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Nikolay K. Vitanov

Publications and source records attributed to Nikolay K. Vitanov.

At least 37 records · Page 2Linked to original sources
arXiv2017

Human migration and the motion of substance in a channel of a network

We study the motion of a substance in a channel of a network that consists of chain of nodes of a network (the nodes can be considered as boxes) and edges that connect the nodes and form the way for motion of the substance. The nodes of the channel can have different "leakage", i.e., some amount of the substance can leave the channel at a node and the rate of leaving may be different for the different nodes of the channel. In addition the nodes close to the end of the channel for some (construction or other) reason may be more "attractive" for the substance in comparison to the nodes around the entry node of the channel. We discuss channels containing infinite or finite number of nodes and obtain the distribution of the substance along the nodes. Two regimes of functioning of the channels are studied: stationary regime and non-stationary regime. The distribution of the substance along the nodes of the channel for the case of stationary regime is a generalization of the Waring distribution (for channel with infinite number of nodes) or generalization of the truncated Waring distribution (for channel with finite number of nodes). In the non-stationary regime of functioning of the channel one observes an exponential increase or exponential decrease of the amount of substance in the nodes. Despite this the asymptotic distribution of the substance among the nodes of the channel in this regime is stationary. The developed theory is applied for a study of the distrribution of migrants in countries that form migration channels.

physics.soc-ph↗
arXiv2017

Solitary wave solutions of nonlinear partial differential equations based on the simplest equation for the function $1/\cosh^n$

The method of simplest equation is applied for obtaining exact solitary traveling-wave solutions of nonlinear partial differential equations that contain monomials of odd and even grade with respect to participating derivatives. The used simplest equation is $f_ξ^2 = n^2(f^2 -f^{(2n+2)/n})$. The developed methodology is illustrated on two examples of classes of nonlinear partial differential equations that contain: (i) only monomials of odd grade with respect to participating derivatives; (ii) only monomials of even grade with respect to participating derivatives. The obtained solitary wave solution for the case (i) contains as particular cases the solitary wave solutions of Korteweg-deVries equation and of a version of the modified Korteweg-deVries equation.

nlin.SI↗
arXiv2017

Numerical Study of Nonlinear Dynamics of a Population System with Time Delay

Mathematical models of interacting populations are often constructed as systems of differential equations, which describe how populations change with time. Below we study one such model connected to the nonlinear dynamics of a system of populations in presence of time delay. The consequence of the presence of the time delay is that the nonlinear dynamics of the studied system become more rich, e.g., new orbits in the phase space of the system arise which are dependent on the time-delay parameters. In more detail we introduce a time delay and generalize the model system of differential equations for the interaction of three populations based on generalized Volterra equations in which the growth rates and competition coefficients of populations depend on the number of members of all populations \cite{Dimitrova2001a},\cite{Dimitrova2001b} and then numerically solve the system with and without time delay. We use a modification of the method of Adams for the numerical solution of the system of model equations with time delay. By appropriate selection of the parameters and initial conditions we show the impact of the delay time on the dynamics of the studied population system.

nlin.CD↗
arXiv2016

On nonlinear waves of the blood flow through arteries

We discuss propagation of traveling waves in a blood filled elastic artery with an axially symmetric dilatation (an idealized aneurysm). The processes in the injured artery are modelled by equations for the motion of the wall of the artery and by equation for the motion of the fluid (the blood). For the case when long-wave approximation holds the model equations are reduced to a version of the perturbed Korteweg-deVries equation. Exact travelling-wave solutions of this equation are obtained by the modified method of simplest equation where the differential equation of Abel is used as a simplest equation. A particular case of the obtained exact solution is discussed from the point of view of arterial mechanics.

physics.flu-dyn↗
arXiv2016

Box model of migration in channels of migration networks

We discuss a box model of migration in channels of networks with possible application for modelling motion of migrants in migration networks. The channel consists of nodes of the network (nodes may be considered as boxes representing countries) and edges that connect these nodes and represent possible ways for motion of migrants. The nodes of the migration channel have different "leakage", i.e. the probability of change of the status of a migrant (from migrant to non-migrant) may be different in the different countries along the channel. In addition the nodes far from the entry node of the channel may be more attractive for migrants in comparison to the nodes around the entry node of the channel. We discuss below channels containing infinite number of nodes. Two regimes of functioning of these channels are studied: stationary regime and non-stationary regime. In the stationary regime of the functioning of the channel the distribution of migrants in the countries of the channel is described by a distribution that contains as particular case the Waring distribution. In the non-stationary regime of functioning of the channel one observes exponential increase or exponential decrease of the number of migrants in the countries of the channel. It depends on the situation in the entry country of the channel for which scenario will be realized. Despite the non-stationary regime of the functioning of the channel the asymptotic distribution of the migrants in the nodes of the channel is stationary. From the point of view of the characteristic features of the migrants we discuss the cases of (i) migrants having the same characteristics and (ii) two classes of migrants that have differences in some characteristic (e.g., different religions).

