Table of Contents Author Guidelines Submit a Manuscript
Discrete Dynamics in Nature and Society
Volume 2015, Article ID 586783, 23 pages
http://dx.doi.org/10.1155/2015/586783
Research Article

Design of Positive, Negative, and Alternating Sign Generalized Logistic Maps

1Engineering Mathematics and Physics Department, Faculty of Engineering, Cairo University, Giza 12613, Egypt
2Nanoelectronics Integrated Systems Center, Nile University, Cairo 12588, Egypt
3Electronics and Electrical Communications Department, Faculty of Engineering, Cairo University, Giza 12613, Egypt

Received 1 April 2015; Accepted 2 June 2015

Academic Editor: Agustin Martin

Copyright © 2015 Wafaa S. Sayed 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. K. M. Cuomo and A. V. Oppenheim, “Circuit implementation of synchronized chaos with applications to communications,” Physical Review Letters, vol. 71, no. 1, pp. 65–68, 1993. View at Publisher · View at Google Scholar · View at Scopus
  2. A. G. Radwan, A. M. Soliman, and A.-L. El-Sedeek, “An inductorless CMOS realization of Chua's circuit,” Chaos, Solitons & Fractals, vol. 18, no. 1, pp. 149–158, 2003. View at Publisher · View at Google Scholar · View at Scopus
  3. A. G. Radwan, A. M. Soliman, and A. El-Sedeek, “MOS realization of the modified Lorenz chaotic system,” Chaos, Solitons & Fractals, vol. 21, no. 3, pp. 553–561, 2004. View at Publisher · View at Google Scholar · View at Scopus
  4. S. Mandal and S. Banerjee, “Analysis and CMOS implementation of a chaos-based communication system,” IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 51, no. 9, pp. 1708–1722, 2004. View at Publisher · View at Google Scholar · View at Scopus
  5. A. G. Radwan, A. M. Soliman, and A. S. Elwakil, “1-D digitally-controlled multiscroll chaos generator,” International Journal of Bifurcation and Chaos, vol. 17, no. 1, pp. 227–242, 2007. View at Publisher · View at Google Scholar · View at Scopus
  6. A. S. Mansingka, A. G. Radwan, and K. N. Salama, “Design, implementation and analysis of fully digital 1-D controllable multiscroll chaos,” in Proceedings of the 23rd International Conference on Microelectronics (ICM '11), pp. 1–5, IEEE, December 2011. View at Publisher · View at Google Scholar · View at Scopus
  7. M. L. Barakat, A. G. Radwan, and K. N. Salama, “Hardware realization of chaos based block cipher for image encryption,” in 2011 23rd International Conference on Microelectronics, ICM 2011, tun, December 2011. View at Publisher · View at Google Scholar · View at Scopus
  8. M. A. Zidan, A. G. Radwan, and K. N. Salama, “Random number generation based on digital differential chaos,” in Proceedings of the 54th IEEE International Midwest Symposium on Circuits and Systems (MWSCAS '11), pp. 1–4, IEEE, August 2011. View at Publisher · View at Google Scholar · View at Scopus
  9. A. S. Mansingka, A. G. Radwan, and K. N. Salama, “Fully digital 1-D, 2-D and 3-D multiscroll chaos as hardware pseudo random number generators,” in Proceedings of the IEEE 55th International Midwest Symposium on Circuits and Systems (MWSCAS '12), pp. 1180–1183, IEEE, August 2012. View at Publisher · View at Google Scholar · View at Scopus
  10. M. A. Zidan, A. G. Radwan, and K. N. Salama, “Controllable V-shape multiscroll butterfly attractor: system and circuit implementation,” International Journal of Bifurcation and Chaos, vol. 22, no. 6, Article ID 1250143, 13 pages, 2012. View at Publisher · View at Google Scholar · View at Scopus
  11. R. M. May, “Simple mathematical models with very complicated dynamics,” Nature, vol. 261, no. 5560, pp. 459–467, 1976. View at Publisher · View at Google Scholar · View at Scopus
  12. P. F. Verhulst, “Notice sur la loi que la population suit dans son accroissement,” Correspondance Mathématique et Physique de l'Observatoire de Bruxelles Publiée Part A: Quetelet, vol. 10, pp. 113–121, 1838. View at Google Scholar
  13. M. J. Feigenbaum, “Quantitative universality for a class of nonlinear transformations,” Journal of Statistical Physics, vol. 19, no. 1, pp. 25–52, 1978. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. A. L. Lloyd, “The coupled logistic map: a simple model for the effects of spatial heterogeneity on population dynamics,” Journal of Theoretical Biology, vol. 173, no. 3, pp. 217–230, 1995. View at Publisher · View at Google Scholar · View at Scopus
