{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,4,8]],"date-time":"2025-04-08T16:07:45Z","timestamp":1744128465069},"reference-count":4,"publisher":"World Scientific Pub Co Pte Lt","issue":"09","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2009,9]]},"abstract":"<jats:p> The \"Q-curves [Formula: see text] have long been observed and studied as the shadowy curves which appear illusively \u2014 not explicitly drawn \u2014 in the familiar orbit diagram of Myrberg's map f<jats:sub>c<\/jats:sub>(x) = x<jats:sup>2<\/jats:sup> + c. We illustrate that Q-curves also appear implicitly, for a different reason, in a computer-drawn bifurcation diagram of x<jats:sup>2<\/jats:sup> + c as well \u2014 by \"bifurcation diagram\" we mean the collection of all periodic points of f<jats:sub>c<\/jats:sub> (attracting, indifferent and repelling) \u2014 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-curve s 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. <\/jats:p><jats:p> 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. <\/jats:p><jats:p> Our algorithm for finding and plotting all periodic points (up to any reasonable period) in the bifurcation diagram is reviewed in an Appendix. <\/jats:p>","DOI":"10.1142\/s0218127409024621","type":"journal-article","created":{"date-parts":[[2009,12,15]],"date-time":"2009-12-15T06:45:55Z","timestamp":1260859555000},"page":"3017-3031","source":"Crossref","is-referenced-by-count":5,"title":["THE SHADOW-CURVES OF THE ORBIT DIAGRAM PERMEATE THE BIFURCATION DIAGRAM, TOO"],"prefix":"10.1142","volume":"19","author":[{"given":"CHIP","family":"ROSS","sequence":"first","affiliation":[{"name":"Department of Mathematics, Bates College, Lewiston, ME 04240, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"MEREDITH","family":"ODELL","sequence":"additional","affiliation":[{"name":"12 Meter Charters, 49 Bowen's Wharf, Newport, RI 02840, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"SARAH","family":"CREMER","sequence":"additional","affiliation":[{"name":"WestEd, 400 Seaport Court, Suite 222, Redwood City, CA 94063, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2012,5,2]]},"reference":[{"key":"p_9","doi-asserted-by":"publisher","DOI":"10.1038\/261459a0"},{"key":"p_11","first-page":"1","volume":"256","author":"Myrberg P. J.","year":"1958","journal-title":"I, II, III; Ann. Academi\u00e6 Scientiarum Fennic\u00e6, Series A"},{"key":"p_12","doi-asserted-by":"publisher","DOI":"10.2307\/2974876"},{"key":"p_14","doi-asserted-by":"publisher","DOI":"10.2307\/2687094"}],"container-title":["International Journal of Bifurcation and Chaos"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218127409024621","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,6]],"date-time":"2019-08-06T17:10:20Z","timestamp":1565111420000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218127409024621"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,9]]},"references-count":4,"journal-issue":{"issue":"09","published-online":{"date-parts":[[2012,5,2]]},"published-print":{"date-parts":[[2009,9]]}},"alternative-id":["10.1142\/S0218127409024621"],"URL":"https:\/\/doi.org\/10.1142\/s0218127409024621","relation":{},"ISSN":["0218-1274","1793-6551"],"issn-type":[{"value":"0218-1274","type":"print"},{"value":"1793-6551","type":"electronic"}],"subject":[],"published":{"date-parts":[[2009,9]]}}}