# zbMATH — the first resource for mathematics

Basin boundary metamorphoses: Changes in accessible boundary orbits. (English) Zbl 0613.58018
The authors investigate what they call ”basin boundary metamorphoses”. As is well known, the boundary of the basin of attraction of a fixed or periodic point for a smooth map is often a fractal. Sometimes these basin boundaries undergo sudden changes as parameters are changed. For example, basin boundaries may change in size or shape at particular parameter values. This has been observed in many maps, including the often-studied Hénon map.
The authors propose one mechanism for these transitions. Often, there are infinitely many unstable periodic points in a basin boundary. However, only one is ”usually” accessible. The authors suggest that when this accessible point changes, the basin boundary undergoes a metamorphosis.
Reviewer: R.Devaney

##### MSC:
 37B99 Topological dynamics 37N99 Applications of dynamical systems 28A75 Length, area, volume, other geometric measure theory 26A18 Iteration of real functions in one variable
Full Text:
##### References:
 [1] Grebogi, C.; Ott, E.; Yorke, J.A., Physica, 7D, 181, (1983) [2] McDonald, S.W.; Grebogi, C.; Ott, E.; Yorke, J.A., Physica, 17D, 125, (1985) [3] Grebogi, C.; McDonald, S.W.; Ott, E.; Yorke, J.A., Phys. lett., 99A, 415, (1983) [4] Grebogi, C.; Ott, E.; Yorke, J.A.; McDonald, S.W.; Grebogi, C.; Ott, E.; Yorke, J.A.; Mira, C.; Gumowski, I.; Mira, C.; Arecchi, F.T.; Badii, R.; Politi, A.; Iansiti, M.; Hu, Q.; Westervelt, R.M.; Tinkham, M.; Holt, R.G.; Schwartz, I.B.; Schwartz, I.B.; Schwartz, I.B.; Gwinn, E.G.; Westervelt, R.M.; Yamaguchi, Y.; Mishima, N.; Decroly, O.; Goldbeter, A.; Takesue, S.; Kaneko, K.; Moon, F.C.; Li, G.-X., Dinamique chaotique, Phys. rev. lett., Phys. lett., C.R. acad. sc. Paris, Phys. rev., Phys. rev. lett., Phys. lett., Phys. lett., J. math. biol., Phys. rev. lett., Phys. lett., Phys. lett., Progr. theor. phys., Phys. rev. lett., 55, 1439, (1985), Cepadues Paris [5] Newhouse, S.E.; Gavrilov, N.K.; Shilnikov, L.P., Publ. math. IHES, Math. USSR sbornik, 17, 467, (1972) [6] K. Alligood, L. Tedeschini-Lalli and J.A. Yorke, private communication. [7] S. Hammel and C. Jones, Jumping stable manifolds for dissipative maps on the plane, preprint.. · Zbl 0686.58008 [8] Grebogi, C.; Ott, E.; Yorke, J.A., Phys. rev. lett., 56, 1011, (1986), A preliminary version of this work is contained in
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.