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This article is cited in 55 scientific papers (total in 55 papers)
The dynamics of strictly non-Volterra quadratic stochastic operators on the 2-simplex
U. U. Zhamilov, U. A. Rozikov Institute of Mathematics and Information Technologies of the National Academy of Sciences of Uzbekistan
Abstract:
An arbitrary strictly non-Volterra quadratic operator on the 2-simplex is shown to have a unique fixed point, which is established as being nonattracting. A description of the $\omega$-limit set of the trajectory of some subclasses of these operators is obtained. Strictly non-Volterra operators, as distinct from the Volterra operators, are shown to have cyclic trajectories. For two particular operators, we show that there exists a cyclic trajectory with period 3. Each trajectory which starts at the boundary of the simplex converges
to this cyclic trajectory, whereas trajectories which begin at an interior point of the simplex (not at the fixed point) must diverge. Furthermore, the $\omega$-limit set of such a trajectory is infinite, and lies at the boundary of the simplex. Also, we study subclasses of strictly non-Volterra operators whose trajectories
tend to a cyclic trajectory with period 2.
Bibliography: 18 titles.
Keywords:
quadratic stochastic operators; simplex; trajectory.
Received: 30.10.2007 and 28.04.2009
Citation:
U. U. Zhamilov, U. A. Rozikov, “The dynamics of strictly non-Volterra quadratic stochastic operators on the 2-simplex”, Mat. Sb., 200:9 (2009), 81–94; Sb. Math., 200:9 (2009), 1339–1351
Linking options:
https://www.mathnet.ru/eng/sm3962https://doi.org/10.1070/SM2009v200n09ABEH004039 https://www.mathnet.ru/eng/sm/v200/i9/p81
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Abstract page: | 594 | Russian version PDF: | 303 | English version PDF: | 12 | References: | 60 | First page: | 10 |
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