Narrow Results

We develop a general approach to setting up and studying classes of quantum dynamical systems close to and structurally similar to systems having specified properties, in particular detailed balance. This is done in terms of transport plans and Wasserstein distances between systems on possibly different observable algebras.

Model reduction aims to simplify complex models by decreasing the number of equations, variables, or parameters while preserving key characteristics. This approach enhances accessibility, comprehensibility, and computational efficiency, enabling a more focused analysis of relevant variables. In this study, we describe the reduction process of a population model that incorporates cancer cell differentiation and its interaction with the immune system, maintaining the fundamental dynamics and evolution of the original model. This led to a substantial reduction in variables and parameters, creating a more efficient model suitable for computational simulations, mathematical analysis, and quantitative understanding of population dynamics. Additionally, we performed a global sensitivity analysis of model parameters using the Sobol and eFast methods, revealing insights into differences and similarities in results from a biological perspective. Our findings emphasize the critical importance of understanding and controlling parameters related to the reproduction and death rates of differentiated cancer cells, as small variations in these parameters can have significant effects on model outcomes. This underscores the importance of thoroughly understanding these essential biological variables and processes in cancer treatment, as they have a significant impact on model outcomes and, consequently, on the development of more effective therapies.

In this paper, we examine the dynamical evolution of flat FRW cosmological model in $f(R,L_m)$ gravity theory. We consider the general form of $f(R,L_m)$ defined as $f(R,L_m)=\mathrm{\Lambda }+\frac{\alpha }{2}R+\beta L_m^n$, where $\mathrm{\Lambda }$, $\alpha$, $\beta$, $n$ are model parameters, with the matter Lagrangian given by $L_m=-p$. We investigate the model through phase plane analysis, actively studying the evolution of cosmological solutions using dynamical systems techniques. To analyze the evolution equations, we introduce suitable transformations of variables and discuss the corresponding solutions by phase-plane analysis. The nature of critical points is analyzed and stable attractors are examined for $f(R,L_m)$ gravity cosmological model. We examine the linear and classical stabilities of the model and discuss it in detail. Further, we investigate the transition stage of the Universe, i.e. from the early decelerating stage to the present accelerating phase of the Universe by evolution of the effective equation of state, $\{r,s\}$ parameters and statefinder diagnostics for the central values of parameters $\alpha ,\beta$ and $n$ constrained using MCMC technique with cosmic chronometer data.

The objective of the this paper is to study the generalized $\alpha$-Tsallis’s entropy on sequential effect algebras (SEA). The generalized $\alpha$-Tsallis’s entropy of the partition in SEA of order $R$, where $R\in {{ℛ}}^{+},R\ne 1$, and its conditional version are investigated with suitable examples. We also examine properties of these entropies. The sub-additive property of Tasallis entropy of has also been achieved. The dynamical system of SEA and its generalized $\alpha$-Tsallis’s entropy for $R>1$, has been defined on SEA and it is achieved that the generalized $\alpha$-Tsallis’s entropy of SEA dynamical system is invariant under the isomorphism.

State explosion is a fundamental problem in the analysis and synthesis of discrete event systems. Continuous Petri nets can be seen as a relaxation of the corresponding discrete model. The expected gains are twofold: improvements in complexity and in decidability. In the case of autonomous nets we prove that liveness or deadlock-freeness remain decidable and can be checked more efficiently than in Petri nets. Then we introduce time in the model which now behaves as a dynamical system driven by differential equations and we study it w.r.t. expressiveness and decidability issues. On the one hand, we prove that this model is equivalent to timed differential Petri nets which are a slight extension of systems driven by linear differential equations (LDE). On the other hand, (contrary to the systems driven by LDEs) we show that continuous timed Petri nets are able to simulate Turing machines and thus that basic properties become undecidable.

MP systems are a class of P systems introduced for modeling metabolic processes. Here a regression method is presented for deducing a MP system exhibiting the dynamics of an observed metabolic system. In the procedure here described the knowledge of the stoichiometry of the system is combined with the log-gain principle of MP systems and is integrated with the Least Square Estimation method and with the stepwise regression approximation.

Recurrent Neural Networks (RNN) have been developed for a better understanding and analysis of open dynamical systems. Still the question often arises if RNN are able to map every open dynamical system, which would be desirable for a broad spectrum of applications. In this article we give a proof for the universal approximation ability of RNN in state space model form and even extend it to Error Correction and Normalized Recurrent Neural Networks.

We study topological properties of attracting sets for automorphisms of ℂ^k. Our main result is that a generic volume preserving automorphism has a hyperbolic fixed point with a dense stable manifold. On the other hand, we show that an attracting set can only contain a neighborhood of the fixed point if it is an attracting fixed point. We will see that the latter does not hold in the non-autonomous setting.

It has been recently discovered that in smooth unfoldings of maps with a rank-one homoclinic tangency there are codimension two laminations of maps with infinitely many sinks. Indeed, these laminations, called Newhouse laminations, occur also in the holomorphic context. In the space of polynomials of ${ℂ}^2$, with bounded degree, there are Newhouse laminations.

We compute all relative dynamical degrees of equivariant dominant rational maps on toric varieties. We use the intersection theory on toric varieties using Minkowski weights. As a result, we see that the relative dynamical degrees can be reduced to the dynamical degrees on toric varieties. Hence these are all algebraic integers.

