Bifurcation points and the “n-squared” approximation and conjecture, illustrated by M.L Frame and K Mitchell

Benoit B. Mandelbrot
Additional contact information
Benoit B. Mandelbrot: Yale University, Mathematics Department

Chapter C6 in Fractals and Chaos, 2004, pp 96-99 from Springer

Abstract: Abstract Foreword to this chapter and the appended figure (2003). The n 2 conjecture advanced in this chapter’s Section 2 was first proven in Guckenheimer & McGehee 1984. The two authors and I were participating in a special year on iteration that Lennart Carleson and Peter W. Jones convened during 1983–1984 at the Mittag-Leffler Institute in Djursholm (Sweden). During a seminar that I was giving, two auditors suddenly stopped listening and started writing furiously. After my talk ended, they rushed up with proofs that turned out to be identical and led to a joint report. They explained the n 2 phenomenon in terms of the normal forms of resonant bifurcations with multiplier exp(2πi/n). More extensive results establish that these stability domains have a limiting shape following rescaling. They are corollaries of the theory of analytic normal forms for parabolic points. See, for example, Shishikura 2000.

Keywords: Bifurcation Point; Stability Domain; Extensive Result; Circular Boundary; Figure Plot (search for similar items in EconPapers)
Date: 2004
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-1-4757-4017-2_6

Ordering information: This item can be ordered from
http://www.springer.com/9781475740172

DOI: 10.1007/978-1-4757-4017-2_6

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().