{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,7,29]],"date-time":"2024-07-29T13:51:17Z","timestamp":1722261077146},"reference-count":19,"publisher":"World Scientific Pub Co Pte Lt","issue":"05","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2008,5]]},"abstract":"<jats:p> There is a one-to-one correspondence between homogeneous quadratic dynamical systems and commutative (possibly nonassociative) algebras. The corresponding theory for continuous systems is well known (c.f. [Markus, 1960; Walcher, 1991; Kinyon &amp; Sagle, 1995]). In this paper the dynamics on the boundary of the basin of attraction of the origin, \u2202 B<jats:sub> Att <\/jats:sub>(0), in homogeneous quadratic discrete dynamical systems is considered. In particular, we consider the dynamical behavior in a family of systems corresponding to a family of algebras [Formula: see text] which admits nilpotents of rank 2 and idempotents. The complete periodicity of a system (and the corresponding algebra) is defined and it is proven that for every n &gt; 2 there are some systems\/algebras from [Formula: see text] which are on \u2202 B<jats:sub>Att<\/jats:sub>(0) completely periodic with period n. The dynamics on \u2202 B<jats:sub> Att <\/jats:sub>(0) is considered via a special class of polynomials P<jats:sub>n<\/jats:sub>, n \u2208 \u2115 \u222a {0, -1}, recursively defined by P<jats:sub>n<\/jats:sub>(\u03b1) = 2\u03b1P<jats:sub>n-2<\/jats:sub>(\u03b1) + P<jats:sub>n-1<\/jats:sub>(\u03b1); P<jats:sub>-1<\/jats:sub>(\u03b1) = 0, P<jats:sub>0<\/jats:sub>(\u03b1) = 1, n \u2208 \u2115. <\/jats:p>","DOI":"10.1142\/s0218127408021087","type":"journal-article","created":{"date-parts":[[2008,7,17]],"date-time":"2008-07-17T08:59:21Z","timestamp":1216285161000},"page":"1425-1433","source":"Crossref","is-referenced-by-count":6,"title":["A FAMILY OF COMPLETELY PERIODIC QUADRATIC DISCRETE DYNAMICAL SYSTEM"],"prefix":"10.1142","volume":"18","author":[{"given":"MILAN","family":"KUTNJAK","sequence":"first","affiliation":[{"name":"Faculty of Electrical Engineering and Computer Science, University of Maribor, Smetanova 17, 2000 Maribor, Slovenia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"MATEJ","family":"MENCINGER","sequence":"additional","affiliation":[{"name":"Faculty of Civil Engineering, IMFM, Jadranska 19, 1000 Ljubljana, Slovenia"},{"name":"University of Maribor, Smetanova 17, 2000 Maribor, Slovenia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2011,11,20]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.1080\/10236190500267970"},{"key":"rf2","volume-title":"An Introduction to Chaotic Dynamical Systems","author":"Devaney R. 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