Jumping stable manifolds for dissipative maps of the plane.

*(English)*Zbl 0686.58008An essential assumption of the authors is that basin boundaries arise ‘typically’ from stable manifolds of saddle points. Other separatrices are ignored. The ‘discontinuity’ argument is inspired by low accuracy numerical computations. A numerical study of a particular \({\mathbb{C}}\to {\mathbb{C}}\) map, due to Ikeda, is used as confirming evidence. For \({\mathbb{C}}\to {\mathbb{C}}\) maps it was already known to Julia (70 years ago) that basin boundaries can have an extremely complicated structure, not necessarily related to saddle-point manifolds. Other non-‘typical’ cases can be found in books, for example [in the reviewer and C. Mira: Recurrences and discrete dynamical systems (Lect. Notes Math. 809, 1980; Zbl 0449.58003)].

Reviewer: I.Gumowski

##### MSC:

58D15 | Manifolds of mappings |

65Q05 | Numerical methods for functional equations (MSC2000) |

PDF
BibTeX
XML
Cite

\textit{S. M. Hammel} and \textit{C. K. R. T. Jones}, Physica D 35, No. 1--2, 87--106 (1989; Zbl 0686.58008)

Full Text:
DOI

##### References:

[1] | K. Alligood and J.A. Yorke, preprint. |

[2] | R. Easton, Trellises with dense homoclinic points, preprint. |

[3] | Grebogi, C.; Ott, E.; Yorke, J.A., Metamorphoses of basin boundaries in nonlinear dynamical systems, Phys. rev. lett., 56, 1011, (1986) |

[4] | C. Grebogi, E. Ott and J.A. Yorke, metamorphoses: changes in accessible boundary orbits, preprints. · Zbl 0613.58018 |

[5] | Guckenheimer, J.; Holmes, P., Nonlinear oscillations, dynamical systems and bifurcations of vector fields, (1983), Springer New York · Zbl 0515.34001 |

[6] | Hammel, S.; Jones, C.; Moloney, J., Global dynamical behavior of the optical field in a ring cavity, J. opt. soc. am. B, 2, 552-564, (1985) |

[7] | Ikeda, K., Opt. commun., 30, 257, (1979) |

[8] | S. Patterson and B. Robinson, The basin of attraction of the sink when creating a horseshoe, preprint. |

[9] | Smale, S., Differentiable dynamical systems, Bull. AMS, 73, 747-817, (1967) · Zbl 0202.55202 |

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.