Nonlinear Dynamics

, Volume 91, Issue 3, pp 1697–1711 | Cite as

Self-similarity and adaptive aperiodic stochastic resonance in a fractional-order system

  • Chengjin Wu
  • Shang Lv
  • Juncai Long
  • Jianhua YangEmail author
  • Miguel A. F. Sanjuán
Original Paper


We investigate the aperiodic stochastic resonance (ASR) in a bistable fractional-order system when the fractional order lies in the interval (0, 2]. We find that a weak aperiodic signal can be amplified and optimized by varying the fractional order in the nonlinear system, no matter whether it has the assistance of the noise or not. We focus mainly on the self-similarity of the response to the input aperiodic signal and the adaptive ASR. The self-similarity is a characteristic that the response of a nonlinear system matches the input signal well, in the absence of the noise excitation. The adaptive ASR is a technique for the optimal ASR to occur by modulating the fractional order, the noise intensity, or the bistable system parameters. In order to make the optimal ASR occur, an adaptive particle swarm optimization (APSO) algorithm is used in this work as the method of parameter optimization. In previous works, only the periodic signal is optimized in a noisy bistable fractional-order system and the ASR induced by the fractional-order system has not been achieved. In engineering and scientific fields, not only periodic signals need to be processed, but also weak aperiodic ones. Moreover, the optimal ASR shows better results based on the APSO algorithm. We believe that the results of this paper might have a positive contribution in the dynamics research.


Self-similarity Aperiodic stochastic resonance Fractional-order calculus Noise APSO algorithm 


