Narrow Results

We consider a generalization of the classical results obtained by Freidlin and Wentzell to consider the case of time-dependent dissipative drifts. We show the convergence of diffusions with multiplicative noise in the zero limit of a diffusivity parameter to the related dynamical systems. We also derive the solution to the associated transport equation as an application.

The stochastic dynamical ${\rho }^4$ equation is utilized as a robust framework for modeling the behavior of complex systems characterized by randomness and nonlinearity, with applications spanning various scientific fields. The aim of this paper is to employ an analytical method to identify stochastic traveling wave solutions of the dynamical ${\rho }^4$ equation. Novel hyperbolic and rational functions are investigated through this method. A Galilean transformation is applied to reformulate the model into a planar dynamical system, which enables a comprehensive qualitative analysis. Additionally, the emergence of chaotic and quasi-periodic patterns following the introduction of a perturbation term is addressed. Simulation results indicate that significant changes in the systems’ dynamic behavior are caused by adjusting the amplitude and frequency parameters. Our findings indicate the impact of the method on system dynamics and its efficacy in analyzing solitons and phase behavior in nonlinear models. These discoveries provide fresh perspectives on how the suggested method can lead to notable shifts in the systems’ dynamic behavior. The effectiveness and practicality of the proposed methodology in scrutinizing soliton solutions and phase visualizations across diverse nonlinear models are underscored by these revelations.

A Universe containing uniform magnetic fields, strings, or domain walls is shown to have an ellipsoidal expansion. This case has motivations from observational cosmology especially the anomaly concerning the low quadrupole amplitude compared to the best-fit $\mathrm{\Lambda }$CDM prediction in Planck data. It is shown that a Universe with eccentricity at decoupling of order $10^{-2}$ can reduce the quadrupole amplitude without affecting higher multipoles of the angular power spectrum of the temperature anisotropy. We study the evolution of ellipsoidal Universes using dynamical system techniques for the first time. The determined critical points vary between saddle and past attractors depending on dark energy state equation parameter $w_{\mathrm{\Lambda }}$, with no future attractors. Important results are shown with numerical integrations of this dynamical system done using several initial conditions. For instance, a tendency for high expansion differences between planar and perpendicular axes is observed which contradicts previous assumption on the evolution behavior of ellipsoid Universes. Considering dark energy as a Chaplygin gas solves this contradiction by controlling cosmic shear evolution.

Dynamical probabilistic P systems are discrete, stochastic, and parallel devices, where the probability values associated with the rules change during the evolution of the system. These systems are proposed as a novel approach to the analysis and simulation of the behavior of complex systems. We introduce all necessary definitions of these systems and of their dynamical aspects, we describe the functioning of the parallel and stochastic algorithm used in computer simulation, and evaluate its time complexity. Finally, we show some applications of dynamical probabilistic P systems for the investigation of the dynamics of the Lotka-Volterra system and of metapopulation systems.

Reaction networks, or equivalently Petri nets, are a general framework for describing processes in which entities of various kinds interact and turn into other entities. In chemistry, where the reactions are assigned ‘rate constants’, any reaction network gives rise to a nonlinear dynamical system called its ‘rate equation’. Here we generalize these ideas to ‘open’ reaction networks, which allow entities to flow in and out at certain designated inputs and outputs. We treat open reaction networks as morphisms in a category. Composing two such morphisms connects the outputs of the first to the inputs of the second. We construct a functor sending any open reaction network to its corresponding ‘open dynamical system’. This provides a compositional framework for studying the dynamics of reaction networks. We then turn to statics: that is, steady state solutions of open dynamical systems. We construct a ‘black-boxing’ functor that sends any open dynamical system to the relation that it imposes between input and output variables in steady states. This extends our earlier work on black-boxing for Markov processes.

In this paper, we describe the commutant of an arbitrary subalgebra A of the algebra of functions on a set X in a crossed product of A with the integers, where the latter act on A by a composition automorphism defined via a bijection of X. The resulting conditions which are necessary and sufficient for A to be maximal abelian in the crossed product are subsequently applied to situations where these conditions can be shown to be equivalent to a condition in topological dynamics. As a further step, using the Gelfand transform, we obtain for a commutative completely regular semi-simple Banach algebra a topological dynamical condition on its character space which is equivalent to the algebra being maximal abelian in a crossed product with the integers.

Landstad–Vaes theory deals with the structure of the crossed product of a ${\mathrm{\text{C}}}^{\ast }$-algebra by an action of locally compact (quantum) group. In particular, it describes the position of original algebra inside crossed product. The problem was solved in 1979 by Landstad for locally compact groups and in 2005 by Vaes for regular locally compact quantum groups. To extend the result to non-regular groups we modify the notion of $G$-dynamical system introducing the concept of weak action of quantum groups on ${\mathrm{\text{C}}}^{\ast }$-algebras. It is still possible to define crossed product (by weak action) and characterize the position of original algebra inside the crossed product. The crossed product is unique up to an isomorphism. At the end we discuss a few applications.

We consider compact group actions on C*- and W*-algebras. We prove results that relate the duality property of the action (as defined in the Introduction) with other relevant properties of the system such as the relative commutant of the fixed point algebras being trivial (called the irreducibility of the inclusion) and also to the Galois correspondence between invariant C*-subalgebras containing the fixed point algebra and the class of closed normal subgroups of the compact group.

An action of a compact, in particular finite group on a C*-algebra is called properly outer if no automorphism of the group that is distinct from identity is implemented by a unitary element of the algebra of local multipliers of the C*-algebra. In this paper, I define the notion of strictly outer action (similar to the definition for von Neumann factors in [S. Vaes, The unitary implementation of a locally compact group action, J. Funct. Anal. 180 (2001) 426–480]) and prove that for finite groups and prime C*-algebras, it is equivalent to the proper outerness of the action. For finite abelian groups this is equivalent to other relevant properties of the action.

A causal, stochastic model of networked computers, based on information theory and nonequilibrium dynamical systems is presented. This provides a simple explanation for recent experimental results revealing the structure of information in network transactions. The model is based on non-Poissonian stochastic variables and pseudo-periodic functions. It explains the measured patterns seen in resource variables on computers in network communities. Weakly non-Poissonian behavior can be eliminated by a conformal scaling transformation and leads to a mapping onto statistical field theory. From this, it is possible to calculate the exact profile of the spectrum of fluctuations. This work has applications to anomaly detection and time-series analysis of computer transactions.

The aim of this paper is to present algorithmic specifications of institutions as an alternative to the pervasive gradient institutions used in mainstream economics. A framework to evaluate the performance of the institutions is proposed using instruments from landscape theory. The problem of how a simple market allocates surplus is studied as an example of application of the framework.

This work deals with a discussion of complex dynamics of the elementary cellular automaton rule 54. An equation which shows some degree of self-similarity is obtained. It is shown that rule 54 exhibits Bernoulli shift and is topologically mixing on its closed invariant subsystem. Finally, many complex Bernoulli shifts are explored for the finite symbolic sequences with periodic boundary conditions.

Epidemics usually spread widely and can cause a great deal of loss to humans. In the real world, vaccination is the principal method for suppressing the spread of infectious diseases. The Susceptible-Infected-Susceptible (SIS) model suggests that voluntary vaccination may affect the spread of an epidemic. Most studies to date have argued that the infection rates of nodes in the SIS model are not heterogeneous. However, in reality, there exist differences in the neighbor network structure and the number of contacts, which may affect the spread of infectious diseases in society. As a consequence, it can be reasonably assumed that the infection rate of the nodes is heterogeneous because of the amount of contact among people. Here, we propose an improved SIS model with heterogeneity in infection rates, proportional to the degree of nodes. By conducting simulations, we illustrate that almost all vaccinated nodes have high degrees when the infection rate is positively correlated with the degree of a node. These vaccinated nodes can divide the whole network into many connected sub-graphs, which significantly slows down the propagation of an epidemic; the heterogeneity of infection rates has a strong inhibitory effect on epidemic transmission. On the other hand, when the infection rate is negatively related to the degrees of the infection rate nodes, it is difficult for most nodes to meet the inoculation conditions, and the number of inoculations is close to zero.

In this paper, we work on the fundamental collocation strategy using the moved Vieta–Lucas polynomials type (SVLPT). A numeral method is used for unwinding the nonlinear Rubella illness Tributes. The quality of the SVLPT is presented. The limited contrast system is used to understand the game plan of conditions. The mathematical model is given to attest the resolute quality and ampleness of the recommended procedure. The oddity and meaning of the outcomes are cleared utilizing a 3D plot. We examine free sickness harmony, security balance point and the presence of a consistently steady arrangement.

