Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2015, Article ID 276059, 13 pages
http://dx.doi.org/10.1155/2015/276059
Research Article

Fractional Dynamics in Calcium Oscillation Model

Faculty of Engineering, King Mongkut’s Institute of Technology Ladkrabang, Chalongkrung Road, Ladkrabang, Bangkok 10520, Thailand

Received 20 April 2014; Revised 8 June 2014; Accepted 1 July 2014

Academic Editor: Shaofan Li

Copyright © 2015 Yoothana Suansook and Kitti Paithoonwattanakij. 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. E. N. Lorenz, “Deterministic non-periodic flow,” Journal of the Atmospheric Sciences, vol. 20, pp. 130–141, 1963. View at Google Scholar
  2. S. H. Strogatz, Nonlinear Dynamics and Chaos, Westview, 2000.
  3. J. G. Lu, “Chaotic dynamics and synchronization of fractional-order Arneodo's systems,” Chaos, Solitons and Fractals, vol. 26, no. 4, pp. 1125–1133, 2005. View at Publisher · View at Google Scholar · View at Scopus
  4. W. Zhang, S. Zhou, H. Li, and H. Zhu, “Chaos in a fractional order Rössler system,” Chaos, Solitons and Fractals, vol. 42, no. 3, pp. 1684–1691, 2009. View at Publisher · View at Google Scholar · View at Scopus
  5. J. G. Lu and G. Chen, “A note on the fractional-order Chen system,” Chaos, Solitons and Fractals, vol. 27, no. 3, pp. 685–688, 2006. View at Publisher · View at Google Scholar · View at Scopus
  6. Z.-M. Ge and C.-Y. Ou, “Chaos in a fractional order modified Duffing system,” Chaos, Solitons and Fractals, vol. 34, no. 2, pp. 262–291, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  7. X. Wang and L. Tian, “Bifurcation analysis and linear control of the Newton-Leipnik system,” Chaos, Solitons and Fractals, vol. 27, no. 1, pp. 31–38, 2006. View at Publisher · View at Google Scholar · View at Scopus
  8. I. Petráš, “A note on the fractional-order Chua's system,” Chaos, Solitons and Fractals, vol. 38, no. 1, pp. 140–147, 2008. View at Publisher · View at Google Scholar · View at Scopus
  9. I. Petráš, “A note on the fractional-order Volta's system,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 2, pp. 384–393, 2010. View at Publisher · View at Google Scholar
  10. C. M. A. Pinto and J. A. T. Machado, “Fractional model for malaria transmission under control strategies,” Computers & Mathematics with Applications, vol. 66, no. 5, pp. 908–916, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. Y. Suansook and K. Paithoonwattanakij, “Dynamic of logistic model at fractional order,” in Proceeding of the International Symposium on Industrial Electronics (IEEE ISIE '09), pp. 718–723, Seoul, Republic Korea, July 2009. View at Publisher · View at Google Scholar · View at Scopus
  12. G.-C. Wu, D. Baleanu, and S.-D. Zeng, “Discrete chaos in fractional sine and standard maps,” Physics Letters A, vol. 378, no. 5-6, pp. 484–487, 2014. View at Publisher · View at Google Scholar
  13. Y. Liu, Y. Xie, Y. Kang et al., “Dynamical characteristics of the fractional-order fitzHugh-Nagumo model neuron,” in Advances in Cognitive Neurodynamics (II), pp. 253–258, 2011. View at Google Scholar
  14. S. Pooseh, H. S. Rodrigues, and D. F. M. Torres, “Fractional derivatives in dengue epidemics,” in Proceedings of the International Conference on Numerical Analysis and Applied Mathematics: Numerical Analysis and Applied Mathematics (ICNAAM '11), vol. 1389, pp. 739–742, September 2011. View at Publisher · View at Google Scholar · View at Scopus
  15. E. Hanert, E. Schumacher, and E. Deleersnijder, “Front dynamics in fractional-order epidemic models,” Journal of Theoretical Biology, vol. 279, pp. 9–16, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. V. Daftardar-Gejji and S. Bhalekar, “Chaos in fractional ordered Liu system,” Computers and Mathematics with Applications, pp. 1117–1127, 2010. View at Google Scholar
