Skip to main content
SpringerLink
Log in

Nonlocal time-delayed feedback control of spatiotemporal patterns: controlling a network of digital phase-locked loops

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

Control of spatially extended systems is an important and challenging topic of research. Previous control techniques rely on local or global feedback control. In this paper, for the first time, we propose the nonlocal time-delayed feedback control technique that can control spatiotemporal patterns of an important class of spatially extended systems, namely the coupled map lattice systems. We demonstrate the proposed technique in an engineering system, namely digital phase-locked loop, which is widely used in communication and signal processing systems, and at the same time its network mimics a physically realizable coupled map lattice system. We carry out stability analysis to derive the condition of controllability. Unlike other control techniques, apart from feedback strength, the proposed control scheme offers an additional control parameter, namely the feedback range that makes this technique more versatile. We show that depending upon these control parameters one can suppress spatiotemporal chaos and other complex patterns in order to establish a synchronized fixed point solution, which is the desired solution in most practical systems. We also derive the optimum value of the control parameter for which the control gives fastest convergence. We believe that the proposed technique will be useful in controlling spatially extended systems in other fields also.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  1. Boccaletti, S., Grebogi, C., Lai, Y.C., Mancini, H., Maza, D.: The control of chaos: theory and applications. Phys. Rep. 329, 103–197 (2000)

    Article  MathSciNet  Google Scholar 

  2. Ott, E., Grebogi, C., Yorke, J.A.: Controlling chaos. Phys. Rev. Lett. 64(11), 1196–1199 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  3. Pyragas, K.: Continuous control of chaos by self-controlling feedback. Phys. Lett. A 170, 421–428 (1992)

    Article  Google Scholar 

  4. Kaneko, K.: Pattern dynamics in spatiotemporal chaos: pattern selection, diffusion of defect and pattern competition intermettency. Phys. D Nonlinear Phenom. 34(1), 1–41 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  5. Kaneko, K.: Coupled Map Lattices: Theory and Experiments. World Scientific, Singapore (1993)

    MATH  Google Scholar 

  6. Gad-El-Hak, M.: Flow Control: Passive, Active, and Reactive Flow Management. Cambridge University Press, Cambridge (2000)

    Book  MATH  Google Scholar 

  7. Liu, P.: Control and synchronization of mandelbrot sets in coupled map lattice. Nonlinear Dyn. 73(1), 299–310 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Wang, D., Liu, S., Zhao, Y., Jiang, C.: Control of the spatial mandelbrot set generated in coupled map lattice. Nonlinear Dyn. 84(3), 1795–1803 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  9. Kuramoto, Y.: Chemical Oscillations, Waves, and Turbulence. Springer, Berlin (1984)

    Book  MATH  Google Scholar 

  10. Bramwell, S.T., Holdsworth, P.C.W., Pinton, J.F.: Universality of rare fluctuations in turbulence and critical phenomena. Nature 396, 552 (1998)

    Article  Google Scholar 

  11. Roy, R., Murphy, T.W., Maier, T.D., Gills, Z., Hunt, E.R.: Dynamical control of a chaotic laser: experimental stabilization of a globally coupled system. Phys. Rev. Lett. 68, 1259 (1992)

    Article  Google Scholar 

  12. Pastur, L., Gostiaux, L., Bortolozzo, U., Boccaletti, S., Ramazza, P.L.: Experimental targeting and control of spatiotemporal chaos in nonlinear optics. Phys. Rev. Lett. 93(6), 063,902 (2004)

    Article  Google Scholar 

  13. Löber, J., Engel, H.: Controlling the position of traveling waves in reaction–diffusion systems. Phys. Rev. Lett. 112, 148,305 (2014)

    Article  Google Scholar 

  14. Weiss, J.N., Chen, P.S., Qu, Z., Karagueuzian, H.S., Garfinkel, A.: Ventricular fibrillation. How do we stop the waves from breaking? Circ. Res. 87, 1103–1107 (2000)