In this paper, we present a recurrent neural network for solving mixed linear complementarity problems (MLCPs) with positive semi-definite matrices. The proposed neural network is derived based on an NCP function and has a low complexity respect to the other existing models. In theoretical and numerical aspects, global convergence of the proposed neural network is proved. As an application, we show that the proposed neural network can be used to solve linear and convex quadratic programming problems. The validity and transient behavior of the proposed neural network are demonstrated by using five numerical examples.

A proposal to study the effect of interaction in an agegraphic dark energy model in DGP brane-world cosmology is presented in this paper. After explaining the details, we proceed to apply the dynamical system approach to the model to analyze its stability. We first constrain model parameters with a variety of independent observational data such as cosmic microwave background anisotropies, baryon acoustic oscillation peaks and observational Hubble data. Then, we obtain the critical points related to different cosmological epochs. In particular, we conclude that in the presence of interaction, dark energy dominated era could be a stable point if model parameters n and $\beta$, obey a given constraint. Also, big rip singularity is avoidable in this model.

In this paper, we extend the notion of Bekenstein–Hawking entropy for a cover of a site. We deduce a new class of discrete dynamical system on a site and we introduce the Bekenstein–Hawking entropy for each member of it. We present an upper bound for the Bekenstein–Hawking entropy of the iterations of a dynamical system. We define a conjugate relation on the set of dynamical systems on a site and we prove that the Bekenstein–Hawking entropy preserves under this relation. We also prove that the twistor correspondence preserves the Bekenstein–Hawking entropy.

Accelerating cosmic expansion is a challenging issue faced by cosmology in the present times. Modified gravity could present a promising choice in order to understand and explain it in its framework. In this context, $f(R)$ models of modified gravity look apparently the most compatible and viable scenario. In this work, we investigate the cosmic dynamics of the late times using a dynamical system approach in $f(R)$ cosmology. The properties associated with the critical points are investigated to understand the system stability by analyzing the dynamical system which describes the cosmological evolution from the perspective of the model under consideration. It is observed that the accelerated cosmic expansion following the phase of matter domination is arrived at in a particular model $f(R)=R^{p}exp(\frac{q}{R})$ under discussion. The geometric curve $m(r)$ also helps figure out some significant properties of the model upon plotting in the $(r,m)$ plane. It also assists substantially to form the dynamical system for the model in question. The analysis of system stability is carried out by finding out the critical points of the dynamical system whose dynamic characteristics are responsible for the stability of the model. It is extended afterward by considering the cosmological constant as dark energy, which proves, however, redundant from the viewpoint of modified gravity in $f(R)$ models. Two cases regarding linear and nonlinear interactions between cosmic fluids are also discussed. At some points, as the analysis shows, we see that accelerated expansion is attained by yielding a viable epoch of matter domination. The results which came out through stability analysis show that the universe is currently subject to accelerating expansion regardless of the dark energy to remain in existence.

The aim of this work is to provide a basis to interpret the dilaton as the dark matter of the universe, in the context of a particular cosmological model derived from type IIB supergravity theory with fluxes. In this theory, the dilaton is usually interpreted as a quintessence field. But, with this alternative interpretation we find that (in this supergravity model) the model gives a similar evolution and structure formation of the universe compared with the ΛCDM model in the linear regime of fluctuations of the structure formation. Some free parameters of the theory are fixed using the present cosmological observations. In the nonlinear regime there are some differences between the type IIB supergravity theory with the traditional CDM paradigm. The supergravity theory predicts the formation of galaxies earlier than the CDM and there is no density cusp in the center of galaxies. These differences can distinguish both models and might give a distinctive feature to the phenomenology of the cosmology coming from superstring theory with fluxes.

The late time accelerated expansion of the universe can be realized using scalar fields with the given self-interacting potentials. Here, we consider a straightforward approach where a three cosmic fluid mixture is assumed. The fluids are standard matter perfect fluid, dark matter, and a scalar field with the role of dark energy. A dynamical system analysis is developed in this context. A central role is played by the equation of state ${\omega }_{\mathrm{\text{eff}}}$ which determines the acceleration phase of the models. Determining the domination of a particular fluid at certain stages of the universe history by stability analysis allows, in principle, to establish the succession of the various cosmological eras.