  17. I. Petráš, “Method for simulation of the fractional order chaotic system,” Acta Montanistica Slovaca, vol. 11, pp. 273–277, 2006. View at Google Scholar
  18. A. Goldbeter, Biochemical Oscillations and Cellular Rhythms: The Molecular Bases of Periodic and Chaotic Behavior, Cambridge University Press, 1996.
  19. M. Falcke and D. Malchow, Understanding Calcium Dynamics: Experiments and Theory, vol. 623 of Lecture Notes in Physics, 2003.
  20. M. J. Berridge, “Calcium oscillations,” The Journal of Biological Chemistry, vol. 265, no. 17, pp. 9583–9586, 1990. View at Google Scholar · View at Scopus
  21. U. Kummer, L. F. Olsen, C. J. Dixon, A. K. Green, E. Bornberg-Bauer, and G. Baier, “Switching from simple to complex oscillations in calcium signaling,” Biophysical Journal, vol. 79, no. 3, pp. 1188–1195, 2000. View at Publisher · View at Google Scholar · View at Scopus
  22. A. Goldbeter, “Computational approaches to cellular rhythms,” Nature, vol. 420, no. 6912, pp. 238–245, 2002. View at Publisher · View at Google Scholar · View at Scopus
  23. M. D. Bootman, M. J. Berridge, and H. L. Roderick, “Calcium signalling: more messengers, more channels, more complexity,” Current Biology, vol. 12, no. 16, pp. R563–R565, 2002. View at Publisher · View at Google Scholar · View at Scopus
  24. J. A. M. Borghans, G. Dupont, and A. Goldbeter, “Complex intracellular calcium oscillations. A theoretical exploration of possible mechanisms,” Biophysical Chemistry, vol. 66, no. 1, pp. 25–41, 1997. View at Publisher · View at Google Scholar · View at Scopus
  25. N. M. Woods, K. S. R. Kuthbertson, and P. H. Cobbold, “Agonist-induced oscillations in hepatocytes,” Cells Calcium, vol. 8, pp. 79–100, 1987. View at Google Scholar
  26. M. J. Berridge, “Unlocking the secrets of cell signaling,” Annual Review of Physiology, vol. 67, pp. 1–21, 2005. View at Publisher · View at Google Scholar · View at Scopus
  27. P. Uhlén and N. Fritz, “Biochemistry of calcium oscillations,” Biochemical and Biophysical Research Communications, vol. 396, no. 1, pp. 28–32, 2010. View at Publisher · View at Google Scholar
  28. M. Perc and M. Marhl, “Sensitivity and flexibility of regular and chaotic calcium oscillations,” Biophysical Chemistry, vol. 104, no. 2, pp. 509–522, 2003. View at Publisher · View at Google Scholar · View at Scopus
  29. P. Shen and R. Larter, “Chaos in intracellular Ca2+ oscillations in a new model for non-excitable cells,” Cell Calcium, vol. 17, no. 3, pp. 225–232, 1995. View at Publisher · View at Google Scholar · View at Scopus
  30. G. Dupont and A. Goldbeter, “One-pool model for Ca2+ oscillations involving Ca2+ and inositol 1,4,5-trisphosphate as co-agonists for Ca2+ release,” Cell Calcium, vol. 14, no. 4, pp. 311–322, 1993. View at Publisher · View at Google Scholar · View at Scopus
  31. T. Meyer and L. Stryer, “Transient calcium release induced by successive increments of inositol 1,4,5-trisphosphate,” Proceedings of the National Academy of Sciences of the United States of America, vol. 87, no. 10, pp. 3841–3845, 1990. View at Publisher · View at Google Scholar · View at Scopus
  32. G. W. de Young and J. Keizer, “A single-pool inositol 1,4,5-trisphosphate-receptor-based model for agonist-stimulated oscillations in Ca2+ concentration,” Proceedings of the National Academy of Sciences of the United States of America, vol. 89, no. 20, pp. 9895–9899, 1992. View at Google Scholar · View at Scopus
  33. Y. Timofeeva, Oscillations and waves in single and multi-cellular system with free calcium [Ph.D. thesis], Loughborough University, 2003.
