Nonlinear Dynamical Analysis of Twitter Time Series
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.
Based on the basic principles of the self-organized criticality theory, we proposed an identifiers of network criticality. The identifiers allow you to determine the subcritical and supercritical phases of Twitter, using only the results of the analysis of the time series of microposts. The most significant result is the existence of two classes of time series of microposts and tweet Ids corresponding to them. The first class of the time series corresponds to the subcritical phase of the network. On the contrary, the second class corresponds to the supercritical phase.
Recently, there has been an increasing number of empirical evidence supporting the hypothesis that spread of avalanches of microposts on social networks, such as Twitter, is associated with some sociopolitical events. Typical examples of such events are political elections and protest movements. Inspired by this phenomenon, we built a phenomenological model that describes Twitter’s self-organization in a critical state. An external manifestation of this condition is the spread of avalanches of microposts on the network. e model is based on a fractional three-parameter self-organization scheme with stochastic sources. It is shown that the adiabatic mode of self-organization in a critical state is determined by the intensive coordinated action of a relatively small number of network users. To identify the critical states of the network and to verify the model, we have proposed a spectrum of three scaling indicators of the observed time series of microposts.
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.