Multidimensional Systems Research Articles

A partial order toolbox for building synthetic indicators of sustainability with ordinal data

This paper designs and implements a statistical toolbox for the construction of synthetic indicators of sustainability and rankings of multidimensional systems of ordinal variables. The toolbox employs results from partial order theory to provide a purely ordinal way to perform the classical steps of indicator construction (i.e., with neither quantification nor aggregation of the input variables) and thus fills a gap in the statistical literature. “Ordinal” non-aggregative procedures are developed to score statistical units and assess the reliability of the final rankings as well as account for exogenous information relating to variable importance. The toolbox is introduced and shown in action through the real example of Sustainable Development Goal 16 (Peace, Justice, and Strong Institutions), specifically as it relates to Political Pluralism and Participation in North and South American countries.

Diagonalize three-dimensional nonlinear chaotic map to encrypt color image

In this paper, multidimensional chaos systems called a trilinear chaotic system for encryption the color image is used. Combining the multi-dimensional chaotic system with encryption algorithms improves the security. The trilinear system is built from logistic and cubic maps and generates six completely random bijections T1x,T2x,T2y,T3x,T3y, and T3z. The map T1x randomly shuffles the image pixels, the two maps T2x and T2y scramble the position of the pixels, and the other three maps change the pixel values. The correlations among the three components of RGB components are reduced, and the security of the algorithm increases. The simulation results demonstrated that the procedure has very large key spaces and a high level of security.

Novel Girsanov correction based Milstein schemes for analysis of nonlinear multi-dimensional stochastic dynamical systems

This work proposes Girsanov corrected explicit, semi-implicit, and implicit Milstein approximations for the solution of nonlinear stochastic differential equations. The solution trajectories provided by the Milstein schemes are corrected by employing the change of measures, aimed at removing the error associated with the diffusion process incurred due to the transformation between two probability measures. The change of measures invoked in the Milstein schemes ensures that the solution from the mapping is measurable with respect to the filtration generated by the error process. The proposed scheme incorporates the error between the approximated mapping and the exact representation as an innovation that is accounted in the Milstein trajectories as an additive term. Numerical demonstrations using parametrically and non-parametrically excited stochastic oscillators, a practical problem involving ring type gyroscope, and a 7-degree-of-freedom nonlinear structural system subjected to both stationary and non-stationary excitation demonstrate the improvement in the solution accuracy for the proposed schemes with much coarser time steps when compared with the classical Milstein approximation with finer time steps.

The Geometry of Misinformation: Embedding Twitter Networks of Users Who Spread Fake News in Geometrical Opinion Spaces

To understand why internet users spread fake news online, many studies have focused on individual drivers, such as cognitive skills, media literacy, or demographics. Recent findings have also shown the role of complex socio-political dynamics, highlighting that political polarization and ideologies are closely linked to a propensity to participate in the dissemination of fake news. Most of the existing empirical studies have focused on the US example by exploiting the self-reported or solicited positioning of users on a dichotomous scale opposing liberals with conservatives. Yet, left-right polarization alone is insufficient to study socio-political dynamics when considering non binary and multi-dimensional party systems, in which relevant ideological stances must be characterized in additional dimensions, relating for example to opposition to elites, government, political parties or mainstream media. In this article we leverage ideological embeddings of Twitter networks in France in multi-dimensional opinions spaces, where dimensions stand for attitudes towards different issues, and we trace the positions of users who shared articles that were rated as misinformation by fact-checkers. In multi-dimensional settings, and in contrast with the US, opinion dimensions capturing attitudes towards elites are more predictive of whether a user shares misinformation. Most users sharing misinformation hold salient anti-elite sentiments and, among them, more so those with radical left- and right-leaning stances. Our results reinforce the importance of enriching one-dimensional left-right analyses, showing that other ideological dimensions, such as anti-elite sentiment, are critical when characterizing users who spread fake news. This lends support to emerging accounts of social drivers of misinformation through political polarization, but also stresses the role of the entanglement between fake news, anti-elite polarization, and the role of scientific authorities in public debate.

Differential Effects of Multidimensional Review Evaluations on Product Sales for Mainstream vs. Niche Products

Despite a large body of literature on online reviews, none have considered the nuanced impacts of how multidimensional reviews affect product sales differently for mainstream vs. niche products. This study seeks to fill this knowledge gap by conducting complementary studies in two product categories (i.e., automobiles and laptops) with different methods (a field study and three lab experiments). Our paper reveals three key insights into the emerging literature and phenomenon on multidimensional review systems: (1) the interdimensional rating variance is more negatively related to product sales for mainstream products than for niche products in the same category, (2) the intradimensional rating valence on the dominant dimension of a product is more positively related to product sales for niche products than for mainstream products, and (3) the intradimensional rating variance on the dominant dimension of a product is more negatively related to product sales for niche products than for mainstream products. Our research provides important managerial implications for both product providers and review platforms.

Estimation of diffusion time with the Shannon entropy approach.

The present work revisits and improves the Shannon entropy approach when applied to the estimation of an instability timescale for chaotic diffusion in multidimensional Hamiltonian systems. This formulation has already been proved efficient in deriving the diffusion timescale in 4D symplectic maps and planetary systems, when the diffusion proceeds along the chaotic layers of the resonance's web. Herein the technique is used to estimate the diffusion rate in the Arnold model, i.e., of the motion along the homoclinic tangle of the so-called guiding resonance for several values of the perturbation parameter such that the overlap of resonances is almost negligible. Thus differently from the previous studies, the focus is fixed on deriving a local timescale related to the speed of an Arnold diffusion-like process. The comparison of the current estimates with determinations of the diffusion time obtained by straightforward numerical integration of the equationsof motion reveals a quite good agreement.

Improving the Feasibility of Moment-Based Safety Analysis for Stochastic Dynamics

Given a dynamical system modelled via stochastic differential equations (SDEs), we evaluate the safety of the system through its exit time moments. Using appropriate semidefinite positive matrix constraints, an SDP moment-based approach can be used to compute moments of the exit time. However, the approach is impeded when analyzing higher dimensional physical systems as the dynamics are limited to polynomials. Computational scalability is also poor as the dimensionality of the state grows, largely due to the combinatorial growth of the optimization program. We propose methods to make feasible the safety analysis of higher dimensional physical systems. The restriction to polynomial dynamics is lifted by using state augmentation which allows one to generate the optimization for a broader class of nonlinear stochastic systems. We then reformulate the constraints to mitigate the computational limitations associated with an increase in state dimensionality. We employ our methods on two example processes to characterize their safety via exit times and show the ability to handle multi-dimensional systems that were previously unsupported by the existing SDP method of moments.

TO ASSESS THE STRUCTURAL AND TECHNOLOGICAL CHANGES IN THE PARAMETERS OF THE PRECISION ROTARY SYSTEM ACCORDING TO THE SPECIFIED FUNCTIONS OF SENSITIVITY AND DEVIATION OF THE SPECTRUM OF OWN FREQUENCIES

The question of precision rotary systems with artificial elements of pretension, which significantly affect the rigidity and displacement of resonance frequencies and, as a result, lead to the appearance of very conditioned sensitivity matrices, has been investigated. The mathematical model of the system is identified. With spatial oscillations of the system, two of the most typical and important are defined: fluctuations in vibration force along the rotor axis and in a direction perpendicular to it. Designed diagnostic operator as a complete sensitivity matrix. The proposed algorithm for the diagnosis of parameters of multidimensional mechanical fluctuation systems using computer technology. In any prefabricated design, the parameters of the system during assembly may differ from the design values and change during operation. In order to predict the consequences of this, it is necessary to know the reaction of the system to these changes. At the same time, in many cases, it is almost impossible to assess experimentally the effect of parameters on the behavior of the system. Therefore, there is a need to study and analyze the effect of changes in parameters on the properties of the system analytically, that is, according to the known mathematical model of the system. Analytical study of the dynamics of such systems is a very difficult task, but determining the sensitivity functions for this type of compound experimentally is another much more complex technological task, so their characteristics should be obtained by different methods of identification. The main purpose of this work is to determine: the identified mathematical model of the system, to offer a methodology and algorithm for the diagnosis of parameters of multidimensional mechanical fluctuation systems using computer technology. Of course, the presented algorithm is most suitable for the diagnosis of parameters of multidimensional fluctuation systems, but of course, in this case, widespread use of modern computer technology is necessary. If in the system it is necessary to determine the degree of influence of various physical parameters on the vector of the state, then it is necessary to consider the matrix of relative sensitivity. Sensitivity matrices allow determining sensitive and invariant parameters to the state vector. This information answers the question of which parameters determine the vibrational characteristics of the object to the greatest extent. In addition, the determination of the sensitivity matrix can significantly simplify the dynamic model of the system, leaving only those parameters that most determine the vibrational state of the system. Keywords: construction, press-threaded connection, dynamic parameters, structural model, deformation, deviation, frequency and amplitude of oscillations, sensitivity of dynamic parameters, algorithm, sensitivity matrix, methodology.

