Nonlinear Dynamical Analysis of Twitter Time Series
The present paper outlines a novel approach to predict popularity of topics for social network Twitter; the method is designed to identify precociously the topics able to demonstrate “explosive” growth in popularity. First of all, the predictive clustering method ascertains real (not written in hash-tags!) topics of tweets and then predicts popularity rates for the topics. The same clustering algorithm is employed both to ascertain the real topic of a message and to cluster segments of time series (in order to predict topics popularity), namely, maximum likelihood adaptive neural system based upon modelling field theory. In the course of wide-ranging simulation, typical variants of “pre-explosive” dynamics were revealed; some of them were turned out to be equal to heuristic techniques to predict topics popularity well known for PR community collaborating with the network (“crab,” “Pesavento’s butterfly,” etc.).
Nowadays, production control problems has been widely studied and a lot of valuable approaches have been implemented. Some work addresses the problem of tracking the uncertain demand in case of uncertain production speeds. The uncertainties are described by deterministic inequalities and the performance is analyzed in from of the worst-case scenario. First, simple mathematical models are introduced and the control problem is formulated. In continuous-time, the cumulative output of a manufacturing machine is the integral of the production speed over time. At the same time, the production speed is bounded from below and above, and hence the manufacturing process can be modeled as an integrator with saturated input. Since the cumulative demand (which is the reference signal to track) is a growing function of time, it is natural to consider control policies that involve integration of the mismatch between the current output and current demand. In the simplest consideration it results in models similar to a double integrator closed by saturated linear feedback with an extra input that models disturbances of a different nature. This model is analyzed and particular attention is devoted to the integrator windup phenomenon: lack of global stability of the system solutions that correspond to the same input signal.
Recent developments in nonlinear science have caused the formation of a new paradigm called the paradigm of complexity. The self-organized criticality theory constitutes the foundation of this paradigm. To estimate the complexity of a microblogging social network, we used one of the conceptual schemes of the paradigm, namely, the system of key signs of complexity of the external manifestations of the system irrespective of its internal structure. Our research revealed all the key signs of complexity of the time series of a number of microposts. We offer a new model of a microblogging social network as a nonlinear random dynamical system with additive noise in three-dimensional phase space. Implementations of this model in the adiabatic approximation possess all the key signs of complexity, making the model a reasonable evolutionary model for a microblogging social network. The use of adiabatic approximation allows us to model a microblogging social network as a nonlinear random dynamical system with multiplicative noise with the power-law in one-dimensional phase space.
The present research is devoted to the application of stochastic bifurcation theory to the early detection of economic bubbles. A nonlinear random dynamical system with the possible appearance of stochastic P-bifurcations with a fat-tailed probability density function is deduced. The possibility of application of chaotic bifurcation theory to the early detection of culminations of economic bubbles is investigated by the example of dot-com bubbles. For the increments of NASDAQ it is shown that the criterion of reaching the culmination for dot-com bubbles is a formation of a bimodal distribution with the subsequent conversion to a unimodal distribution as a result of codimension one P-bifurcation – a triple equilibrium point.
In this paper, using the theory of differential games developed the algorithm of constructing guaranteed control for nonlinear systems. Through the transition from the nonlinear model to model the linear form with parameters that depend on the state, will move from finding solutions to equations of Hamilton-Jacobi-Bellman equation to the Riccati equation. Moreover, the appointment of the "worst settings" will make the matrix not depend on the state that will allow you to solve Riccati equation and to synthesize the required control. The example of construction controls for non-linear mathematical model of the human immune system in the presence of the HIV. Built management controls the supply of antiretroviral drugs to maintain healthy cells of the human immune system at the required level to maximize and facilitate the life of the patient.