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Metamorphoses: Sudden jumps in basin boundaries

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Abstract

In some invertible maps of the plane that depend on a parameter, boundaries of basins of attraction are extremely sensitive to small changes in the parameter. A basin boundary can jump suddenly, and, as it does, change from being smooth to fractal. Such changes are calledbasin boundary metamorphoses. We prove (under certain non-degeneracy assumptions) that a metamorphosis occurs when the stable and unstable manifolds of a periodic saddle on the boundary undergo a homoclinic tangency.

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Authors and Affiliations

  1. Department of Mathematical Sciences, George Mason University, 22030, Fairfax, VA, USA

    Kathleen T. Alligood

  2. Dipartimento di Matematica, Universita di Roma “La Sapienza”, I-00185, Rome, Italy

    Laura Tedeschini-Lalli

  3. Department of Mathematics, University of Maryland, 20742, College Park, MD, USA

    James A. Yorke

  4. Institute for Physical Science and Technology, University of Maryland, 20742, College Park, MD, USA

    James A. Yorke

Authors
  1. Kathleen T. Alligood
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  2. Laura Tedeschini-Lalli
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  3. James A. Yorke
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Additional information

Communicated by J. N. Mather

This research was supported in part by grants and contracts from the Defense Advanced Research Projects Agency, the Consiglio Nazionale delle Ricerche (Comitato per le Matematiche), and the National Science Foundation

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Alligood, K.T., Tedeschini-Lalli, L. & Yorke, J.A. Metamorphoses: Sudden jumps in basin boundaries. Commun.Math. Phys. 141, 1–8 (1991). https://doi.org/10.1007/BF02100002

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  • DOI: https://doi.org/10.1007/BF02100002

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