    Article  Google Scholar 

  15. Banerjee, T., Paul, B., Sarkar, B.C.: Spatiotemporal dynamics of a digital phase-locked loop based coupled map lattice system. Chaos 24(1), 013,116 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. Paul, B., Banerjee, T., Sarkar, B.C.: Spatiotemporal dynamics of a network of coupled time-delay digital tanlock loops. Int. J. Bifurc. Chaos 26(05), 1650,076 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  17. Rakshit, S., Ray, A., Bera, B.K., Ghosh, D.: Synchronization and firing patterns of coupled rulkov neuronal map. Nonlinear Dyn. 94(2), 785–805 (2018)

    Article  Google Scholar 

  18. Wang, Xy, Qin, X.: A new pseudo-random number generator based on cml and chaotic iteration. Nonlinear Dyn. 70(2), 1589–1592 (2012)

    Article  MathSciNet  Google Scholar 

  19. Wang, Xy, Bao, Xm: A novel block cryptosystem based on the coupled chaotic map lattice. Nonlinear Dyn. 72(4), 707–715 (2013)

    Article  MathSciNet  Google Scholar 

  20. Lv, X., Liao, X., Yang, B.: A novel pseudo-random number generator from coupled map lattice with time-varying delay. Nonlinear Dyn. 94(1), 325–341 (2018)

    Article  Google Scholar 

  21. Nettesheim, S., von Oertzen, A., Rotermund, H.H., Ertl, G.: Reaction diffusion patterns in the catalytic cooxidation on pt(110): front propagation and spiral waves. J. Chem. Phys. 98(12), 9977–9985 (1993)

    Article  Google Scholar 

  22. Lu, W., Yu, D., Harrison, R.G.: Instabilities and tracking of travelling wave patterns in a three-level laser. Int. J. Bifur. Chaos 08(09), 1769–1775 (1998)

    Article  Google Scholar 

  23. Zhang, G., Jiang, D., Cheng, S.S.: 3-Periodic traveling wave solutions for a dynamical coupled map lattice. Nonlinear Dyn. 50(1), 235–247 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  24. Zengrong, L., Wei, Z., Xin, H.: Is there chaotic synchronization in space extend systems? Nonlinear Dyn. 12(4), 319–326 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  25. Yang, K.L., Wang, C.J.: Coexistence of attractors and effects of noise on coupled piecewise maps. Nonlinear Dyn. 79(1), 377–385 (2015)

    Article  MathSciNet  Google Scholar 

  26. Vasegh, N.: Spatiotemporal and synchronous chaos in accumulated coupled map lattice. Nonlinear Dyn. 89, 1089–1097 (2017)

    Article  MATH  Google Scholar 

  27. Gluckman, B.J., Nguyen, H., Weinstein, S.L., Schiff, S.J.: Adaptive electric field control of epileptic seizures. J. Neurosci. 21, 590–600 (2001)

    Article  Google Scholar 

  28. Paramananda, P., Hildebrand, M., Eiswirth, E.: Controlling turbulence in coupled map lattice systems using feedback techniques. Phys. Rev. E 56, 239 (1997)

    Article  Google Scholar 

  29. Gade, P.M.: Feedback control in coupled map lattices. Phys. Rev. E. 57(6), 14897–14900 (1998)

    Article  Google Scholar 

  30. Sepulchre, J.A., Babloyantz, A.: Controlling chaos in a network of oscillators. Phys. Rev. E 48, 945–950 (1993)

    Article  Google Scholar 

  31. Auerbach, D.: Controlling extended systems of chaotic elements. Phys. Rev. Lett. 72, 1184–1187 (1994)

    Article  Google Scholar 

  32. Gang, H., Zhilin, Q.: Controlling spatiotemporal chaos in coupled map lattice systems. Phys. Rev. Lett. 72(1), 68–71 (1994)

    Article  Google Scholar 

  33. Astakhov, V., Anishchenko, V., Shabunin, A.: Controlling spatiotemporal chaos in a chain of the coupled logistic maps. IEEE Trans. Circuits Syst. Part I Fundam. Theory Appl. 42(6), 352–357 (1995)

    Article  Google Scholar 

  34. Lu, W., Yu, D., Harrison, R.G.: Control of patterns in spatiotemporal chaos in optics. Phys. Rev. Lett. 76(18), 3316–3319 (1996)

    Article  Google Scholar 

  35. Lu, W., Yu, D., Harrison, R.G.: Tracking periodic patterns into spatiotemporal chaotic regimes. Phys. Rev. Lett. 78, 4375–4378 (1997)