  15. S. H. Strogatz, Nonlinear Dynamics And Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, Westview Press, 2014.
  16. E. Scholl, Nonlinear Spatio-Temporal Dynamics and Chaos in Semiconductors, vol. 10, Cambridge University Press, 2001.
  17. N. K. Pareek, V. Patidar, and K. K. Sud, “Image encryption using chaotic logistic map,” Image and Vision Computing, vol. 24, no. 9, pp. 926–934, 2006. View at Publisher · View at Google Scholar · View at Scopus
  18. W. Yao, P. Ye, and X. Li, “An effective privacy-preserving algorithm based on logistic map and rubik's cube transformation,” Discrete Dynamics in Nature and Society, vol. 2014, Article ID 178585, 11 pages, 2014. View at Publisher · View at Google Scholar · View at Scopus
  19. S. C. Phatak and S. S. Rao, “Logistic map: a possible random-number generator,” Physical Review E, vol. 51, no. 4, 1995. View at Publisher · View at Google Scholar · View at Scopus
  20. L. Kocarev and G. Jakimoski, “Logistic map as a block encryption algorithm,” Physics Letters. A, vol. 289, no. 4-5, pp. 199–206, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  21. A. Kanso and N. Smaoui, “Logistic chaotic maps for binary numbers generations,” Chaos, Solitons & Fractals, vol. 40, no. 5, pp. 2557–2568, 2009. View at Publisher · View at Google Scholar · View at Scopus
  22. V. Patidar, K. K. Sud, and N. K. Pareek, “A pseudo random bit generator based on chaotic logistic map and its statistical testing,” Informatica, vol. 33, no. 4, pp. 441–452, 2009. View at Google Scholar · View at MathSciNet
  23. N. Singh and A. Sinha, “Chaos-based secure communication system using logistic map,” Optics and Lasers in Engineering, vol. 48, no. 3, pp. 398–404, 2010. View at Publisher · View at Google Scholar · View at Scopus
  24. M. Suneel, “Electronic circuit realization of the logistic map,” Sadhana, vol. 31, no. 1, pp. 69–78, 2006. View at Google Scholar
  25. J. Gu and S. Chen, “Nonlinear analysis on traffic flow based on catastrophe and chaos theory,” Discrete Dynamics in Nature and Society, vol. 2014, Article ID 535167, 11 pages, 2014. View at Publisher · View at Google Scholar · View at Scopus
  26. D. A. Hsieh, “Chaos and nonlinear dynamics: application to financial markets,” The Journal of Finance, vol. 46, no. 5, pp. 1839–1877, 1991. View at Publisher · View at Google Scholar
  27. N. Basalto, R. Bellotti, F. De Carlo, P. Facchi, and S. Pascazio, “Clustering stock market companies via chaotic map synchronization,” Physica A: Statistical Mechanics and Its Applications, vol. 345, no. 1-2, pp. 196–206, 2005. View at Publisher · View at Google Scholar · View at Scopus
  28. T. Chen, J. He, and Q. Yin, “Dynamics evolution of credit risk contagion in the CRT market,” Discrete Dynamics in Nature and Society, vol. 2013, Article ID 206201, 9 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  29. K. T. Alligood, T. D. Sauer, and J. A. Yorke, Chaos: An Introduction to Dynamical Systems, Springer, 1996.
