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 ω-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 ω-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.
Citation:
U. U. Zhamilov, U. A. Rozikov, “The dynamics of strictly non-Volterra quadratic stochastic operators on the 2-simplex”, Sb. Math., 200:9 (2009), 1339–1351
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\paper The dynamics of strictly non-Volterra quadratic stochastic operators on the 2-simplex
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\yr 2009
\vol 200
\issue 9
\pages 1339--1351
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U. U. Jamilov, Kh. O. Khudoyberdiev, “On the dynamics of non-Volterra quadratic operators corresponding to permutations”, Journal of Difference Equations and Applications, 30:3 (2024), 336
Uygun Jamilov, “Regular dynamics in a quadratic model”, Math Methods in App Sciences, 2024
B. Dzh. Mamurov, “Vypuklaya kombinatsiya dvukh kvadratichnykh stokhasticheskikh operatorov, deistvuyuschikh na 2D-simplekse”, Izv. vuzov. Matem., 2023, no. 7, 66–70
F. F. Eshmatov, U. U. Jamilov, Kh. O. Khudoyberdiev, “Discrete time dynamics of a SIRD reinfection model”, Int. J. Biomath., 16:05 (2023)
B. J. Mamurov, “A Convex Combination of Two Quadratic Stochastic Operators Acting in the 2D Simplex”, Russ Math., 67:7 (2023), 55
Sh. B. Abdurakhimova, U. A. Rozikov, “Dynamical System of a Quadratic Stochastic Operator with Two Discontinuity Points”, Math. Notes, 111:5 (2022), 676–687
M. Saburov, Kh. Saburov, “Prilozheniya kvadratichnykh stokhasticheskikh operatorov k nelineinym problemam konsensusa”, Nauka — tekhnologiya — obrazovanie — matematika — meditsina, SMFN, 68, no. 1, Rossiiskii universitet druzhby narodov, M., 2022, 110–126
Jamilov U.U. Khamrayev A.Yu., “On Dynamics of Volterra and Non-Volterra Cubic Stochastic Operators”, Dynam. Syst., 37:1 (2022), 66–82
Uygun U. Jamilov, Elbek Kh. Ziyodullaev, Springer Proceedings in Mathematics & Statistics, 390, Infinite Dimensional Analysis, Quantum Probability and Applications, 2022, 341
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Khayotjon O. Khudoyberdiev, Springer Proceedings in Mathematics & Statistics, 390, Infinite Dimensional Analysis, Quantum Probability and Applications, 2022, 369
K. A. Aralova, U. U. Jamilov, “The dynamics of superposition of non-Volterra quadratic stochastic operators”, Journal of Difference Equations and Applications, 28:9 (2022), 1178
Jamilov U. Reinfelds A., “A Family of Volterra Cubic Stochastic Operators”, J. Convex Anal., 28:1 (2021), 19–30
Jamilov U. Ladra M., “Evolution Algebras and Dynamical Systems of a Worm Propagation Model”, Linear Multilinear Algebra, 2021
Ganikhodjaev N.N. Pah C.H. Rozikov U., “Dynamics of Quadratic Stochastic Operators Generated By China'S Five Element Philosophy”, J. Differ. Equ. Appl., 27:8 (2021), 1173–1192
Jamilov U.U., Kurganov K.A., “On a Non-Volterra Cubic Stochastic Operator”, Lobachevskii J. Math., 42:12 (2021), 2800–2807
U. U. Jamilov, “Certain Polynomial Stochastic Operators”, Math. Notes, 109:5 (2021), 828–831
Khamrayev A.Yu., “On the Dynamics of a Quasistrictly Non-Volterra Quadratic Stochastic Operator”, Ukr. Math. J., 71:8 (2020), 1273–1281
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