    Article  Google Scholar 

  36. Gauthier, D.J., Sukow, D.W., Concannon, H.M., Socolar, J.E.S.: Stabilizing unstable periodic orbits in a fast diode resonator using continuous time-delay autosynchronization. Phys. Rev. E 50, 2343–2346 (1994)

    Article  Google Scholar 

  37. Bielawski, S., Derozier, D., Glorieux, P.: Controlling unstable periodic orbits by a delayed continuous feedback. Phys. Rev. E 49, R971–R974 (1994)

    Article  Google Scholar 

  38. Fukuyama, T., Shirahama, H., Kawai, Y.: Dynamical control of the chaotic state of the current-driven ion acoustic instability in a laboratory plasma using delayed feedback. Phys. Plasmas 9(11), 4525–4529 (2002)

    Article  Google Scholar 

  39. Parmananda, P., Madrigal, R., Rivera, M., Nyikos, L., Kiss, I.Z., Gáspár, V.: Stabilization of unstable steady states and periodic orbits in an electrochemical system using delayed-feedback control. Phys. Rev. E 59, 5266–5271 (1999)

    Article  Google Scholar 

  40. Bleich, M.E., Socolar, J.E.S.: Controlling spatiotemporal dynamics with time-delay feedback. Phys. Rev. E 54(1), 17–20 (1996)

    Article  Google Scholar 

  41. Parekh, N., Parthasarathy, S., Sinha, S.: Global and local control of spatiotemporal chaos in coupled map lattices. Phys. Rev. Lett. 81, 1401 (1998)

    Article  Google Scholar 

  42. Battogtokh, D., Mikhailov, A.: Controlling turbulence in the complex ginzburglandau equation. Phys. D 90, 84–95 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  43. Junge, L., Parlitz, U.: Synchronization and control of coupled ginzburg-landau equations using local coupling. Phys. Rev. E 61, 3736–3742 (2000)

    Article  Google Scholar 

  44. Parekh, N., Sinha, S.: Controllability of spatiotemporal systems using constant pinnings. Phys. A 318, 200–212 (2005)

    Article  MATH  Google Scholar 

  45. Grigoriev, R.O., Cross, M.C., Schuster, H.G.: Pinning control of spatiotemporal chaos. Phys. Rev. Lett. 79, 2795–2798 (1997)

    Article  Google Scholar 

  46. Sinha, S., Gupte, N.: Targeting spatiotemporal patterns in extended systems with multiple coexisting attractors. Phys. Rev. E 64, 015,203 (2001)

    Article  Google Scholar 

  47. Sinha, S., Gupte, N.: Adaptive control of spatially extended systems: targeting spatiotemporal patterns and chaos. Phys. Rev. E 58, R5221–R5224 (1998)

    Article  Google Scholar 

  48. Huang, W.: Stabilizing coupled map lattice systems with adaptive adjustment. Phys. Rev. E. 66, 036,222 (2002)

    Article  Google Scholar 

  49. De, S., Sinha, S.: Effect of switching links in networks of piecewise linear maps. Nonlinear Dyn. 81, 1741–1749 (2015)

    Article  MathSciNet  Google Scholar 

  50. Grigoriev, R.O., Cross, M.C.: Controlling physical systems with symmetries. Phys. Rev. E. 57, 1550–1554 (1998)

    Article  Google Scholar 

  51. Jensen, S.J., Schwab, M., Denz, C.: Manipulation, stabilization, and control of pattern formation using fourier space filtering. Phys. Rev. Lett. 81, 1614–1617 (1998)

    Article  Google Scholar 

  52. Zhang, X., Shen, K.: Controlling spatiotemporal chaos via phase space compression. Phys. Rev. E 63, 046,212 (2001)

    Article  Google Scholar 

  53. Gardner, F.M.: Phaselock Techniques, 2nd edn. Wiley, New york (1979)

    Google Scholar 

  54. Best, R.E.: Phase-Locked Loops. McGraw-Hill, New York (2003)

    Google Scholar 

  55. Leonov, G.A., Kuznetsov, N.V.: Nonlinear Mathematical Models of Phase-Locked Loops, Stability and Oscillations. Cambridge Scientific Publisher, Cambridge (2014)