  1. 1.
    Benzi, R., Sutera, A., Vulpiani, A.: The mechanism of stochastic resonance. J. Phys. A Math. Gen. 14, L453–L457 (1981)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Benzi, R., Parisi, G., Sutera, A., Vulpiani, A.: Stochastic resonance in climatic change. Tellus 34, 10–16 (1982)CrossRefzbMATHGoogle Scholar
  3. 3.
    Benzi, R., Parisi, G., Sutera, A., Vulpiani, A.: A theory of stochastic resonance in climatic change. SIAM J. Appl. Math. 43, 565–578 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Huelga, S.F., Plenio, M.B.: Stochastic resonance phenomena in quantum many-body systems. Phys. Rev. Lett. 98, 170601 (2007)CrossRefzbMATHGoogle Scholar
  5. 5.
    Zhong, W.R., Shao, Y.Z., He, Z.H.: Pure multiplicative stochastic resonance of a theoretical anti-tumor model with seasonal modulability. Phys. Rev. E 73, 060902 (2006)CrossRefGoogle Scholar
  6. 6.
    Hirano, Y., Segawa, Y., Kawai, T., Matsumoto, T.: Stochastic resonance in a molecular redox circuit. J. Phys. Chem. C 117, 140–145 (2013)CrossRefGoogle Scholar
  7. 7.
    Mondal, D., Muthukumar, M.: Stochastic resonance during a polymer translocation process. J. Chem. Phys. 144, 144901 (2016)CrossRefGoogle Scholar
  8. 8.
    Valenti, D., Fiasconaro, A., Spagnolo, B.: Stochastic resonance and noise delayed extinction in a model of two competing species. Physica A 331, 477–486 (2004)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Hänggi, P.: Stochastic resonance in biology. How noise can enhance detection of weak signals and help improve biological information processing. ChemPhysChem 3, 285–290 (2002)CrossRefGoogle Scholar
  10. 10.
    McDonnell, M.D., Abbott, D.: What is stochastic resonance? Definitions, misconceptions, debates, and its relevance to biology. PLoS Comput. Biol. 5, e1000348 (2009)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Berdichevsky, V., Gitterman, M.: Multiplicative stochastic resonance in linear systems: analytical solution. Europhys. Lett. 36, 161–165 (1996)CrossRefzbMATHGoogle Scholar
  12. 12.
    Jia, Y., Yu, S.N., Li, J.R.: Stochastic resonance in a bistable system subject to multiplicative and additive noise. Phys. Rev. E 62, 1869–1878 (2000)CrossRefGoogle Scholar
  13. 13.
    Zheng, R., Nakano, K., Hu, H., Su, D., Cartmell, M.P.: An application of stochastic resonance for energy harvesting in a bistable vibrating system. J. Sound Vib. 333, 2568–2587 (2014)CrossRefGoogle Scholar
  14. 14.
    Mitaim, S., Kosko, B.: Adaptive stochastic resonance. Proc. IEEE 86, 2152–2183 (1998)CrossRefGoogle Scholar
  15. 15.
    Mitaim, S., Kosko, B.: Adaptive stochastic resonance in noisy neurons based on mutual information. IEEE Trans. Neural Netw. 15, 1526–1540 (2004)CrossRefGoogle Scholar
  16. 16.
    Lei, Y., Han, D., Lin, J., He, Z.: Planetary gearbox fault diagnosis using an adaptive stochastic resonance method. Mech. Syst. Signal Process. 38, 113–124 (2013)CrossRefGoogle Scholar
  17. 17.
    Collins, J.J., Chow, C.C., Imhoff, T.T.: Aperiodic stochastic resonance in excitable systems. Phys. Rev. E 52, R3321 (1995)CrossRefGoogle Scholar
  18. 18.
    Collins, J.J., Chow, C.C., Capela, A.C., lmhoff, T.T.: Aperiodic stochastic resonance. Phys. Rev. E 54, 5575–5584 (1996)CrossRefGoogle Scholar
  19. 19.
    Collins, J.J., Chow, C.C., Imhoff, T.T.: Stochastic resonance without tuning. Nature 376, 236–238 (1995)CrossRefGoogle Scholar
  20. 20.
    Kohar, V., Sinha, S.: Noise-assisted morphing of memory and logic function. Phys. Lett. A 376, 957–962 (2012)CrossRefzbMATHGoogle Scholar
  21. 21.
    Duan, F., Rousseau, D., Chapeau-Blondeau, F.: Residual aperiodic stochastic resonance in a bistable dynamic system transmitting a suprathreshold binary signal. Phys. Rev. E 69, 011109 (2004)CrossRefGoogle Scholar
  22. 22.
    Duan, F., Abbott, D.: Binary modulated signal detection in a bistable receiver with stochastic resonance. Physica A 376, 173–190 (2007)CrossRefGoogle Scholar
  23. 23.
    Liu, J., Li, Z., Guan, L., Pan, L.: A novel parameter-tuned stochastic resonator for binary PAM signal processing at low SNR. IEEE Commun. Lett. 18, 427–430 (2014)CrossRefGoogle Scholar
  24. 24.
    Liu, J., Li, Z.: Binary image enhancement based on aperiodic stochastic resonance. IET Image Process. 9, 1033–1038 (2015)CrossRefGoogle Scholar
  25. 25.
    Collins, J.J., Imhoff, T.T., Grigg, P.: Noise-enhanced information transmission in rat SA1 cutaneous mechanoreceptors via aperiodic stochastic resonance. J. Neurophysiol. 76, 642–645 (1996)CrossRefGoogle Scholar
  26. 26.
    Sun, S., Lei, B.: On an aperiodic stochastic resonance signal processor and its application in digital watermarking. Signal Process. 88, 2085–2094 (2008)CrossRefzbMATHGoogle Scholar
  27. 27.
    Li, X., Cao, G., Liu, H.: Aperiodic signals processing via parameter-tuning stochastic resonance in a photorefractive ring cavity. AIP Adv. 4, 047111 (2014)CrossRefGoogle Scholar
  28. 28.
    Barbay, S., Giacomelli, G., Marin, F.: Experimental evidence of binary aperiodic stochastic resonance. Phys. Rev. Lett. 85, 4652–4655 (2000)CrossRefGoogle Scholar
  29. 29.
    Barbay, S., Giacomelli, G., Marin, F.: Stochastic resonance in vertical cavity surface emitting lasers. Phys. Rev. E 61, 157–166 (2000)CrossRefGoogle Scholar
  30. 30.
    Barbay, S., Giacomelli, G., Marin, F.: Noise-assisted transmission of binary information: theory and experiment. Phys. Rev. E 63, 051110 (2001)CrossRefGoogle Scholar
  31. 31.
