Analysis and comparison of multiple-delay schemes for controlling unstable fixed points of discrete maps

Joshua E. S. Socolar; Daniel J. Gauthier

doi:10.1103/PhysRevE.57.6589

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Analysis and comparison of multiple-delay schemes for controlling unstable fixed points of discrete maps

Joshua E. S. Socolar and Daniel J. Gauthier

Physics Department and Center for Nonlinear and Complex Systems, Duke University, Durham, North Carolina 27708

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Phys. Rev. E 57, 6589 – Published 1 June, 1998

DOI: https://doi.org/10.1103/PhysRevE.57.6589

Abstract

We investigate theoretically the stabilization of a fixed point of a discrete one-dimensional nonlinear map by applying small perturbations to an accessible system parameter or variable. The size of the perturbations is determined in real time using feedback schemes incorporating only the dynamical state of the system and its state at previous iterates without making a comparison to a reference state. In particular, we compare and contrast two algorithms: extended time-delay autosynchronization, which uses an infinite series of past iterates with weights that decay by a factor of $R$ with each time step, and $N$ -time-delay autosynchronization, which uses an average of $N$ past iterates with equal weights. The range of feedback parameters that successfully stabilize the fixed point and the robustness of the schemes to noise are determined. It is found that the domain of control for the two schemes is similar for appropriately matched values of $R$ and $N$ , and that $N$ -time-delay autosynchronization tends to be less sensitive to noise.

References (21)

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[c1] E. Ott and M. Spano, Phys. Today 48 (5), 34 (1995).

[c2] E. Ott, C. Grebogi, and J. A. Yorke, Phys. Rev. Lett. 64, 1196 (1990).

[c3] F. J. Romeiras, C. Grebogi, E. Ott, and W. P. Dayawansa, Physica D 58, 165 (1992).

[c4] K. Pyragas, Phys. Lett. A 170, 421 (1992).

[c5] S. Bielawski, D. Derozier, and P. Glorieux, Phys. Rev. E 49, R971 (1994).

[c6] D. J. Gauthier, D. W. Sukow, H. M. Concannon, and J. E. S. Socolar, Phys. Rev. E 50, 2343 (1994).

[c7] J. E. S. Socolar, D. W. Sukow, and D. J. Gauthier, Phys. Rev. E 50, 3245 (1994).

[c8] M. E. Bleich and J. E. S. Socolar, Phys. Lett. A 210, 87 (1996).

[c9] M. E. Bleich and J. E. S. Socolar, Phys. Rev. E 54, R17 (1996).

[c10] M. E. Bleich, D. Hochheiser, J. V. Moloney, and J. E. S. Socolar, Phys. Rev. E 55, 2119 (1997).

[c11] D. W. Sukow, M. E. Bleich, D. J. Gauthier, and J. E. S. Socolar, Chaos 7, 560 (1997).

[c12] K. Hall, D. J. Christini, M. Tremblay, J. J. Collins, L. Glass, and J. Billette, Phys. Rev. Lett. 78, 4518 (1997).

[c13] D. J. Gauthier and J. E. S. Socolar, Phys. Rev. Lett. 79, 4938 (1997).

[c14] K. Pyragas, Phys. Lett. A 206, 323 (1995).

[c15] G. W. Flake, G.-Z. Sun, Y.-C. Lee, and H.-H. Chen, in Advances in Neural Information Processing, edited by J. D. Cowan, G. Tesauro, and J. Alspector (Morgan Publishers, San Mateo, CA, 1994), p. 647.

[c16] D. J. Christini, V. In, M. L. Spano, W. L. Ditto, and J. J. Collins, Phys. Rev. E 56, R3749 (1997).

[c17] D. J. Christini and J. J. Collins, IEEE Trans. Circuits Syst. 44, 1027 (1997).

[c18] R. W. Rollins, P. Parmananda, and P. Sherard, Phys. Rev. E 47, R3003 (1993).

[c19] D. A. Egolf and J. E. S. Socolar, Phys. Rev. E 57, 5271 (1998).

[c20] See, for example, J. C. Doyle, B. A. Francis, and A. R. Tannenbaum, Feedback Control Theory (Macmillan, New York, 1992).

[c21] L. N. Trefethen, A. E. Trefethen, S. C. Reddy, and T. A. Driscoll, Science 261, 578 (1993).

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Analysis and comparison of multiple-delay schemes for controlling unstable fixed points of discrete maps

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References (21)

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