properties and their equivalence,” Discrete and Continuous Dynamical Systems A, vol. 13, no. 2, pp. 533–540, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. E. S. van Vleck, “Numerical shadowing using componentwise bounds and a sharper fixed point result,” SIAM Journal on Scientific Computing, vol. 22, no. 3, pp. 787–801, 2000. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. S. Reich, “Backward error analysis for numerical integrators,” SIAM Journal on Numerical Analysis, vol. 36, no. 5, pp. 1549–1570, 1999. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. T. M. Apostol, Mathematical Analysis, Addison-Wesley, Reading, Mass, USA, 1958.
  12. Q. L. Zhang, G. Z. Dai, J. Lam, L. Q. Zhang, and M. De La Sen, “Asymptotic stability and stabilization of descriptor systems,” Acta Automatica Sinica, vol. 24, no. 2, pp. 208–211, 1998. View at Google Scholar · View at MathSciNet
  13. M. de la Sen, “Fundamental properties of linear control systems with after-effect. I. The continuous case,” Mathematical and Computer Modelling, vol. 10, no. 7, pp. 473–489, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  14. R. A. Maher and R. Samir, “Robust stability of a class of unstable systems under mixed uncertainty,” Journal of Control Science and Engineering, vol. 2011, Article ID 970962, 8 pages, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  15. D. Boutat, “Extended nonlinear observer normal forms for a class of nonlinear dynamical systems,” International Journal of Robust and Nonlinear Control, 2013. View at Publisher · View at Google Scholar
  16. M. Benchohra and M. Ziane, “Impulsive evolution inclusions with state-dependent delay and multivalued jumps,” Electronic Journal of Qualitative Theory of Differential Equations, no. 42, pp. 1–21, 2013. View at Google Scholar · View at MathSciNet
  17. C. Tunç and M. Ateş, “Boundedness of solutions to differential equations of fourth order with oscillatory restoring and forcing terms,” Discrete Dynamics in Nature and Society, vol. 2013, Article ID 758796, 5 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  18. C. Tunc and M. Gozen, “Stability and uniform boundedness in multidelay functional differential equations of third order,” Abstract and Applied Analysis, vol. 2013, Article ID 248717, 7 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  19. J. Diblík, M. Fečkan, and M. Pospíšil, “Representation of a solution of the Cauchy problem for an oscillating system with multiple delays and pairwise permutable matrices,” Abstract and Applied Analysis, vol. 2013, Article ID 931493, 10 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  20. B. Baculikova, J. Dzurina, and Y. V. Rogovchenko, “Oscillation of third order trinomial delay differential equations,” Applied Mathematics and Computation, vol. 218, no. 13, pp. 7023–7033, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  21. J. Baštinec, L. Berezansky, J. Diblík, and Z. Šmarda, “On the critical case in oscillation for differential equations with a single delay and with several delays,” Abstract and Applied Analysis, vol. 2010, Article ID 417869, 20 pages, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  22. M. Hasanbulli and Y. V. Rogovchenko, “Oscillation criteria for second order nonlinear neutral differential equations,” Applied Mathematics and Computation, vol. 215, no. 12, pp. 4392–4399, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  23. Z. Han, Y. Zhao, Y. Sun, and C. Zhang, “Oscillation for a class of fractional differential equation,” Discrete Dynamics in Nature and Society, vol. 2013, Article ID 390282, 6 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  24. J. C. Soto and M. Delasen, “Nonlinear oscillations in nonperiodic sampling systems,” Electronics Letters, vol. 20, no. 20, pp. 816–818, 1984. View at Publisher · View at Google Scholar · View at Scopus
  25. M. de la Sen, “Oscillatory behavior in linear difference equations under unmodeled dynamics and parametrical errors,” Mathematical Problems in Engineering, vol. 2007, Article ID 25692, 18 pages, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  26. J. C. Soto and M. Delasen, “On the derivation and analysis of a non-linear model for describing a class of adaptive sampling laws,” International Journal of Control, vol. 42, no. 6, pp. 1347–1368, 1985. View at Publisher · View at Google Scholar · View at Scopus
  27. J. Chacón, J. Sánchez, A. Visioli, L. Yebra, and S. Dormido, “Characterization of limit cycles for self-regulating and integral processes with PI control and send-on-delta sampling,” Journal of Process Control, vol. 23, no. 6, pp. 826–838, 2013. View at Publisher · View at Google Scholar · View at Scopus
  28. R. Memarbashi and H. Rasuli, “Notes on the dynamics of nonautonomous discrete dynamical systems,” Journal of Advanced Research in Dynamical and Control Systems, vol. 6, no. 2, pp. 8–17, 2014. View at Google Scholar
  29. F. Mazenc, M. Malisoff, and M. de Querioz, “Tracking control and robustness analysis for a nonlinear model of human heart rate during exercise,” Automatica, vol. 47, no. 5, pp. 968–974, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus