arXiv subjects
Publications and source records attributed to U. A. Rozikov.
We study a discrete-time dynamical system of wild mosquito population with parameters: $β$ - the birth rate of adults; $α$ - maximum emergence rate; $μ>0$ - the death rate of adults; $γ$ - Allee effects. We prove that if $γ\geq\frac{α(β-μ)}{μ^2}$ then the mosquito population dies and if $γ<\frac{α(β-μ)}{μ^2}$ holds then extinction or survival of the mosquito population depends on their initial state.
In this short note we study a dynamical system generated by a two-parametric quadratic operator mapping 3-dimensional simplex to itself. This is an evolution operator of the frequencies of gametes in a two-locus system. We find the set of all (a continuum set) fixed points and show that each fixed point is non-hyperbolic. We completely describe the set of all limit points of the dynamical system. Namely, for any initial point (taken from the 3-dimensional simplex) we find an invariant set containing the initial point and a unique fixed point of the operator, such that the trajectory of the initial point converges to this fixed point.
For the SOS (solid-on-solid) model with an external field and with spin values from the set of all integers on a Cayley tree each (gradient) Gibbs measure corresponds to a boundary law (an infinite-dimensional vector function defined on vertices of the Cayley tree) satisfying a non-linear functional equation. Recently some translation-invariant and height-periodic (non-normalisable) solutions to the equation are found. Here our aim is to find non-height-periodic and non-normalisable boundary laws for the SOS model. By such a solution one can construct a non-probability Gibbs measure. We find explicitly several non-normalisable boundary laws. Moreover, we reduce the problem to solving of a non-linear, second-order difference equation. We give analytic and numerical analysis of the difference equation.
In this paper we investigate generalized Gibbs measure (GGM) for $p$-adic Hard-Core(HC) model with a countable set of spin values on a Cayley tree of order $k\geq 2$. This model is defined by $p$-adic parameters $λ_i$, $i\in \mathbb N$. We analyze $p$-adic functional equation which provides the consistency condition for the finite-dimensional generalized Gibbs distributions. Each solutions of the functional equation defines a GGM by $p$-adic version of Kolmogorov's theorem. We define $p$-adic Gibbs distributions as limit of the consistent family of finite-dimensional generalized Gibbs distributions and show that, for our $p$-adic HC model on a Cayley tree, such a Gibbs distribution does not exist. Under some conditions on parameters $p$, $k$ and $λ_i$ we find the number of translation-invariant and two-periodic GGMs for the $p$-adic HC model on the Cayley tree of order two.
We describe fixed points of an infinite dimensional non-linear operator related to a hard core (HC) model with a countable set $\mathbb{N}$ of spin values on the Cayley tree. This operator is defined by a countable set of parameters $λ_{i}>0$, $a_{ij}\in \{0,1\}$, $ i,j \in \mathbb{N}$. We find a sufficient condition on these parameters under which the operator has unique fixed point. When this condition is not satisfied then we show that the operator may have up to five fixed points. Also, we prove that every fixed point generates a normalisable boundary law and therefore defines a Gibbs measure for the given HC-model.
For the solid-on-solid (SOS) model with spin values from the set of all integers on a Cayley tree we give gradient Gibbs measures (GGMs). Such a measure corresponds to a boundary law (which is an infinite-dimensional vector-valued function defined on vertices of the Cayley tree) satisfying an infinite system of functional equations. We give several concrete GGMs of boundary laws which are independent from vertices of the Cayley tree and (as an infinite-dimensional vector) have periodic, (non-)mirror symmetric coordinates.
We consider a hard core (HC) model with a countable set $\mathbb{Z}$ of spin values on the Cayley tree. This model is defined by a countable set of parameters $λ_{i}>0, i \in \mathbb{Z}\setminus\{0\}$. For all possible values of parameters, we give limit points of the dynamical system generated by a function which describes the consistency condition for finite-dimensional measures. Also, we prove that every periodic Gibbs measure for the given model is either translation-invariant or periodic with period two. Moreover, we construct uncountable set of Gibbs measures for this HC model.
In this paper, we study the HC-model with a countable set $\mathbb Z$ of spin values on a Cayley tree of order $k\geq 2$. This model is defined by a countable set of parameters (that is, the activity function $λ_i>0$, $i\in \mathbb Z$). A functional equation is obtained that provides the consistency condition for finite-dimensional Gibbs distributions. Analyzing this equation, the following results are obtained: - Let $Λ=\sum_iλ_i$. For $Λ=+\infty$ there are no translation-invariant Gibbs measures (TIGM) and no two-periodic Gibbs measures (TPGM); - For $Λ<+\infty$, the uniqueness of TIGM is proved; - Let $Λ_{\rm cr}(k)=\frac{k^k}{(k-1)^{k+1}}$. If $0<Λ\leqΛ_{\rm cr}$, then there is exactly one TPGM that is TIGM; - For $Λ>Λ_{\rm cr}$, there are exactly three TPGMs, one of which is TIGM.