physics.soc-ph↗
arXiv2016

Box model for channels of human migration

We discuss a mathematical model of migration channel based on the truncated Waring distribution. The truncated Waring distribution is obtained for a more general model of motion of substance through a channel containing finite number of boxes. The model is applied then for case of migrants moving through a channel consisting of finite number of countries or cities. The number of migrants in the channel strongly depends on the number of migrants that enter the channel through the country of entrance. It is shown that if the final destination country is very popular then large percentage of migrants may concentrate there.

physics.soc-ph↗
arXiv2015

Modified method of simplest equation for obtaining exact analytical solutions of nonlinear partial differential equations: Further development of methodology with two applications

We discuss the application of a variant of the method of simplest equation for obtaining exact traveling wave solutions of a class of nonlinear partial differential equations containing polynomial nonlinearities. As simplest equation we use differential equation for a special function that contains as particular cases trigonometric and hyperbolic functions as well as the elliptic function of Weierstrass and Jacobi. We show that for this case the studied class of nonlinear partial differential equations can be reduced to a system of two equations containing polynomials of the unknown functions. This system may be further reduced to a system of nonlinear algebraic equations for the parameters of the solved equation and parameters of the solution. Any nontrivial solution of the last system leads to a traveling wave solution of the solved nonlinear partial differential equation. The methodology is illustrated by obtaining solitary wave solutions for the generalized Korteweg-deVries equation and by obtaining solutions of the higher order Korteweg-deVries equation.

nlin.SI↗
arXiv2015

Test of two hypotheses explaining the size of populations in a system of cities

Two classical hypotheses are examined about the population growth in a system of cities: Hypothesis 1 pertains to Gibrat's and Zipf's theory which states that the city growth-decay process is size independent; Hypothesis 2 pertains to the so called Yule process which states that the growth of populations in cities happens when (i) the distribution of the city population initial size obeys a log-normal function, (ii) the growth of the settlements follows a stochastic process. The basis for the test is some official data on Bulgarian cities at various times. This system was chosen because (i) Bulgaria is a country for which one does not expect biased theoretical conditions; (ii) the city populations were determined rather precisely. The present results show that: (i) the population size growth of the Bulgarian cities is size dependent, whence Hypothesis 1 is not confirmed for Bulgaria; (ii) the population size growth of Bulgarian cities can be described by a double Pareto log-normal distribution, whence Hypothesis 2 is valid for the Bulgarian city system. It is expected that this fine study brings some information and light on other, usually considered to be more pertinent, city systems in various countries.

physics.soc-ph↗
arXiv2014

Nonlinear analysis of time series of vibration data from a friction brake: SSA, PCA, and MFDFA

We use the methodology of singular spectrum analysis (SSA), principal component analysis (PCA), and multi-fractal detrended fluctuation analysis (MFDFA), for investigating characteristics of vibration time series data from a friction brake. SSA and PCA are used to study the long time-scale characteristics of the time series. MFDFA is applied for investigating all time scales up to the smallest recorded one. It turns out that the majority of the long time-scale dynamics, that is presumably dominated by the structural dynamics of the brake system, is dominated by very few active dimensions only and can well be understood in terms of low dimensional chaotic attractors. The multi-fractal analysis shows that the fast dynamical processes originating in the friction interface are in turn truly multi-scale in nature.

nlin.CD↗
arXiv2014

Solitary wave solutions for nonlinear partial differential equations containing monomials of odd and even grades with respect to participating derivatives

We apply the method of simplest equation for obtaining exact solitary traveling-wave solutions of nonlinear partial differential equations that contain monomials of odd and even grade with respect to participating derivatives. We consider first the general case of presence of monomials of the both (odd and even) grades and then turn to the two particular cases of nonlinear equations that contain only monomials of odd grade or only monomials of even grade. The methodology is illustrated by numerous examples.

nlin.SI↗
arXiv2014

Primacy & Ranking of UEFA Soccer Teams from Biasing Organizing Rules

A question is raised on whether some implied regularity or structure, as found in soccer team ranking by the Union of European Football Associations (UEFA), is due to implicit game result value or score competition conditions. The analysis is based on considerations about complex systems, i.e. searching whether power or other simple law fits are appropriate to describe some internal dynamics. It is observed that the ranking is specifically organized: a major class made of a few teams emerges after each game season. Other classes which apparently have regular sizes subsequently occur. Thus, the notion of Sheppard primacy index is envisaged to describe the findings. Additional primacy indices are discussed for enhancing the features. These measures can be used to sort out peer classes in more general terms. A very simplified toy model containing ingredients of the UEFA ranking rules suggests that such peer classes are an extrinsic property of the ranking, as obtained in many nonlinear systems under boundary condition constraints.

