On b-bistochastic quadratic stochastic operators (original) (raw)
2014
A quadratic stochastic operator (QSO) is usually used to present the time evolution of differing species in biology. Some quadratic stochastic operators have been studied by Lotka and Volterra. The general problem in the nonlinear operator theory is to study the behavior of operators. This problem was not fully finished even for quadratic stochastic operators which are the simplest nonlinear operators. To study this problem, several classes of QSO were investigated. In this paper, we study the ξ(a)–QSO defined on 2D simplex. We first classify ξ(a)–QSO into 2 non-conjugate classes. Further, we investigate the dynamics of these classes of such operators.
Quadratic Stochastic Operators and Processes: Results and Open Problems
Infinite Dimensional Analysis, Quantum Probability and Related Topics, 2011
The history of the quadratic stochastic operators can be traced back to the work of Bernshtein (1924). For more than 80 years, this theory has been developed and many papers were published. In recent years it has again become of interest in connection with its numerous applications in many branches of mathematics, biology and physics. But most results of the theory were published in non-English journals, full text of which are not accessible. In this paper we give all necessary definitions and a brief description of the results for three cases: (i) discrete-time dynamical systems generated by quadratic stochastic operators; (ii) continuous-time stochastic processes generated by quadratic operators; (iii) quantum quadratic stochastic operators and processes. Moreover, we discuss several open problems.
On Quadratic Stochastic Operators Having Three Fixed Points
Journal of Physics: Conference Series, 2016
We knew that a trajectory of a linear stochastic operator associated with a positive square stochastic matrix starting from any initial point from the simplex converges to a unique fixed point. However, in general, the similar result for a quadratic stochastic operator associated with a positive cubic stochastic matrix does not hold true. In this paper, we provide an example for the quadratic stochastic operator with positive coefficients in which its trajectory may converge to different fixed points depending on initial points.
On a class of separable quadratic stochastic operators
Lobachevskii Journal of Mathematics, 2011
The purpose of this paper is to investigate a class of separable quadratic stochastic operators. Each separable quadratic stochastic operator (SQSO) depends on two quadratic matrices A and B, which have some relations. In this paper we proved that for each skew symmetric matrix A the corresponding SQSO is a linear operator. We also proved that non linear Volterra QSOs are not SQSOs. For a fixed matrix A we also discussed some properties of the set of all the corresponding matrices B of SQSOs.
Classification of ξ ( s ) -Quadratic Stochastic Operators on 2D simplex
Journal of Physics: Conference Series, 2013
A quadratic stochastic operator (in short QSO) is usually used to present the time evolution of differing species in biology. Some QSO has been studied by Lotka and Volterra. The general problem in the nonlinear operator theory is to study the behavior of operators. This problem was not fully finished even for the quadratic stochastic operators. To study this problem it was investigated several classes of such QSO. In this paper we study ξ (s) -QSO class of operators. We study such kind of operators on 2D simplex. We first classify these ξ (s) -QSO into 20 classes. Further, we investigate the dynamics of one class of such operators.
On ξa -quadratic stochastic operators on 2-D simplex
2014
A quadratic stochastic operator (QSO) is usually used to present the time evolution of differing species in biology. Some quadratic stochastic operators have been studied by Lotka and Volterra. The general problem in the nonlinear operator theory is to study the behavior of operators. This problem was not fully finished even for quadratic stochastic operators which are the simplest nonlinear operators. To study this problem, several classes of QSO were investigated. In this paper, we study the ξ (a)-QSO defined on 2D simplex. We first classify ξ (a)-QSO into 2 non-conjugate classes. Further, we investigate the dynamics of these classes of such operators.
ON DYNAMICS OF ξ S QUADRATIC STOCHASTIC OPERATORS
International Journal of Modern Physics: Conference Series, 2012
In this research we introduce a new class of quadratic stochastic operators called ξ s -QSO which are defined through coefficient of the operator from measure-theoretic (namely we are looking the coefficient as the measures which are absolute continuous or singular) point of view. We also study the limiting behaviour of ξ s -QSO defined on 2D-simplex. We first describe ξ s -QSO on 2Dsimplex and classify them with respect to the conjugacy and renumeration of the coordinates. We find six non-isomorphic classes of such operators. Moreover, we investigate the behaviour of each operator from three classes and prove convergence of trajectories of these classes and study their certain properties. We showed trajectories of two classes converge to the equilibrium. For the third class, it is established only the negative trajectories converge to the equilibrium.
On Volterra and orthoganality preserving quadratic stochastic operators
A quadratic stochastic operator (in short QSO) is usually used to present the time evolution of differing species in biology. Some quadratic stochastic operators have been studied by Lotka and Volterra. In the present paper, we first give a simple characterization of Volterra QSO in terms of absolutely continuity of discrete measures. Moreover, we provide its generalization in continuous setting. Further, we introduce a notion of orthogonal preserving QSO, and describe such kind of operators defined on two dimensional simplex. It turns out that orthogonal preserving QSOs are permutations of Volterra QSO. The associativity of genetic algebras generated by orthogonal preserving QSO is studied too.
On Three-Dimensional Mixing Geometric Quadratic Stochastic Operators
Mathematics and Statistics, 2021
It is widely recognized that the theory of quadratic stochastic operator frequently arises due to its enormous contribution as a source of analysis for the investigation of dynamical properties and modeling in diverse domains. In this paper, we are motivated to construct a class of quadratic stochastic operators called mixing quadratic stochastic operators generated by geometric distribution on infinite state space X. We also study regularity of such operators by investigating of the limit behavior for each case of the parameter. Some of non-regular cases proved for a new definition of mixing operators by using the shifting definition, where the new parameters satisfy the shifted conditions. A mixing quadratic stochastic operator was established on 3-partitions of the state space X and considered for a special case of the parameter ε. We found that the mixing quadratic stochastic operator is a regular transformation for 1 1 2 4 ε < < and is a non-regular for 1 4 ε <. Also, the trajectories converge to one of the fixed points. Stability and instability of the fixed points were investigated by finding of the eigenvalues of Jacobian matrix at these fixed points. We approximate the parameter ε by the parameter 6 r , where we established the regularity of the quadratic stochastic operators for some inequalities that satisfy 6 r. We conclude this paper by comparing with previous studies where we found some of such quadratic stochastic operators will be non-regular.
On nonhomogeneous geometric quadratic stochastic operators
Turkish Journal of Mathematics
In this paper, we construct a nonhomogeneous geometric quadratic stochastic operator generated by 2partition ξ on countable state space X = Z *. The limiting behavior of such operator is studied. We have proved that such operator possesses the regular property.