Will the Real Bifurcation Diagram Please Stand up!

Ross, Chip; Sorensen, Jody

doi:10.2307/2687094

Cited by 2 publications

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“…2. This picture, drawn using our algorithm 1 in [ Ross & Sorensen, 2000] shows all periodic points (up through period 12 here), not just the attracting ones seen in the orbit diagram. The periodic points themselves form curves having very elongated "⊃"-shapes which we shall call "P -curves".…”

Section: Q-curves and P -Curvesmentioning

confidence: 99%

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The Shadow-Curves of the Orbit Diagram Permeate the Bifurcation Diagram, Too

Ross

Odell²,

Cremer

2009

Int. J. Bifurcation Chaos

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The "Q-curves" Q 1 (c) = c, Q 2 (c) = c 2 + c, . . . , Q n (c) = (Q n−1 (c)) 2 + c = f n c (0) have long been observed and studied as the shadowy curves which appear illusively -not explicitly drawnin the familiar orbit diagram of Myrberg's map f c (x) = x 2 + c. We illustrate that Q-curves also appear implicitly, for a different reason, in a computer-drawn bifurcation diagram of x 2 + c as well -by "bifurcation diagram" we mean the collection of all periodic points of f c (attracting, indifferent and repelling) -these collections form what we call "P -curves". We show Q-curves and P -curves intersect in one of two ways: At a superattracting periodic point on a P -curve, the infinite family of Q-curves which intersect there are all tangent to the P -curve. At a Misiurewicz point, no tangencies occur at these intersections; the slope of the P -curve is the fixed point of a linear system whose iterates give the slopes of the Q-curves.We also introduce some new phenomena associated with c sin x illustrating briefly how its two different families of Q-curves interact with P -curves.Our algorithm for finding and plotting all periodic points (up to any reasonable period) in the bifurcation diagram is reviewed in an Appendix.

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Section: Q-curves and P -Curvesmentioning

confidence: 99%

“…Readers may find useful a bifurcation diagram which distinguishes attracting and repelling periodic points, and a brief discussion of our algorithm (detailed in [Ross & Sorensen, 2000]) used to find these points. The bifurcation diagram in Fig.…”

Section: Appendixmentioning

confidence: 99%

The Shadow-Curves of the Orbit Diagram Permeate the Bifurcation Diagram, Too

Ross

Odell²,

Cremer

2009

Int. J. Bifurcation Chaos

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Sprinkler Bifurcations and Stability

Sorensen

Rykken

2010

The College Mathematics Journal

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Will the Real Bifurcation Diagram Please Stand up!

Cited by 2 publications

References 0 publications

The Shadow-Curves of the Orbit Diagram Permeate the Bifurcation Diagram, Too

The Shadow-Curves of the Orbit Diagram Permeate the Bifurcation Diagram, Too

Sprinkler Bifurcations and Stability

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