  34. S. Schuster, M. Marhl, and T. Höfer, “Modelling of simple and complex calcium oscillations from single-cell responses to intercellular signalling,” European Journal of Biochemistry, vol. 269, no. 5, pp. 1333–1355, 2002. View at Publisher · View at Google Scholar · View at Scopus
  35. K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, NY, USA, 1974. View at MathSciNet
  36. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993. View at MathSciNet
  37. I. Podlubny, Fractional Differential Equations, vol. 198, Academic Press, San Diego, Calif, USA, 1999. View at MathSciNet
  38. M. S. Tavazoei and M. Haeri, “Limitations of frequency domain approximation for detecting chaos in fractional order systems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 4, pp. 1299–1320, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  39. V. Daftardar-Gejji and H. Jafari, “Adomian decomposition: a tool for solving a system of fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 301, no. 2, pp. 508–518, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  40. J. Sabatier, O. P. Agrawal, and J. A. T. Machado, Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, Dordrecht, The Netherlands, 2007.
  41. S. J. Singh and A. Chatterjee, “Three classes of FDEs amenable to approximation using a Galerkin technique,” in Advances in Fractional Calculus, Springer, 2007. View at Google Scholar
  42. J. Sabatier, M. Merveillaut, R. Malti, and A. Oustaloup, “How to impose physically coherent initial conditions to a fractional system?” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 5, pp. 1318–1326, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  43. N. Bertrand, J. Sabatier, O. Briat, and J.-M. Vinassa, “Fractional non-linear modelling of ultracapacitors,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 5, pp. 1327–1337, 2010. View at Publisher · View at Google Scholar · View at Scopus
  44. R. L. Bagley and P. J. Torvik, “A theoretical basis for the application of fractional calculus to viscoelasticity,” Journal of Rheology, vol. 27, no. 3, pp. 201–210, 1983. View at Publisher · View at Google Scholar · View at Scopus
  45. J. Sabatier, H. C. Nguyen, C. Farges et al., “Fractional models for thermal modeling and temperature estimation of a transistor junction,” Advances in Difference Equations, vol. 201, Article ID 687363, 12 pages, 2011. View at Publisher · View at Google Scholar
  46. I. Podlubny, “Geometric and physical interpretation of fractional integration and fractional differentiation,” Fractional Calculus & Applied Analysis, vol. 5, no. 4, pp. 367–386, 2002. View at Google Scholar · View at MathSciNet
  47. J. A. Tenreiro Machado, “A Probabilistic Interpretation of the fractional-order differentiation,” Fractional Calculus & Applied Analysis, vol. 6, no. 1, pp. 73–80, 2003. View at Google Scholar
  48. M. Du, Z. Wang, and H. Hu, “Measuring memory with the order of fractional derivative,” Scientific Reports, vol. 3,article 3431, 2013. View at Publisher · View at Google Scholar
  49. C. Ionescu, R. de Keyser, J. Sabatier, A. Oustaloup, and F. Levron, “Low frequency constant-phase behavior in the respiratory impedance,” Biomedical Signal Processing and Control, vol. 6, no. 2, pp. 197–208, 2011. View at Publisher · View at Google Scholar · View at Scopus
  50. Z. Odibat and S. Momani, “An Algorithm for the numerical solution of differential equations of fractional order,” Journal of Applied Mathematics & Informatics, vol. 26, no. 1-2, pp. 15–27, 2008. View at Google Scholar
  51. D. Matignon, “Stability results for fractional differential equations with applications to control processing,” in Proceedings of the Computational Engineering in Systems Applications, vol. 2, pp. 963–968, Lille, France, 1996.