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Most complex systems are intrinsically dynamic in nature. The evolution of a dynamic complex system is typically represented as a sequence of snapshots, where each snapshot describes the configuration of the system at a particular instant…

We present an algorithm for the simulation of the exact real-time dynamics of classical many-body systems with discrete energy levels. In the same spirit of kinetic Monte Carlo methods, a stochastic solution of the master equation is found,…

In this review, we present some advanced algorithms and programs used in our scientific school with short description of types of astrophysical systems, which we study. However, we discuss mainly mathematical methods, which may be applied…

The characterization of dynamical processes in living systems provides important clues for their mechanistic interpretation and link to biological functions. Thanks to recent advances in microscopy techniques, it is now possible to…

In this paper we present a method of discrete modeling and analysis of multi-level dynamics of complex large-scale hierarchical dynamic systems subject to external dynamic control mechanism. In a model each state describes parallel dynamics…

We present a framework for fingerprint matching based on marked point process models. An efficient Monte Carlo algorithm is developed to calculate the marginal likelihood ratio for the hypothesis that two observed prints originate from the…

Proposed to study the dynamics of physiological systems in which the evolution depends on the state in a previous time, the Mackey-Glass model exhibits a rich variety of behaviors including periodic or chaotic solutions in vast regions of…

Piecewise smooth systems are intensively studied today in many application areas, such as economics, finance, engineering, biology, and ecology. In this work, we consider a class of one-dimensional piecewise linear discontinuous maps with a…

In this chapter, we utilize dynamical systems to analyze several aspects of machine learning algorithms. As an expository contribution we demonstrate how to re-formulate a wide variety of challenges from deep neural networks, (stochastic)…

We introduce a two-dimensional discrete-time dynamical system which represents the evolution of an angle and angular velocity. While the angle evolves by a fixed amount in every step, the evolution of the angular velocity is governed by a…

Different models of random walks on the dual graphs of compact urban structures are considered. Analysis of access times between streets helps to detect the city modularity. The statistical mechanics approach to the ensembles of lazy random…

Discrete models have a long tradition in engineering, including finite state machines, Boolean networks, Petri nets, and agent-based models. Of particular importance is the question of how the model structure constrains its dynamics. This…

We give a survey on classical and recent applications of dynamical systems to number theoretic problems. In particular, we focus on normal numbers, also including computational aspects. The main result is a sufficient condition for…

In this work, we systematically investigate linear multi-step methods for differential equations with memory. In particular, we focus on the numerical stability for multi-step methods. According to this investigation, we give some…

The study of neural computation aims to understand the function of a neural system as an information processing machine. Neural systems are undoubtedly complex, necessitating principled and automated tools to abstract away details to…

This paper is concerned with the study of the stability of dynamical systems evolving on time scales. We first {formalize the notion of matrix measures on time scales, prove some of their key properties and make use of this notion to study…

We study the out of equilibrium dynamics of several models exhibiting aging. We attempt at identifying various types of aging systems using a phase space point of view: we introduce a trial classification, based on the overlap between two…

We investigate slow non-equilibrium dynamical processes in two-dimensional $q$--state Potts model with both ferromagnetic and $\pm J$ couplings. Dynamical properties are characterized by means of the mean-flipping time distribution. This…

Dynamical maps describe general transformations of the state of a physical system, and their iteration can be interpreted as generating a discrete time evolution. Prime examples include classical nonlinear systems undergoing transitions to…

This work is devoted to modelling and identification of the dynamics of the inter-sectoral balance of a macroeconomic system. An approach to the problem of specification and identification of a weakly formalized dynamical system is…