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Noninvertibility and the structure of basins of attraction in a model adaptive control system

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Summary

We present a two-dimensional, nonlinear map, arising from a simple adaptive control problem, which exhibits disconnected boundaries separating the basins of attraction of its coexisting attractors. We perform a detailed study of the relation between this phenomenon and the noninvertible nature of the map and demonstrate how the complex basin structure is caused by a change in the number of preimages of points along a stable manifold.

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References

  • Adomaitis, R. A., and I. G. Kevrekidis (1990). On the global bifurcation characteristics of adaptive systems.Proc. 11th IFAC World Congress, Tallinn, Estonia,4, 299–304.

    Google Scholar 

  • de la Llave, R., R. A. Adomaitis, and I. G. Kevrekidis (1991). The dynamics of the rotating logistic map. Manuscript in preparation.

  • Doedel, E. (1986). AUTO: Software for continuation and bifurcation problems in ordinary differential equations. AUTO 86 user manual.

  • Golden, M. P., and B. E. Ydstie (1988). Bifurcation in model reference adaptive control systems.Systems & Control Letters,11, 413–430.

    Google Scholar 

  • Goodwin, G. C., and K. S. Sin (1984).Adaptive Filtering, Prediction, and Control, Prentice Hall, Englewood Cliffs, NJ.

    Google Scholar 

  • Grebogi, C., E. Ott, and J. A. Yorke (1986). Metamorphoses of basin boundaries in nonlinear dynamical systems.Phys. Rev. Lett.,56, 1011–1014.

    Google Scholar 

  • Gumowski, I., and C. Mira (1980). Recurrences and discrete dynamic systems.Lect. Notes in Math.,809, Springer-Verlag Berlin.

    Google Scholar 

  • Kevrekidis, I. G., and M. S. Jolly (1987). On the use of interactive graphics in the numerical study of chemical dynamics. Paper no. 22c, 1987 AIChE Annual Meeting, New York.

    Google Scholar 

  • Mareels, I. M. Y., and R. R. Bitmead (1986). Nonlinear dynamics in adaptive control: chaotic and periodic stabilization.Automatica,22, 641–655.

    Google Scholar 

  • Mareels, I. M. Y., and R. R. Bitmead (1988). Nonlinear dynamics in adaptive control: chaotic and periodic stabilization—II. Analysis.Automatica,24, 485–497.

    Google Scholar 

  • Marotto, F. R. (1978). Snap-back repellers imply chaos in ℝn.J. Math. Anal. Appl.,63, 199–223.

    Google Scholar 

  • Yamaguchi, Y., and K. Tanikawa (1989). Change of the fractal dimension of basin boundaries in a two-dimensional cubic map.Phys. Lett. A,142, 95–98.

    Google Scholar 

  • Wiggins, S. (1988).Global Bifurcations and Chaos, Appl. Math. Sciences, vol.73, Springer-Verlag, New York.

    Google Scholar 

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Authors and Affiliations

  1. Department of Chemical Engineering, Princeton University, 08544, Princeton, NJ

    Raymond A. Adomaitis & Ioannis G. Kevrekidis

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  1. Raymond A. Adomaitis
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  2. Ioannis G. Kevrekidis
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Communicated by Stephen Wiggins

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Adomaitis, R.A., Kevrekidis, I.G. Noninvertibility and the structure of basins of attraction in a model adaptive control system. J Nonlinear Sci 1, 95–105 (1991). https://doi.org/10.1007/BF01209149

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  • DOI: https://doi.org/10.1007/BF01209149

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