Dynamics of Lebesgue Quadratic Stochastic Operator with Nonnegative Integers Parameters Generated by 2-Partition
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Keywords:
quadratic stochastic operator; nonlinear operator; Lebesgue measure; infinite state space; trajectory behaviorAbstract
The theory of quadratic stochastic operator (QSO) has been significantly developed since it was introduced in 1920s by Bernstein on population genetics. Over the century, many researchers have studied the behavior of such nonlinear operators by considering different classes of QSO on finite and infinite state spaces. However, all these studies do not comprehensively represent the core problem of QSO; i.e., the trajectory behavior. Recently, a class of QSO called Lebesgue QSO has been introduced and studied. Such an operator got its name based on Lebesgue measure which serves as a probability measure of the QSO. The conditions of the Lebesgue QSO have allowed us to consider the possibility of introducing a new measure for such QSO. This research presents a new class
of Lebesgue QSO with nonnegative integers parameters generated by a measurable 2-partition on the continual state space X = [ , 0 1]. This research aims to study the trajectory behavior of the QSO by reducing its infinite variables into a mapping of one-dimensional simplex. The behavior of such operators will be investigated computationally and analytically, where the computational results
conform to the analytical results. We will apply measure and probability theory as well as functional analysis to describe the limit behavior of such QSO. We show that for the new class of Lebesgue QSO generated by a 2-partition, one could analyze the behavior of such an operator by describing the existence of fixed points and periodic points of period-2. These results demonstrate that such
Lebesgue QSO generated by a measurable 2-partition can be a regular or nonregular transformation dependence of fixed parameters
References
A. Alsarayreh, I. Qaralleh and M. Z. Ahmad, ξ( ) as -Quadratic stochastic operators in two-dimensional simplex and their behavior, JP J. Algebra, Number Theory Appl., 2017, 39, 737–770.
S. N. Bernstein, The solution of a mathematical problem related to the theory of heredity, Uchn Zap. NI Kaf Ukr Otd Mat., 1924, 1, 83–115.
N. Ganikhodjaev and N. Z. A. Hamzah, On Poisson Nonlinear Transformations, Sci. World J, 2014, 2014, 1–7.
N. Ganikhodjaev and N. Z. A. Hamzah, On Volterra quadratic stochastic operators with continual state space, AIP Conf. Proc., 2015, 1660, 1–7.
N. Ganikhodjaev and N. Z. A. Hamzah, On Gaussian Nonlinear Transformations, AIP Conf. Proc., 2015, 1682(040009).
N. Ganikhodjaev and N. Z. A. Hamzah, Geometric quadratic stochastic operator on countable infinite set, AIP Conf. Proc., 2015, 1643, 706–712.
N. Ganikhodjaev and N. Z. A. Hamzah, Quadratic Stochastic Operators on Segment [0, 1] and Their Limit Behavior, Indian J Sci Technol, 2015, 8.
N. Ganikhodjaev, R. Muhitdinov and M. Saburov, On Lebesgue nonlinear transformations, Bull. Korean Math. Soc.,
, 2017, 607–618.
R. Ganikhodzhaev, F. Mukhamedov and U. Rozikov, Quadratic Stochastic Operators and Processes: Results and Open Problems, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 2011, 14, 279–335.
N. Z. A. Hamzah and N. Ganikhodjaev, On non-ergodic Gaussian quadratic stochastic operators, AIP Conf. Proc., 2018, 1974(030021).
A. J. Hardin and U. A. Rozikov, A Quasi-strictly Non-Volterra Quadratic Stochastic Operator, Qual. Theory Dyn. Syst., 2019, 18, 1013–1029.
S. N. Karim, N. Z. A. Hamzah and N. Ganikhodjaev, A Class of Geometric Quadratic Stochastic Operator on Countable State Space and Its Regularity, Mal. J. Fund. Appl. Sci., 2019, 15, 872–877.
S. N. Karim, N. Z. A. Hamzah and N. Ganikhodjaev, Regularity of Geometric quadratic stochastic operator generated by 2-partition of infinite points, Mal. J. Fund. Appl. Sci., 2020, 16, 281–285.
S. N. Karim, N. Z. A. Hamzah, N. N. M. Fauzi and N. Ganikhodjaev, New Class of 2-Partition Poisson Quadratic Stochastic Operators on Countable State Space, J. Phys.: Conf. Ser., 2021, 1988, 012080
S. N. Karim, N. Z. A. Hamzah and N. Ganikhodjaev, On nonhomogeneous geometric quadratic stochastic operators, Turk. J. Math., 2022, 46, 1397–1407
F. Khaled and P.C. Hee, On Three-Dimensional Mixing Geometric Quadratic Stochastic Operators, Math. Stat., 2021, 9, 151–158.
Y. I. Lyubich, Iterations of quadratic maps, In Mathematical Economics and Functional Analysis (Moscow,1974).
F. Mukhamedov, M. Saburov and A. H. Mohd Jamal, On Dynamics of ξs -Quadratic stochastic operators on 2-D simplex, Int. J. Mod. Phys. Conf. Ser., 2012, 09, 299–307.
F. Mukhamedov, I. Qaralleh and W. N. F. A. Wan Rozali, On ξa-Quadratic stochastic operators on 2-D simplex, Sains Malays., 2014, 43, 1275–1281.
M. Saburov and N. A. Yusof, On uniqueness of fixed points of quadratic stochastic operators on a 2D simplex, Methods Funct. Anal. Topol., 2018, 24, 255–264