    Google Scholar 

  56. Lindsey, W.C., Chie, C.M.: A survey of digital phase lock loops. Proc. IEEE 69, 410–431 (1981)

    Article  Google Scholar 

  57. Banerjee, T., Sarkar, B.C.: Chaos, intermittency and control of bifurcation in a ZC2-DPLL. Int. J. Electron. 96, 717–732 (2009)

    Article  Google Scholar 

  58. Banerjee, T., Sarkar, B.C.: Conventional and extended time-delayed feedback controlled zero-crossing digital phase locked loop. Int. J. Bifurc. Chaos 22(12), 1230,044 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  59. Leonov, G.A., Kuznetsov, N.V., Yuldashev, M.V., Yuldashev, R.V.: Hold-in, pull-in, and lock-in ranges of PLL circuits: rigorous mathematical definitions and limitations of classical theory. IEEE Trans. Circuit Syst. I(62), 2454–2464 (2015)

    Article  MathSciNet  Google Scholar 

  60. Leonov, G.A., Seledzhi, S.M.: Stability and bifurcations of phase locked loops for digital signal processors. Int. J. Bifurc. Chaos 15, 1347–1360 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  61. Martinez, R.D., Compton, R.C.: Electronic beam steering of active arrays with phase-locked loops. IEEE Microw. Guid. Lett. 4, 166–168 (1994)

    Article  Google Scholar 

  62. Maccarini, P.F., Buckwalter, J.: Coupled phase-locked loop arrays for beam steering. IEEE MTT-S Int. Microw. Symp. Digest (2003). https://doi.org/10.1109/MWSYM.2003.1210464

    Article  Google Scholar 

  63. York, R.A., Itoh, T.: Injection and phase-locking techniques for beam control. IEEE Trans. Microw. Theory Tech. 46, 1920–1929 (1998)

    Article  Google Scholar 

  64. Banerjee, T., Sarkar, B.C.: Chaos and bifurcation in a third-order digital phase locked loop. AEU Int. J. Electron. Commun. 62, 86–91 (2008)

    Article  Google Scholar 

  65. Paul, B., Banerjee, T.: Chimeras in digital phase-locked loops. Chaos 29, 013102 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  66. Viana, R.L., Batista, A.M., Batista, C.A.S., Iarosz, K.C.: Lyapunov spectrum of chaotic maps with a long-range coupling mediated by a diffusing substance. Nonlinear Dyn. 87(3), 1589–1601 (2017)

    Article  MATH  Google Scholar 

  67. Kuznetsov, N.V., Alexeeva, T.A., Leonov, G.A.: Invariance of Lyapunov exponents and Lyapunov dimension for regular and irregular linearizations. Nonlinear Dyn. 85, 195–201 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  68. Kuznetsov, N.V., Leonov, G.A., Mokaev, T.N., Prasad, A., Shrimali, M.D.: Finite-time Lyapunov dimension and hidden attractor of the Rabinovich system. Nonlinear Dyn. 92, 267–285 (2018)

    Article  MATH  Google Scholar 

  69. Ørstavik, S., Carretero-González, R., Stark, J.: Estimation of intensive quantities in spatio-temporal systems from time-series. Phys. D 147, 204–220 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  70. Rosin, D.P., Rontani, D., Gauthier, D.J.: Synchronization of coupled Boolean phase oscillators. Phys. Rev. E 89, 042907 (2014)

    Article  Google Scholar 

  71. Rosin, D.P.: Dynamics of complex autonomous Boolean networks. TU Berlin and Duke University, Ph.D. Thesis, unpublished (2014)

  72. Rosin, D.P.: Dynamics of Complex Autonomous Boolean Networks. Springer, Cham (2015)

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Chaos and Complex Systems Research Laboratory, Department of Physics, University of Burdwan, Burdwan, West Bengal, 713 104, India

    Bishwajit Paul & Tanmoy Banerjee

Authors
  1. Bishwajit Paul
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tanmoy Banerjee
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tanmoy Banerjee.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Paul, B., Banerjee, T. Nonlocal time-delayed feedback control of spatiotemporal patterns: controlling a network of digital phase-locked loops. Nonlinear Dyn 96, 811–823 (2019). https://doi.org/10.1007/s11071-019-04823-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-019-04823-y

Keywords

Search

Navigation