Abstract: A broad range of physical, biological and social phenomena evade the description within the framework of traditional analytic functions. Properties of disordered materials, diffusion in anisotropic systems, stock market crashes, power grid failures, earthquakes or physiological time series are all characterized by long-term memory, spatial heterogeneity, non-stationarity and nonergodicity. Those inherent signatures of complexity require a new approach extending beyond classical models, and the fractional calculus provides one framework for such new way of thinking.?In this talk we will explore reasons why the traditional calculus is not sufficient to capture the full range of dynamics found in natural and man-made processes. We will introduce few definitions fundamental to the field and we will illustrate how fractional calculus approach can provide more natural description of complex phenomena than traditional methods.

Abstract: Many dynamical processes on real world networks display complex temporal patterns as, for instance, a fat-tailed distribution of inter-events times, leading to heterogeneous waiting times between events. In this work, we focus on distributions whose average inter-event time diverges, and study its impact on the dynamics of random walkers on networks. The process can naturally be described, in the long time limit, in terms of Riemann-Liouville fractional derivatives. We show that all the dynamical modes possess, in the asymptotic regime, the same power law relaxation, which implies that the dynamics does not exhibit time-scale separation between modes, and that no mode can be neglected versus another one, even for long times. Our results are then confirmed by numerical simulations.

Abstract: Random evolutions provide a framework for exploring the probability interpretation of fractional operators. In this context, fractional order dynamical systems represent the average behavior of integer order dynamical systems subject to fluctuations from either internal or external sources. Many complex systems consist of inherently noisy elements and cannot be isolated from their surrounding environment without losing practical meaning. Therefore it is important to study random evolutions that lead to fractional macroscopic equations in order to be able to recognize when a fractional calculus description is appropriate. We plan to discuss this issue at an introductory level with illustrations that include the continuous time random walk and the harmonic oscillator with a randomly modulated frequency (e.g., from being coupled to a network of oscillators).

Abstract: A wide variety of systems require filtering of signals by which noise is dampened so that a signal of interest can be extracted. In control systems, iterative filtering of a stochastic process is needed to estimate state variables and apply those measurements to control signals. A celebrated example is the Kalman filter [1], which uses linear approximation and covariance estimates to iteratively improve estimates of random variables. Each iteration of the Kalman filter consists of a prediction step, in which the mean and covariance of the random process is forecast, and an update step, in which a measurement of the mean and covariance is used to correct errors in the prediction. While the Kalman filter is optimal for linear signals with Gaussian noise, and can work effectively for a broader range of signal processing applications, complex systems with nonlinear signals and non-Gaussian noise impose unique challenges. Approaches to these challenges include the extended Kalman filter, which splices linear approximations, and the particle filter, which generalizes filter design to non-parametric distributions. An important improvement was achieved by Zakai [2], who showed that a nonlinear filtering problem can be reduced to a linear stochastic partial differential equation by allowing the distribution to be unnormalized. L?vy processes [3], which are characterized by heavy-tail distributions, are particularly challenging to model because the jumps and ?L?vy flights? have many of the characteristics of deterministic processes despite their stochastic origin. Fractional calculus provides a methodology to design filters for stochastic processes driven by L?vy processes. This is achieved by generalizing the Zakai equation using pseudo differential operators with fractional order. A particularly interesting application of the fractional Zakai equation is the modeling of time-change processes, in which time is also treated as a random variable [4].

Abstract: As the first step in control and optimization the flow of crowds of pedestrians, modeling of dynamic crowd has received increasing attention in recent years. In this talk, the model of crowds of pedestrians with large numbers is derived from agents that interact with neighbors in micro-scale based on the CTRW (continuous time random walk) framework. We also derived the model in macro-scale based on conservation laws. To some extent, the macroscopic models obtained using the above two methods coincide with each other where long-range interactions and short-range interactions have played an important role in deriving the fractional order macroscopic model. Relationship between microscopic model and macroscopic model revealed in this talk from the viewpoint of interactions, which will contribute to the analysis and control of flocking of large crowds so that some tragedies such as stampede could be avoided. For efficient evacuation of crowds in some bounded area, a micro-macro dynamic model of fractional order is studied using fractional mean field games theory and a fractional forward backward model has been constructed in the end. The obtained dynamic models of fractional order in different scales have been tied and connected with each other from the view point of interaction. Some other interesting work that undergoing is also included in this talk such as the boundary control based on initial distribution of pedestrians and boundary control based on the spatial structure.

Abstract: Universal critical behavior in networks is determined by the structure of a network and the symmetry of a given model. Many real networks exhibiting a small-world property, are scale-free. The problem of studying dynamic processes in networks with such universal properties raises a question: do there exist common patterns in the dynamics of various complex systems? We derive an equation of motion for the order parameter in the small world networks with temporal memory and obtain a fractional differential equation for the order parameter. We analyze phase transitions induced by noise in such systems. We discuss the applicability of the obtained results to analyze dynamic processes in the brain network.