Table of Contents Author Guidelines Submit a Manuscript
The Scientific World Journal
Volume 2014, Article ID 239407, 12 pages
http://dx.doi.org/10.1155/2014/239407
Research Article

Dynamical Tangles in Third-Order Oscillator with Single Jump Function

1Department of Radio Electronics, Brno University of Technology, Purkynova 118, 612 00 Brno, Czech Republic
2Department of Theoretical Electrotechnics and Electrical Measurement, Technical University of Kosice, Letna 9, 042 00 Kosice, Slovakia

Received 16 July 2014; Revised 18 September 2014; Accepted 18 September 2014; Published 3 December 2014

Academic Editor: Esteban Tlelo-Cuautle

Copyright © 2014 Jiří Petržela 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. J. C. Sprott, Chaos and Time-series Analysis, Oxford University Press, Oxford, UK, 2003. View at MathSciNet
  2. H. G. E. Hentschel and I. Procaccia, “The infinite number of generalized dimensions of fractals and strange attractors,” Physica D: Nonlinear Phenomena, vol. 8, no. 3, pp. 435–444, 1983. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  3. R. F. Gans, “When is cutting chaotic?” Journal of Sound and Vibration, vol. 188, no. 1, pp. 75–83, 1995. View at Publisher · View at Google Scholar · View at Scopus
  4. C. P. Silva, “Shil'nikov's theorem—a tutorial,” IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 40, no. 10, pp. 675–682, 1993. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  5. L. O. Chua, M. Komuro, and T. Matsumoto, “The double scroll family. I. Rigorous proof of chaos,” IEEE Transactions on Circuits and Systems, vol. 33, no. 11, pp. 1072–1097, 1986. View at Publisher · View at Google Scholar · View at MathSciNet
  6. V. Spany, P. Galajda, M. Guzan, L. Pivka, and M. Olej{\'a}r, “Chua's singularities: great miracle in circuit theory,” International Journal of Bifurcation and Chaos, vol. 20, no. 10, pp. 2993–3006, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. J. R. Piper and J. C. Sprott, “Simple autonomous chaotic circuits,” IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 57, no. 9, pp. 730–734, 2010. View at Publisher · View at Google Scholar · View at Scopus
  8. T. Kapitaniak, Chaotic Oscillators, World Scientific, 1992. View at Publisher · View at Google Scholar · View at MathSciNet
  9. C. W. Wu and L. O. Chua, “On linear topological conjugacy of Lur'e systems,” IEEE Transactions on Circuits and Systems. I. Fundamental Theory and Applications, vol. 43, no. 2, pp. 158–161, 1996. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. J. Pospisil, J. Brzobohaty, Z. Kolka, and J. Horska, “New canonical state models of Chua's circuit family,” Radioengineering, vol. 8, pp. 1–5, 1999. View at Google Scholar
  11. J. C. Sprott, “Simple chaotic systems and circuits,” American Journal of Physics, vol. 68, no. 8, pp. 758–763, 2000. View at Publisher · View at Google Scholar · View at Scopus
  12. J. C. Sprott, Elegant Chaos: Algebraically Simple Chaotic Flows, World Scientific, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  13. J. C. Sprott, “A new class of chaotic circuit,” Physics Letters A, vol. 266, no. 1, pp. 19–23, 2000. View at Publisher · View at Google Scholar · View at Scopus
  14. L. M. Kocarev and T. D. Stojanovski, “Linear conjugacy of vector fields in Lur'e form,” IEEE Transactions on Circuits and Systems. I. Fundamental Theory and Applications, vol. 43, no. 9, pp. 782–785, 1996. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. J. Petrzela, “Three-segment piecewise-linear vector fields with orthogonal eigenspaces,” Acta Electrotechnica et Informatica, vol. 9, pp. 44–50, 2009. View at Google Scholar
