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In these lecture notes we discuss recent progress in the rigorous derivation of effective evolution equations for the description of the dynamics of quantum mechanical many-body systems.
In this paper we consider second order evolution equations with unbounded dynamic feedbacks. Under a regularity assumption we show that observability properties for the undamped problem imply decay estimates for the damped problem. We…
Robust hyperbolicity and stability results for linear partial differential equations with delay will be given and, as an application, the effect of small delays to the asymptotic properties of feedback systems will be analyzed.
This chapter focuses on variable maturation delay or, more precisely, on the mathematical description of a size-structured population consuming an unstructured resource. When the resource concentration is a known function of time, we can…
The evolution of a composite closed system using the integral wave equation with the kernel in the form of path integral is considered. It is supposed that a quantum particle is a subsystem of this system. The evolution of the reduced…
This paper proposes a systems approach to social sciences based on mathematical framework derived from a generalization of the mathematical kinetic theory and on theoretical tools of game theory. Social systems are modeled as a living…
Delay coordinates are a widely used technique to pass from observations of a dynamical system to a representation of the dynamical system as an embedding in Euclidean space. Current proofs show that delay coordinates of a given dynamical…
Time series of observables measured from complex systems do often exhibit non-normal statistics, their statistical distributions (PDF's) are not gaussian and often skewed, with roughly exponential tails. Departure from gaussianity is…
Driven by the explosion of data and the impact of real-world networks, a wide array of mathematical models have been proposed to understand the structure and evolution of such systems, especially in the temporal context. Recent advances in…
In sustained growth with random dynamics stationary distributions can exist without detailed balance. This suggests thermodynamical behavior in fast growing complex systems. In order to model such phenomena we apply both a discrete and a…
Systems evolving according to the standard concept of biological or technological evolution are often described by catalytic evolution equations. We study the structure of these equations and find a deep relationship to classical…
We study the exponential stability of evolutionary equations. The focus is laid on second order problems and we provide a way to rewrite them as a suitable first order evolutionary equation, for which the stability can be proved by using…
We construct stable periodic solutions for a simple form nonlinear delay differential equation (DDE) with a periodic coefficient. The equation involves one underlying nonlinearity with the multiplicative periodic coefficient. The well-known…
We formulate the notion of continuous evolution algebra in terms of differentiable matrix-valued functions, to then study those such algebras arising as solutions of ODE problems. Given their dependence on natural bases, matrix Lie groups…
The mutual influence of dynamics and structure is a central issue in complex systems. In this paper we study by simulation slow evolution of network under the feedback of a local-majority-rule opinion process. If performance-enhancing local…
In this manuscript, we investigate a fractional stochastic neutral differential equation with time delay, which includes both deterministic and stochastic components. Our primary objective is to rigorously prove the existence of a unique…
In this review I give a summary of the state-of-the-art for what concerns the chemo-dynamical numerical modelling of galaxies in general and of dwarf galaxies in particular. In particular, I focus my attention on (i) initial conditions;…
We investigate the evolution of non-linear density perturbations by taking into account the effects of deviations from spherical symmetry of a system. Starting from the standard spherical top hat model in which these effects are ignored, we…
Dispersal is a well recognized driver of ecological and evolutionary dynamics, and simultaneously an evolving trait. Dispersal evolution has traditionally been studied in single-species metapopulations so that it remains unclear how…
We study the combined effects of periodically varying carrying capacity and survival rates on the fish population in the ocean (sea). We introduce the Getz type delay differential equation model with a control parameter which describes how…