  30. R. Matthews, “On the derivation of a ‘chaotic’ encryption algorithm,” Cryptologia, vol. 13, no. 1, pp. 29–42, 1989. View at Publisher · View at Google Scholar · View at MathSciNet
  31. Z. Elhadj and J. C. Sprott, “The effect of modulating a parameter in the logistic map,” Chaos, vol. 18, no. 2, Article ID 023119, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  32. R. Vázquez-Medina, A. Díaz-Méndez, J. L. del Río-Correa, and J. López-Hernández, “Design of chaotic analog noise generators with logistic map and MOS QT circuits,” Chaos, Solitons & Fractals, vol. 40, no. 4, pp. 1779–1793, 2009. View at Publisher · View at Google Scholar · View at Scopus
  33. E. A. Levinsohn, S. A. Mendoza, and E. Peacock-López, “Switching induced complex dynamics in an extended logistic map,” Chaos, Solitons & Fractals, vol. 45, no. 4, pp. 426–432, 2012. View at Publisher · View at Google Scholar · View at Scopus
  34. A. G. Radwan, “On some generalized discrete logistic maps,” Journal of Advanced Research, vol. 4, no. 2, pp. 163–171, 2013. View at Publisher · View at Google Scholar · View at Scopus
  35. A. G. Radwan and S. K. Abd-El-Hafiz, “Image encryption using generalized tent map,” in Proceedings of the IEEE 20th International Conference on Electronics, Circuits, and Systems (ICECS '13), pp. 653–656, IEEE, December 2013. View at Publisher · View at Google Scholar · View at Scopus
  36. A. G. Radwan, S. K. Abd-El-Hafiz, and S. H. AbdElHaleem, “An image encryption system based on generalized discrete maps,” in Proceedings of the 21st IEEE International Conference on Electronics, Circuits and Systems (ICECS '14), pp. 283–286, IEEE, Marseille, France, December 2014. View at Publisher · View at Google Scholar
  37. D. Orrell, L. Smith, J. Barkmeijer, and T. N. Palmer, “Model error in weather forecasting,” Nonlinear Processes in Geophysics, vol. 8, no. 6, pp. 357–371, 2001. View at Publisher · View at Google Scholar · View at Scopus
  38. A. Trevisan and L. Palatella, “Chaos and weather forecasting: the role of the unstable subspace in predictability and state estimation problems,” International Journal of Bifurcation and Chaos, vol. 21, no. 12, pp. 3389–3415, 2011. View at Publisher · View at Google Scholar · View at Scopus
  39. D. Dangoisse, P. Glorieux, and D. Hennequin, “Chaos in a CO2 laser with modulated parameters: experiments and numerical simulations,” Physical Review A, vol. 36, no. 10, article 4775, 1987. View at Publisher · View at Google Scholar · View at Scopus
  40. M. D. Bernardo and F. Vasca, “Discrete-time maps for the analysis of bifurcations and chaos in DC/DC converters,” IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 47, no. 2, pp. 130–143, 2000. View at Publisher · View at Google Scholar · View at Scopus
  41. A. N. Kolmogorov, Foundations of the Theory of Probability, Chelsea Publishing Company, New York, NY, USA, 1950. View at MathSciNet
  42. A. Khrennikov, Interpretations of Probability, Walter De Gruyter, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  43. Y. D. Han, W. Y. Hwang, and I. G. Koh, “Explicit solutions for negative-probability measures for all entangled states,” Physics Letters A, vol. 221, no. 5, pp. 283–286, 1996. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  44. T. Curtright and C. Zachos, “Negative probability and uncertainty relations,” Modern Physics Letters A: Particles and Fields, Gravitation, Cosmology, Nuclear Physics, vol. 16, no. 37, pp. 2381–2385, 2001. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  45. H. F. Hofmann, “How to simulate a universal quantum computer using negative probabilities,” Journal of Physics. A. Mathematical and Theoretical, vol. 42, no. 27, Article ID 275304, pp. 275–304, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  46. M. O. Scully, H. Walther, and W. Schleich, “Feynman's approach to negative probability in quantum mechanics,” Physical Review A, vol. 49, no. 3, pp. 1562–1566, 1994. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  47. E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Physical Review, vol. 40, no. 5, pp. 749–759, 1932. View at Publisher · View at Google Scholar · View at Scopus
  48. P. Dirac, “Bakerian lecture. The physical interpretation of quantum mechanics,” Proceedings of the Royal Society of London. Series A, vol. 180, no. 980, pp. 1–40, 1942. View at Google Scholar
  49. R. P. Feynman, “Negative probability,” in Quantum Implications: Essays in Honour of David Bohm, F. D. Peat and B. Hiley, Eds., pp. 235–248, Routledge & Kegan Paul, 1987. View at Google Scholar
  50. W. Mückenheim, “A review of extended probabilities,” Physics Reports: A Review Section of Physics Letters, vol. 133, no. 6, pp. 337–401, 1986. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  51. T. Rothman and E. C. Sudarshan, “Hidden variables or positive probabilities?” International Journal of Theoretical Physics, vol. 40, no. 8, pp. 1525–1543, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  52. C. Ferrie and J. Emerson, “Frame representations of quantum mechanics and the necessity of negativity in quasi-probability representations,” Journal of Physics A: Mathematical and Theoretical, vol. 41, no. 35, Article ID 352001, 11 pages, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  53. M. Gell-Mann and J. B. Hartle, “Decoherent histories quantum mechanics with one real fine-grained history,” Physical Review A: Atomic, Molecular, and Optical Physics, vol. 85, no. 6, Article ID 062120, 2012. View at Publisher · View at Google Scholar · View at Scopus
  54. P. Dirac, “Quantum electrodynamics,” Communications of the Dublin Institute for Advanced Studies Series A, vol. 1, pp. 1–36, 1943. View at Google Scholar
  55. R. A. Jarrow and S. M. Turnbull, “Pricing derivatives on financial securities subject to credit risk,” The Journal of Finance, vol. 50, no. 1, pp. 53–85, 1995. View at Publisher · View at Google Scholar
  56. D. Duffie and K. J. Singleton, “Modeling term structures of defaultable bonds,” Review of Financial Studies, vol. 12, no. 4, pp. 687–720, 1999. View at Publisher · View at Google Scholar · View at Scopus
  57. E. G. Haug, “Why so negative to negative probabilities?” Wilmott Magazine, pp. 34–38, 2004. View at Google Scholar
  58. J. C. Cox, S. A. Ross, and M. Rubinstein, “Option pricing: a simplified approach,” Journal of Financial Economics, vol. 7, no. 3, pp. 229–263, 1979. View at Publisher · View at Google Scholar · View at Scopus
  59. M. Burgin and G. Meissner, “Negative probabilities in financial modeling,” Wilmott, vol. 2012, no. 58, pp. 60–65, 2012. View at Publisher · View at Google Scholar
  60. J. T. Ambadan and K. B. Joseph, “Asymmetrical mirror bifurcations in logistic map with a discontinuity at zero,” in Proceedings of the National Conference on Nonlinear Systems and Dynamics (NCNSD '06), February 2006.
  61. Z.-L. Zhu, W. Zhang, K.-W. Wong, and H. Yu, “A chaos-based symmetric image encryption scheme using a bit-level permutation,” Information Sciences, vol. 181, no. 6, pp. 1171–1186, 2011. View at Publisher · View at Google Scholar · View at Scopus