  16. U. Feldmann and W. Schwarz, “Linear conjugacy of n-dimensional piecewise linear systems,” IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 41, no. 2, pp. 190–192, 1994. View at Publisher · View at Google Scholar · View at Scopus
  17. J. Pospisil, Z. Kolka, J. Horska, and J. Brzobohaty, “Simplest ODE equivalents of Chua's equations,” International Journal of Bifurcation and Chaos, vol. 10, no. 1, pp. 1–23, 2000. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. A. I. Mees and P. B. Chapman, “Homoclinic and heteroclinic orbits in the double scroll attractor,” IEEE Transactions on Circuits and Systems, vol. 34, no. 9, pp. 1115–1120, 1987. View at Publisher · View at Google Scholar · View at MathSciNet
  19. J. Petrzela, “On the strategic orbits in third-order oscillator with jump nonlinearity,” International Journal of Algebra, vol. 4, no. 1–4, pp. 197–207, 2010. View at Google Scholar · View at MathSciNet
  20. T. R. O. Medrano, M. S. Baptista, and I. L. Caldas, “Homoclinic orbits in a piecewise system and their relation with invariant sets,” Physica D: Nonlinear Phenomena, vol. 186, no. 3-4, pp. 133–147, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  21. T. Zhou, G. Chen, and Q. Yang, “Constructing a new chaotic system based on the Shilnikov criterion,” Chaos, Solitons & Fractals, vol. 19, no. 4, pp. 985–993, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. K. Grygiel and P. Szlachetka, “Lyapunov exponents analysis of autonomous and nonautonomous sets of ordinary differential equations,” Acta Physica Polonica B, vol. 26, no. 8, pp. 1321–1331, 1995. View at Google Scholar · View at MathSciNet
  23. J. Petržela, Z. Hruboš, and T. Gotthans, “Modeling deterministic chaos using electronic circuits,” Radioengineering, vol. 20, no. 2, pp. 438–444, 2011. View at Google Scholar · View at Scopus
  24. J. Petrzela, Z. Kolka, and S. Hanus, “Simple chaotic oscillator: from mathematical model to practical experiment,” Radioengineering, vol. 15, pp. 6–11, 2006. View at Google Scholar
  25. T. Gotthans, J. Petrzela, Z. Hrubos, and G. Baudoin, “Parallel particle swarm optimization on chaotic solutions of dynamical systems,” in Proceedings of the 22nd International Conference on Radioelektronika (RADIOELEKTRONIKA '12), pp. 1–4, 2012.
  26. R. Brown, “Generalizations of the Chua equations,” IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 40, no. 11, pp. 878–884, 1993. View at Publisher · View at Google Scholar
  27. W. J. Grantham and B. Lee, “A chaotic limit cycle paradox,” Dynamics and Control, vol. 3, no. 2, pp. 159–173, 1993. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  28. M. Itoh, “Synthesis of electronic circuits for simulating nonlinear dynamics,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 11, no. 3, pp. 605–653, 2001. View at Publisher · View at Google Scholar · View at Scopus
  29. A. S. Elwakil and M. P. Kennedy, “Construction of classes of circuit-independent chaotic oscillators using passive-only nonlinear devices,” IEEE Transactions on Circuits and Systems. I. Fundamental Theory and Applications, vol. 48, no. 3, pp. 289–307, 2001. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  30. Z. Elhadj and J. C. Sprott, “Some open problems in chaos theory and dynamics,” International Journal of Open Problems in Computer Science and Mathematics, vol. 4, no. 2, pp. 1–10, 2011. View at Google Scholar · View at MathSciNet
  31. E. Zeraoulia, Models and Applications of Chaos Theory in Modern Sciences, CRC Press, New York, NY, USA, 2011. View at MathSciNet
  32. T. Gotthans and Z. Hruboš, “Multi grid chaotic attractors with discrete Jumps,” Journal of Electrical Engineering, vol. 64, no. 2, pp. 118–122, 2013. View at Publisher · View at Google Scholar · View at Scopus