Nikolay K. Vitanov

Modified method of simplest equation and its application to nonlinear PDEs

Abstract We search for traveling-wave solutions of the class of PDEs ∑ p = 1 N 1 A p ( Q ) ∂ p Q ∂ t p + ∑ r = 2 N 2 B r ( Q ) ∂ Q ∂ t r + ∑ s = 1 N 3 C s ( Q ) ∂ s Q ∂ x s + ∑ u = 2 N 4 D u ( Q ) ∂ Q ∂ x u + F ( Q ) = 0 where A p ( Q ) , B r ( Q ) , C s ( Q ) , D u ( Q ) and F ( Q ) are polynomials of Q. The basis of the investigation is a modification of the method of simplest equation. The equations of Bernoulli, Riccati and the extended tanh-function equation are used as simplest equations. The obtained general results are illustrated by obtaining exact solutions of versions of the generalized Kuramoto–Sivashinsky equation, reaction–diffusion equation with density-dependent diffusion, and the reaction-telegraph equation.

Verhulst–Lotka–Volterra (VLV) model of ideological struggle

A general model for opinion formation and competition, like in ideological struggles is formulated. The underlying set is a closed one, like a country but in which the population size is variable in time. Several ideologies compete to increase their number of adepts. Such followers can be either converted from one ideology to another or become followers of an ideology though being previously ideologically-free. A reverse process is also allowed. We consider two kinds of conversion: unitary conversion, e.g. by means of mass communication tools, or binary conversion, e.g. by means of interactions between people. It is found that the steady state,when it exists, depends on the number of ideologies. Moreover when the number of ideologies increases some tension arises between them. This tension can change in the course of time. We propose to measure the ideology tensions through an appropriately defined scale index.

Adaptation and its impact on the dynamics of a system of three competing populations.

We investigate how the adaptation of the competition coefficients of the competing populations for the same limited resource influences the system dynamics in the regions of the parameter space, where chaotic motion of Shilnikov kind exists. We present results for two characteristic values of the competition coefficient adaptation factor α∗. The first value α∗=−0.05 belongs to the small interval of possible negative values of α∗. For this, α∗ a transition to chaos by period-doubling bifurcations occurs and a window of periodic motion exists between the two regions of chaotic motion. With increasing α∗, the system becomes more dissipative and the number of the windows of periodic motion increases. When α∗=1.0, a region of transient chaos is observed after the last window of periodic motion. We verify the picture of the system dynamics by power spectra, histograms and autocorrelations and calculate the Lyapunov exponents and Kaplan–Yorke dimension. Finally we discuss the eligibility of the investigated system for a topological analysis.

On nonlinear population waves

We discuss a model system of partial differential equations for description of the spatio-temporal dynamics of interacting populations. We are interested in the waves caused by migration of the populations. We assume that the migration is a diffusion process influenced by the changing values of the birth rates and coefficients of interaction among 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 4th order. We investigate this particular case and obtain two kinds of solutions: (i) approximate solution for small value of the ratio between the coefficient of diffusion and the wave velocity and (ii) exact solutions which describe nonlinear kink and solitary waves. In an appropriate phase space the kinks correspond to a connection between two states represented by a saddle point and a stable node. Finally we derive conditions for the asymptotic stability of the obtained solutions.

Influence of adaptation on the nonlinear dynamics of a system of competing populations

Abstract We investigate the nonlinear dynamics of a system of populations competing for the same limited resource assuming that they can adapt their growth rates and competition coefficients with respect to the number of individuals of each population. The adaptation leads to an enrichment of the nonlinear dynamics of the system which is demonstrated by a discussion of new orbits in the phase space of the system, completely dependent on the adaptation parameters, as well as by an investigation of the influence of the adaptation parameters on the dynamics of a strange attractor of the model system of ODEs.

