Global dynamics of triangular maps

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Nonlinear Analysis: Theory, Methods & Applications, ISSN: 0362-546X, Vol: 104, Page: 75-83

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Balreira, Eduardo C; Elaydi, Saber; Luis, Rafael
Elsevier BV
Mathematics; Fibers; Mathematical techniques; Nonlinear analysis; Orbits; Competition model; Euclidean spaces; Global dynamics; Global stability; Globally asymptotically stable; Omega-limit set; Sharkovsky's Theorem; Triangular maps; Chaos theory; Physical Sciences and Mathematics
article description
We consider continuous triangular maps on IN, where I is a compact interval in the Euclidean space R. We show, under some conditions, that the orbit of every point in a triangular map converges to a fixed point if and only if there is no periodic orbit of prime period two. As a consequence we obtain a result on global stability, namely, if there are no periodic orbits of prime period 2 and the triangular map has a unique fixed point, then the fixed point is globally asymptotically stable. We also discuss examples and applications of our results to competition models.