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IEEE Transactions on Microwave Theory and Techniques
Volume 48 Number 11, November 2000

Table of Contents for this issue

Complete paper in PDF format

Chaotic Dynamics in Coupled Microwave Oscillators

Rajeev J. Ram, Ralph Sporer, Hans-Richard Blank and Robert A. York Senior Member, IEEE

Page 1909.

Abstract:

This paper describes an investigation into possible chaotic behavior in a coupled-oscillator system and the possible control of this behavior for communications. The established mathematical models for these oscillator arrays are demonstrated to exhibit chaos when the coupling strength between oscillators is below the range for phase locking. The complexity and predictability of the array dynamics are analyzed by means of standard dynamical measures such as the Lyapunov exponents, the Kolmogorov-Sinai entropy, and the attractor dimension. We show that chaos in these oscillator arrays is low dimensional and well characterized; both necessary conditions for control and possible exploitation of chaos. Finally, the method of occasional proportional feedback is used to stabilize the output from the array while the array is still in the chaotic regime. Possible applications of these chaotic transmitters are also discussed.

References

  1. J. J. Lynch, H. C. Chang and R. A. York, "Coupled-oscillator arrays and scanning techniques,"in Active and Quasi-Optical Arrays for Solid-State Power Combining, R. York, and Z. Popović, Eds. New York: Wiley, 1997, ch. 4 .
  2. H.-C. Chang, X. Cao, U. Mishra and R. A. York, "Phase noise in coupled oscillators: Theory and experiment", IEEE Trans. Microwave Theory Tech., vol. 45, pp.  604-615, May  1997.
  3. H.-C. Chang, X. Cao, M. J. Vaughan, U. K. Mishra and R. A. York, "Phase noise in externally injection-locked oscillator arrays", IEEE Trans. Microwave Theory Tech., vol. MTT-45, pp.  2035-2042, Nov.  1997.
  4. S. Hayes, C. Grebogi and E. Ott, "Communicating with chaos", Phys. Rev. Lett., vol. 70, pp.  3031-3034, 1993.
  5. T. Shinbrot, C. Grebogi, E. Ott and J. A. Yorke, "Using small perturbations to control chaos", Nature, vol. 363, pp.  411-417, June  1993.
  6. S. Hayes, C. Grebogi, E. Ott and A. Mark, "Experimental control of chaos for communication", Phys. Rev. Lett., vol. 73, pp.  1781-1784, Sept.  26, 1994.
  7. R. A. York, "Nonlinear analysis of phase relationships in quasi-optical oscillator arrays", IEEE Trans. Microwave Theory Tech. (Special Issue), vol. 41, pp.  1799-1809, Oct.  1993.
  8. K. Kurokawa, "Injection-locking of solid-state microwave oscillators", Proc. IEEE, vol. 61, pp.  1386-1409, Oct.  1973.
  9. V. Van der Pol and J. Van der Mark, "Frequency demultiplication", Nature, vol. 120, p.  363, 1927.
  10. T. Yamada, K. Fukushima and T. Yazaki, "Chaos in an electronic circuit-Experiment on coupled oscillator system", Phase Trans., vol. 29, pp.  15-20, 1990 .
  11. C. Bracikowski and R. Roy, "Chaos", vol. 1, p.  1749, 1991.
  12. H. G. Winful and L. Rahman, "Synchronized and spatiotemporal chaos in arrays of coupled lasers", Phys. Rev. Lett., vol. 65, pp.  1575-1578, 1990.
  13. Z. Gills, C. Iwata, R. Roy, I. Schwarz and I. Triandaf, "Tracking unstable steady states: Extending the stability regime of a multimode laser system", Phys. Rev. Lett., vol. 69, pp.  3169-3172, 1992.
  14. R. Roy, T. W. Murphy, T. D. Maier and Z. Gills, "Dynamical control of a chaotic laser: Experimental stabilization of a globally coupled system", Phys. Rev. Lett., vol. 68, pp.  1259-1262,  Mar.  1992.
  15. E. R. Hunt, "Stabilizing high-period orbits in a chaotic system: The diode resonator", Phys. Rev. Lett., vol. 67, p.  1953, 1991.