Knowledge Epidemics and Population Dynamics Models for Describing Idea Diffusion

The diffusion of ideas is often closely connected to the creation and diffusion of knowledge and to the technological evolution of society. Because of this, knowledge creation, exchange and its subsequent transformation into innovations for improved welfare and economic growth is briefly described from a historical point of view. Next, three approaches are discussed for modeling the diffusion of ideas in the areas of science and technology, through (i) deterministic, (ii) stochastic, and (iii) statistical approaches. These are illustrated through their corresponding population dynamics and epidemic models relative to the spreading of ideas, knowledge and innovations. The deterministic dynamical models are considered to be appropriate for analyzing the evolution of large and small societal, scientific and technological systems when the influence of fluctuations is insignificant. Stochastic models are appropriate when the system of interest is small but when the fluctuations become significant for its evolution. Finally statistical approaches and models based on the laws and distributions of Lotka, Bradford, Yule, Zipf–Mandelbrot, and others, provide much useful information for the analysis of the evolution of systems in which development is closely connected to the process of idea diffusion.

New class of running-wave solutions of the (2+1)-dimensional sine-Gordon equation

A new class of running-wave solutions of the (2+1)-dimensional sine-Gordon equation is investigated. The obtained waves require two spatial dimensions for their propagation, i.e. they generalize solutions of the (2+0)-dimensional sine-Gordon equation. The parameters of the waves strongly depend on the wave amplitude and there exist forbidden areas for the wavenumber and frequency. The obtained solutions describe a new class of Josephson waves whose velocity is smaller than the Swihart velocity. If omega =0 the running waves are reduced to the self-consistent phase, current and magnetic field distributions in a large two-dimensional Josephson junction. The self-restriction coefficient for the Josephson current corresponding to one of the structures is calculated.

Dynamical consequences of adaptation of the growth rates in a system of three competing populations

We investigate the nonlinear dynamics of a system of populations competing for the same limited resource for the case where each of the populations adapts its growth rate to the total number of individuals in all populations. We consider regions of parameter space where chaotic motion of the Shilnikov kind exists and present results for two characteristic values of the growth ratio adaptation factor r*: r* = -0.15 and 5. Negative r* can lead to vanishing of regions of chaotic motion and to a stabilization of a fixed point of the studied model system of differential equations. Positive r* lead to changes of the shape of the bifurcation diagrams in comparison with the bifurcation diagrams for the case without adaptation. For the case r* = 5 we observe transition to chaos by period-doubling bifurcations, windows of periodic motion between the regions of chaotic motion and a region of transient chaos after the last window of periodic motion. The Lyapunov dimension for the chaotic attractors is close to two and the Lyapunov spectrum has a structure which allows a topological analysis of the attractors of the investigated system.

ON TRAVELLING WAVES AND DOUBLE-PERIODIC STRUCTURES IN TWO-DIMENSIONAL SINE-GORDON SYSTEMS

Exact travelling-wave solutions of the (2 + 1)-dimensional sine - Gordon equation possessing a velocity smaller than the velocity of the linear waves in the correspondent model system are obtained. The dependence of their dispersion relations and allowed areas for the wave parameters on the wave amplitude are discussed. The obtained waves contain as particular cases static structures consisting of elementary cells with zero topological charge. The self-consistent parameters of one static structure are calculated. The obtained structures require minima spatial system sizes for their existence. As an illustration the obtained results are applied for a description of structures in spin systems with an anisotropy created by a magnetic field or by a crystal anisotropy field.

Multifractal analysis of the long-range correlations in the cardiac dynamics of Drosophila melanogaster

By means of the multifractal detrended fluctuation analysis (MFDFA) we investigate long-range correlations in the interbeat time series of heart activity of Drosophila melanogaster—the classical object of research in genetics. Our main investigation tool are the fractal spectra f(α) and h(q) by means of which we trace the correlation properties of Drosophila heartbeat dynamics for three consequent generations of species. We observe that opposite to the case of humans the time series of the heartbeat activity of healthy Drosophila do not have scaling properties. Time series from species with genetic defects can be long-range correlated. Different kinds of genetic heart defects lead to different shape of the fractal spectra. The fractal heartbeat dynamics of Drosophila is transferred from generation to generation.