physics.soc-ph↗
arXiv2014

Ranking structures and Rank-Rank Correlations of Countries. The FIFA and UEFA cases

Ranking of agents competing with each other in complex systems may lead to paradoxes according to the pre-chosen different measures. A discussion is presented on such rank-rank, similar or not, correlations based on the case of European countries ranked by UEFA and FIFA from different soccer competitions. The first question to be answered is whether an empirical and simple law is obtained for such (self-) organizations of complex sociological systems with such different measuring schemes. It is found that the power law form is not the best description contrary to many modern expectations. The stretched exponential is much more adequate. Moreover, it is found that the measuring rules lead to some inner structures, in both cases.

physics.soc-ph↗
arXiv2013

On systems of interacting populations influenced by multiplicative white noise

We discuss a model of a system of interacting populations for the case when: (i) the growth rates and the coefficients of interaction among the populations depend on the populations densities: and (ii) the environment influences the growth rates and this influence can be modelled by a Gaussian white noise. The system of model equations for this case is a system of stochastic differential equations with: (i) deterministic part in the form of polynomial nonlinearities; and (ii) state-dependent stochastic part in the form of multiplicative Gaussian white noise. We discuss both the cases when the formal integration of the stochastic differential equations leads: (i) to integrals of Ito kind; or (ii) to integrals of Stratonovich kind. The systems of stochastic differential equations are reduced to the corresponding Fokker-Planck equations. For the Ito case and for the case of 1 population am analytic results is obtained for the stationary PDF of the the population density. For the case of more than one population and for the both Ito case and Stratonovich case the detailed balance conditions are not satisfied and because of this exact analytic solutions of the corresponding Fokker-Plank equations for the stationary PDFs for the population densities are not known. We obtain approximate solutions for this case by the methodology of the adiabatic elimination.

nlin.CD↗
arXiv2013

On the dynamics of interacting populations in presence of state dependent fluctuations

We discuss several models of the dynamics of interacting populations. The models are constructed by nonlinear differential equations and have two sets of parameters: growth rates and coefficients of interaction between populations. We assume that the parameters depend on the densities of the populations. In addition the parameters can be influenced by different factors of the environment. This influence is modelled by noise terms in the equations for the growth rates and interaction coefficients. Thus the model differential equations become stochastic. In some particular cases these equations can be reduced to a Foker-Plancnk equation for the probability density function of the densities of the interacting populations.

nlin.AO↗
arXiv2013

Waves and distributions connected to systems of interacting populations

We discuss two cases that can be connected to the dynamics of interacting populations: (I.) density waves for the case or negligible random fluctuations of the populations densities, and (II.) probability distributions connected to the model equations for of spatially averaged populations densities for the case of significant random fluctuations of the independent quantity that can be associated with the population density. For the case (I.) we consider model equations containing polynomial nonlinearities. Such nonlinearities can arise as a consequence of interaction among the populations (for the case of large population densities) or as a result of a Taylor series expansion (for the case of small density of interacting populations). In the both cases we can apply the modified method of simplest equation to obtain exact traveling-wave solutions connected to migration of population members. Such solutions are obtained for systems consisting of 1 or 3 populations respectively. For the case (II.) we discuss model equations of the Fokker-Planck kind for the evolution of the statistical distributions of population densities. We derive several stationary distributions for the population density and calculate the expected exit time connected to the extinction of the studied population.

math-ph↗
arXiv2012

On Nonlinear Waves in the Spatio-Temporal Dynamics of Interacting Populations

In this paper the spatial-temporal dynamics of the members of interacting populations is described by nonlinear partial differential equations. We consider the migration as a diffusion process influenced by the changing values of the birth rates and coefficients of interaction between the populations. For the particular case of one population and one spatial dimension the general model is reduced to analytically tractable PDE with polynomial nonlinearity up to third order. By applying the modified method of simplest equation to the described model we obtain an analytical solution which describes nonlinear kink and solitary waves in the population dynamics.

nlin.SI↗
arXiv2012

Application of the modified method of simplest equation for obtaining exact traveling-wave solutions for the extended Korteweg - de Vries equation and generalized Camassa-Holm equation

The modified method of simplest equation is applied to the extended Korteweg - de Vries equation and to generalized Camassa - Holm equation. Exact traveling wave solutions of these two nonlinear partial differential equations are obtained. The equations of Bernoulli, Riccati and the extended tanh - equation are used as simplest equations. Some of the obtained solutions correspond to surface water waves.

nlin.SI↗