    Chizhevsky, V.N., Giacomelli, G.: Vibrational resonance and the detection of aperiodic binary signals. Phys. Rev. E 77, 051126 (2008)CrossRefGoogle Scholar
  32. 32.
    Hu, G., Gong, D., Yang, C., Qing, G., Li, R.: Stochastic resonance in a nonlinear system driven by an aperiodic force. Phys. Rev. A 46, 3250–3254 (1992)CrossRefGoogle Scholar
  33. 33.
    Yu, T., Zhang, L., Luo, M.K.: Stochastic resonance in the fractional Langevin equation driven by multiplicative noise and periodically modulated noise. Phys. Scr. 88, 045008 (2013)CrossRefzbMATHGoogle Scholar
  34. 34.
    Litak, G., Borowiec, M.: On simulation of a bistable system with fractional damping in the presence of stochastic coherence resonance. Nonlinear Dyn. 77, 681–686 (2014)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Zhong, S., Ma, H., Peng, H., Zhang, L.: Stochastic resonance in a harmonic oscillator with fractional-order external and intrinsic dampings. Nonlinear Dyn. 82, 535–545 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Monje, C.A., Vinagre, B.M., Feliu, V., Chen, Y.: Tuning and auto-tuning of fractional order controllers for industry applications. Control Eng. Pract. 16, 798–812 (2008)CrossRefGoogle Scholar
  37. 37.
    Bohannan, G.W.: Analog fractional order controller in temperature and motor control applications. J. Vib. Control 14, 1487–1498 (2008)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Diethelm, K., Freed, A.D.: On the solution of nonlinear fractional-order differential equations used in the modeling of viscoplasticity. In: Keil, F., Mackens, W., Voß, H., Werther, J. (eds.) Scientific Computing in Chemical Engineering II, pp. 217–224. Springer, Berlin (1999)CrossRefGoogle Scholar
  39. 39.
    Meral, F.C., Royston, T.J., Magin, R.: Fractional calculus in viscoelasticity: an experimental study. Commun. Nonlinear Sci. Numer. Simul. 15, 939–945 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Sheng, H., Chen, Y., Qiu, T.: Fractional Processes and Fractional-Order Signal Processing: Techniques and Applications. Springer, London (2011)zbMATHGoogle Scholar
  41. 41.
    Yau, H.T., Wu, S.Y., Chen, C.L., Li, Y.C.: Fractional-Order chaotic self-synchronization-based tracking faults diagnosis of ball bearing systems. IEEE Trans. Ind. Electron. 63, 3824–3833 (2016)CrossRefGoogle Scholar
  42. 42.
    Craiem, D., Magin, R.L.: Fractional order models of viscoelasticity as an alternative in the analysis of red blood cell (RBC) membrane mechanics. Phys. Biol. 7, 013001 (2010)CrossRefGoogle Scholar
  43. 43.
    Shen, Y.J., Wei, P., Yang, S.P.: Primary resonance of fractional-order van der Pol oscillator. Nonlinear Dyn. 77, 1629–1642 (2014)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Shen, Y.J., Wei, P., Sui, C.Y., Yang, S.P.: Subharmonic resonance of van der Pol oscillator with fractional-order derivative. Math. Probl. Eng. 2014, 738087 (2014)MathSciNetGoogle Scholar
  45. 45.
    Yang, J.H., Zhu, H.: Vibrational resonance in Duffing systems with fractional-order damping. Chaos 22, 013112 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Petráš, I.: Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation. Higher Education Press, Beijing (2011)CrossRefzbMATHGoogle Scholar
  47. 47.
    Monje, C.A., Chen, Y.Q., Vinagre, B.M., Xue, D., Feliu, V.: Fractional-Order Systems and Controls: Fundamentals and Applications. Springer, London (2010)CrossRefzbMATHGoogle Scholar
  48. 48.
    Nayfeh, A.H., Balachandran, B.: Applied Nonlinear Dynamics: Analytical, Computational and Experimental Methods. Wiley, Weinheim (2008)zbMATHGoogle Scholar
  49. 49.
    Thompson, J.M.T., Stewart, H.B.: Nonlinear Dynamics and Chaos. Wiley, New York (2002)zbMATHGoogle Scholar
  50. 50.
    Yang, J.H., Sanjuán, M.A.F., Liu, H.G., Litak, G., Li, X.: Stochastic P-bifurcation and stochastic resonance in a noisy bistable fractional-order system. Commun. Nonlinear Sci. 41, 104–117 (2016)MathSciNetCrossRefGoogle Scholar
  51. 51.
    Zhu, W.Q.: Random Vibration, pp. 371–372. Science Press, Beijing (1992)Google Scholar
  52. 52.
    Poli, R., Kennedy, J., Blackwell, T.: Particle swarm optimization. Swarm Intell. 1, 33–57 (2007)CrossRefGoogle Scholar
  53. 53.
    Eberhart, R.C., Shi, Y.: Particle swarm optimization: developments, applications and resources. In: Proceeding of IEEE Congress on Evolutionary Computation, pp. 81–86. Seoul, Korea (2001)Google Scholar
  54. 54.
    Deb, K., Pratap, A., Agarwal, S., Meyarivan, T.: A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans. Evol. Comput. 6, 182–197 (2002)CrossRefGoogle Scholar
  55. 55.
    Zhan, Z.H., Zhang, J., Li, Y., Chung, H.S.H.: Adaptive particle swarm optimization. IEEE Trans. Syst. Man Cybern. B 39, 1362–1381 (2009)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2017

Authors and Affiliations

  1. 1.School of Mechatronic EngineeringChina University of Mining and TechnologyXuzhouChina
  2. 2.Department of Mechanical EngineeringUniversity of MichiganAnn ArborUSA
  3. 3.Jiangsu Key Laboratory of Mine Mechanical and Electrical EquipmentChina University of Mining and TechnologyXuzhouChina
  4. 4.Nonlinear Dynamics, Chaos and Complex Systems Group, Departamento de FísicaUniversidad Rey Juan CarlosMóstolesSpain
  5. 5.Department of Applied InformaticsKaunas University of TechnologyKaunasLithuania
  6. 6.Institute for Physical Science and TechnologyUniversity of MarylandCollege ParkUSA

Personalised recommendations