We introduce a type of zero-dimensional dynamical system (a pair consisting of a totally disconnected compact metrizable space along with a homeomorphism of that space), which we call “fiberwise essentially minimal”, that is a class that includes essentially minimal systems and systems in which every orbit is minimal. We prove that the associated crossed product $C^{\ast }$-algebra of such a system is an A$𝕋$-algebra. Under the additional assumption that the system has no periodic points, we prove that the associated crossed product $C^{\ast }$-algebra has real rank zero, which tells us that such $C^{\ast }$-algebras are classifiable by $K$-theory. The associated crossed product $C^{\ast }$-algebras to these nontrivial examples are of particular interest because they are non-simple (unlike in the minimal case).

In this paper, we revisit the Mackey–Glass model for blood-forming process which was proposed to describe the spontaneous fluctuations of the blood cell counts in normal individuals and the first stage of chronic myelocytic (or granylocytic) leukemia (CML). We obtain the bifurcation diagram as a function of the time delay parameter and show that the onset of leukemia is related to instabilities associated to the presence of periodic windows in the midst of a chaotic regime. We also introduce a very simple modification in the death rate parameter in order to simulate the accumulation of cells and the progressive increase of the minima counts experimentally observed in the final stage of the disease in CML patients. The bifurcation diagram as a function of the death rate parameter is also obtained and we discuss the effects of treatments like leukapheresis.

The problem of knowing and characterizing the transitive behavior of a given cellular automaton is a very interesting topic. This paper provides a matrix representation of the global dynamics in reversible one-dimensional cellular automata with a Welch index 1, i.e., those where the ancestors differ just at one end. We prove that the transitive closure of this matrix shows diverse types of transitive behaviors in these systems. Part of the theorems in this paper are reductions of well-known results in symbolic dynamics. This matrix and its transitive closure were computationally implemented, and some examples are presented.

A simple two-dimensional system is introduced which suggests a qualitative dynamical relationship between (1) stock market prices in the presence of nonlinear trend-followers and nonlinear value investors, (2) the world human population with a competition between a population-dependent growth rate and a nonlinear dependence on a finite carrying capacity and (3) the failure of materials subjected to a time-varying stress with a competition between positive geometrical feedback on the damage variable and nonlinear healing. Our model keeps three key ingredients (inertia, nonlinear positive and negative feedbacks) that compete to give rise to singularities in finite time decorated by accelerating oscillations.

Science and technology have a fundamental role in the economic development. Although this statement is generally well accepted, the internal mechanisms which are responsible for these interactions are not clear. In the last decade, dealing with this problem, many models have been proposed. In this paper, we introduce a model that creates an artificial world economy that is a network of countries. Each country has its own national system of innovation and the interactions between countries are given by functions that connect the competitiveness of their prices and their technological capabilities. Starting from different configurations, the artificial world economy self-organizes itself and creates a hierarchies of countries.

In this paper, we present a mechanism of generation of a class of switched dynamical system without equilibrium points that generates a chaotic attractor. The switched dynamical systems are based on piecewise linear (PWL) systems. The theoretical results are formally given through a theorem and corollary which give necessary and sufficient conditions to guarantee that a linear affine dynamical system has no equilibria. Numerical results are in accordance with the theory.

Using a recently proposed directed random walk test, we systematically investigate the popular random number generator RANLUX developed by Lüscher and implemented by James. We confirm the good quality of this generator with the recommended luxury level. At a smaller luxury level (for instance equal to 1) resonances are observed in the random walk test. We also find that the lagged Fibonacci and Subtract-with-Carry recipes exhibit similar failures in the random walk test. A revised analysis of the corresponding dynamical systems leads to the observation of resonances in the eigenvalues of Jacobi matrix.

It is the purpose of the present article to show that so-called network models, originally designed to describe static properties of disordered electronic systems, can be easily generalized to quantum-dynamical models, which then allow for an investigation of dynamical and spectral aspects. This concept is exemplified by the Chalker–Coddington model for the quantum Hall effect and a three-dimensional generalization of it. We simulate phase coherent diffusion of wave packets and consider spatial and spectral correlations of network eigenstates as well as the distribution of (quasi-)energy levels. Apart from that, it is demonstrated how network models can be used to determine two-point conductances. Our numerical calculations for the three-dimensional model at the Metal-Insulator transition point delivers, among others, an anomalous diffusion exponent of η=3-D₂=1.7±0.1. The methods presented here in detail have been used partially in earlier work.

The problem of separation of different signals in the Cosmic Microwave Background (CMB) radiation using the difference in their statistics is analyzed. Considering samples of sequences which model the CMB as a superposition of signals, we show how the Kolmogorov stochasticity parameter acts as a relevant descriptor, either qualitatively or quantitatively, to distinguish the statistical properties of the cosmological and secondary signals.

The inflaton coupling to a vector field via the $f{(\phi )}^2{F}_{\mu \nu }{F}^{\mu \nu }$ term is used in several contexts in the literature, such as to generate primordial magnetic fields, to produce statistically anisotropic curvature perturbation, to support anisotropic inflation, and to circumvent the $\eta$-problem. In this work, I perform dynamical analysis of this system allowing for the most general Bianchi I initial conditions. I also confirm the stability of attractor fixed points along phase–space directions that had not been investigated before.