For SOS (solid-on-solid) model with external field and with spin values from the set of all integers, on a Cayley tree we give gradient Gibbs measures (GGMs). Such a measure corresponds to a boundary law (a function defined on vertices of Cayley tree) satisfying an infinite system of functional equations. We give several concrete GGMs which correspond to periodic boundary laws.
There is one-to-one correspondence between quadratic operators (mapping $\mathbb R^m$ to itself) and cubic matrices. It is known that any quadratic operator corresponding to a stochastic (in a fixed sense) cubic matrix preserves the standard simplex. In this paper we find conditions on the (non-stochastic) cubic matrix ensuring that corresponding quadratic operator preserves simplex. Moreover, we construct several quadratic non-stochastic operators which generate chaotic dynamical systems on the simplex. These chaotic behaviors are \emph{splitted} meaning that the simplex is partitioned into uncountably many invariant (with respect to quadratic operator) subsets and the restriction of the dynamical system on each invariant set is chaos in the sense of Devaney.
The Susceptible-Exposed-Infectious-Recovered (SEIR) model is applied in several countries to ascertain the spread of the coronavirus disease 2019 (COVID-19). We consider discrete-time SEIR epidemic model in a closed system which does not account for births or deaths, total population size under consideration is constant. This dynamical system generated by a non-linear evolution operator depending on four parameters. Under some conditions on parameters we reduce the evolution operator to a quadratic stochastic operator (QSO) which maps 3-dimensional simplex to itself. We show that the QSO has uncountable set of fixed points (all laying on the boundary of the simplex). It is shown that all trajectories of the dynamical system (generated by the QSO) of the SEIR model are convergent (i.e. the QSO is regular). Moreover, we discuss the efficiency of the model for Uzbekistan.
In this paper we consider a population consisting of two species, dynamics of which is defined by a quadratic stochastic operator with variable coefficients, making it discontinuous operator at two points. This operator depends on three parameters. It is shown that under suitable conditions on the parameters this operator may have fixed points, convergence of trajectories and there may exist periodic points.
In this paper we give a systematic review of the theory of Gibbs measures of Potts model on Cayley trees (developed since 2013) and discuss many applications of the Potts model to real world situations: mainly biology, physics, and some examples of alloy behavior, cell sorting, financial engineering, flocking birds, flowing foams, image segmentation, medicine, sociology etc.
In this paper we study dynamical systems generated by a gonosomal evolution operator of a bisexual population. We find explicitly all (uncountable set) of fixed points of the operator. It is shown that each fixed point has eigenvalues less or equal to 1. Moreover, we show that each trajectory converges to a fixed point, i.e. the operator is reqular. There are uncountable family of invariant sets each of which consisting unique fixed point. Thus there is one-to-one correspondence between such invariant sets and the set of fixed points. Any trajectory started at a point of the invariant set converges to the corresponding fixed point.
We consider a new model which consists of a DNA together with a RNA. Here we assume that DNA is from a mammal or bird but RNA comes from a virus. To study thermodynamic properties of this model we use methods of statistical mechanics, namely, the theory of Gibbs measures. We use these measures to describe phases (states) of the DNA-RNA system. Using a Markov chain (corresponding to Gibbs measure) we give conditions (on temperature) of DNA-RNA renaturation.
In this paper we study $p$-adic dynamical systems generated by the function $f(x)={a\over x^2}$ in the set of complex $p$-adic numbers. We find an explicit formula for the $n$-fold composition of $f$ for any $n\geq 1$. Using this formula we give fixed points, periodic points, basin of attraction and Siegel disk of each fixed (periodic) point depending on parameters $p$ and $a$.
We study the discrete-time dynamical systems of a model of wild mosquito population with distinct birth (denoted by $β$) and death (denoted by $μ$) rates. The case $μ=β$ was considered in our previous work. In this paper we prove that for $β<μ$ the mosquito population will die and for $β>μ$ the population will survive, namely, the number of the larvaes goes to infinite and the number of adults has finite limit ${α\over μ}$, where $α>0$ is the maximum emergence rete.
We define a DNA as a sequence of $1, 2$'s and embed it on a path of Cayley tree in such a way that each vertex of the Cayley tree belongs only to one of DNA and each DNA has its own countably many set of neighboring DNAs. The Hamiltonian of this set of DNAs is a model with two spin values considered as DNA base pairs. We describe translation invariant Gibbs measures (TIGM) of the model on the Cayley tree of order two and use them to study thermodynamic properties of the model of DNAs. We show that there is a critical temperature $T_{\rm c}$ such that (i) if temperature $T\geq T_{\rm c}$ then there exists unique TIGM; (ii) if $T<T_{\rm c}$ then there are three TIGMs. Each TIGM gives a phase of the set of DNAs. In case of very high and very low temperatures we give stationary distributions and typical configurations of the model.