  16. K. D. Stephan, "Inter-injection locked oscillators for power combining and phased arrays", IEEE Trans. Microwave Theory Tech., vol. MTT-34, pp.  1017-1025, Oct.  1986.
  17. R. A. York and R. C. Compton, "Quasi-optical power-combining using mutually synchronized oscillator arrays", IEEE Trans. Microwave Theory Tech., vol. 39, pp.  1000-1009, June  1991.
  18. V. Van der Pol, "The nonlinear theory of electric oscillators", Proc. IRE, vol. 22, pp.  1051-1085, Sept.  1934.
  19. N. Koppel, "Toward a theory of modeling central pattern generators", Neural Control of Rhythmic Movements in Vertebrates, 1988.
  20. D. Ruelle and F. Takens, "On the nature of turbulence", Commun. Math. Phys., vol. 20, pp.  167-192, 1971.
  21. P. Grassberger, "Do climatic attractors exist?", Nature, vol. 323, pp.  609-612, 1986.
  22. S. P. Layne, G. Mayer-Kress and J. Holzfuss, "Problems associated with dimensional analysis of EEG data,"in Dimensions and Entropies in Chaotic Systems, G. Mayer-Kress, Ed. Berlin: Germany: Springer-Verlag, 1986.
  23. E. Ott, T. Sauer, and J. A. Yorke, Eds., Coping with Chaos: Analysis of Chaotic Data and the Exploitation of Chaotic Systems, New York: Wiley, 1995.
  24. P. Grassberger, T. Schreiber and C. Schaffrath, "Non-linear time sequence analysis", Int. J. Bifurcation and Chaos, Apr.   1991.
  25. H.-R. Blank, Ph.D. dissertation, Friedrich Alexander Univ., Erlangen, Germany, 1994.
  26. F. Takens, "Lecture notes in math", vol. 898, 1980.
  27. C. W. Simm, M. L. Sawley, F. Skiff and A. Pochelon, "On the analysis of experimental signals for evidence of deterministic chaos", Helv. Phys. Acta, vol. 60, pp.  510-551,  1987.
  28. P. Grassberger and I. Procaccia, "Characterization of strange attractors", Phys. Rev. Lett., vol. 50, p.  346, 1983.
  29. P. Grassberger and I. Procaccia, "Estimation of the Kolmogorov entropy from a chaotic signal", Phys. Rev. Lett., vol. 50, p.  346, 1983.
  30. M. Frank, H.-R. Blank, J. Heindl, M. Kaltenhauser, H. Kochner, W. Kreische, S. Poscher, R. Sporer and T. Wagner, "Improvement of K2 -entropy calculations by means of dimension scaled distances", Phys. Rev. D. Part. Fields, vol. 65, p.  359, 1993.
  31. R. Stoop and P. F. Meier, "Evaluation of Lyapunov exponents and scaling functions from time series", J. Opt. Soc. Amer. B, Opt. Phys., vol. 5, no. 5, p.  1037, 1988 .
  32. R. Stoop and J. Parisi, "Calculation if Lyapunov exponents avoiding spurious elements", Phys. Rev. D, Part. Fields, vol. 50, pp.  89-94, 1991 .
  33. J. W. Havstad and C. J. Ehlers, "Attractor dimensions of nonstationary dynamical systems from small datasets", Phys. Rev. A, Gen. Phys., vol. 39, p.  845, 1989.
  34. T. Buzug, T. Reimers and G. Pfister, "Optimal reconstruction of strange attractors from purely geometrical arguments", Europhys. Lett., vol. 13, p.  605, 1990.
  35. J. Theiler, "Some comments on the correlation dimension of 1/f noise", Phys. Lett. A, vol. 155, p.  480, 1991.
  36. T. Schreiber, "Extremely simple nonlinear noise reduction method", Phys. Rev. E, Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top., vol. 47, p.  2401,  1993.
  37. J. Theiler, "Spurious dimension from correlation algorithms applied to limited time series data", Phys. Rev. A, vol. 34, p.  2427, 1986.
  38. J.-P. Eckmann and D. Ruelle, "Fundamental limitations in estimating dimensions and Lyaponov exponents in dynamical systems", Phys. Rev. D, Part. Fields, vol. 56, p.  185, 1992.
  39. R. J. Ram, R. Sporer, H.-R. Blank, P. Maccarini, H.-C. Chang and R. A. York, "Chaos in microwave antenna arrays", in IEEE Int. Microwave Symp. Dig., 1996, pp.  1875-1878.