Rate-induced tipping: thresholds, edge states and connecting orbits

Rate-induced tipping (R-tipping) occurs when time-variation of input parameters of a dynamical system interacts with system timescales to give genuine nonautonomous instabilities. Such instabilities appear as the input varies at some critical rates and cannot, in general, be understood in terms of autonomous bifurcations in the frozen system with a fixed-in-time input. This paper develops an accessible mathematical framework for R-tipping in multidimensional nonautonomous dynamical systems with an autonomous future limit. We focus on R-tipping via loss of tracking of base attractors that are equilibria in the frozen system, due to crossing what we call regular R-tipping thresholds. These thresholds are anchored at infinity by regular R-tipping edge states: compact normally hyperbolic invariant sets of the autonomous future limit system that have one unstable direction, orientable stable manifold, and lie on a basin boundary. We define R-tipping and critical rates for the nonautonomous system in terms of special solutions that limit to a compact invariant set of the autonomous future limit system that is not an attractor. We focus on the case when the limit set is a regular edge state, introduce the concept of edge tails, and rigorously classify R-tipping into reversible, irreversible, and degenerate cases. The central idea is to use the autonomous dynamics of the future limit system to analyse R-tipping in the nonautonomous system. We compactify the original nonautonomous system to include the limiting autonomous dynamics. Considering regular R-tipping edge states that are equilibria allows us to prove two results. First, we give sufficient conditions for the occurrence of R-tipping in terms of easily testable properties of the frozen system and input variation. Second, we give necessary and sufficient conditions for the occurrence of reversible and irreversible R-tipping in terms of computationally verifiable (heteroclinic) connections to regular R-tipping edge states in the autonomous compactified system.

Quantum Simulations of Interacting Systems with Broken Time-Reversal Symmetry

Many-body systems of quantum interacting particles in which time-reversal symmetry is broken give rise to a variety of rich collective behaviors and are, therefore, a major target of research in modern physics. Quantum simulators can potentially be used to explore and understand such systems, which are often beyond the computational reach of classical simulation. Of these, platforms with universal quantum control can experimentally access a wide range of physical properties. However, simultaneously achieving strong programmable interactions, strong time-reversal symmetry breaking, and high-fidelity quantum control in a scalable manner is challenging. Here, we realize quantum simulations of interacting, time-reversal-broken quantum systems in a universal trapped-ion quantum processor. Using a recently proposed, scalable scheme, we implement time-reversal-breaking synthetic gauge fields, shown for the first time in a trapped-ion chain, along with unique coupling geometries, potentially extendable to simulation of multidimensional systems. Our high-fidelity single-site resolution in control and measurement, along with highly programmable interactions, allow us to perform full state tomography of a ground state showcasing persistent current and to observe dynamics of a time-reversal-broken system with nontrivial interactions. Our results open a path toward simulation of time-reversal-broken many-body systems with a wide range of features and coupling geometries.4 MoreReceived 4 July 2022Accepted 4 April 2023DOI:https://doi.org/10.1103/PhysRevX.13.021021Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasQuantum simulationQuantum verificationPhysical SystemsTrapped ionsAtomic, Molecular & OpticalQuantum Information

Multidimensional mobile mapping and integrated approach for the digitalisation of underground transport infrastructure

The tunnel industry has started focusing on the maintenance and management challenges of an existing infrastructure. It is an urgent matter in industrialised countries, where the stakeholders’ attention is increasing at a fast pace considering the incidents and the disruptions caused by improper monitoring and maintenance. This paper presents an innovative methodology to survey and inspect existing railway tunnels through multi-dimensional mobile mapping systems. The proposed approach belongs to the digital strategies for infrastructure maintenance. An integrated multidimensional survey system (ARCHITA) allows for collecting information necessary for the diagnostics of a structure with non-destructive tests. Linear cameras, thermographic cameras, and ground-penetrating radars acquire data to be digitalised and manipulated in different IT environments. The results, in terms of the collected data on structural defects, allow for a new approach for the Management and Identification of the Risk for Existing Tunnels (MIRET). The innovative approach aims at a smart integration of information and models for the Facility Management of the transport system. The workflow for the digitalisation and diagnosis from mobile mapping data has been implemented on two 40km-long metro tunnels.

POLE PLACEMENT OF LINEAR MULTIDIMENSIONAL SYSTEMS USING UNLIMITED RANK OUTPUT FEEDBACK

The problem of placing poles in linear multidimensional systems using a feedback matrix of unlimited rank is considered. The effect of output feedback of indefinite rank of the characteristic polynomial of a system with several variables is studied. The results are then used to obtain a recurrent pole placement algorithm without any restrictions on the rank of the output feedback matrix used. The algorithm is based on the pseudo-inverse concept of obtaining solutions to systems of linear equations using the least squares method and is computationally efficient.

Optimal water allocation integrated with water supply, replenishment, and spill in the in-series reservoir based on an improved decomposition and dynamic programming aggregation method

Abstract In humid regions with the monsoon climate, seasonal water shortages and water spills occur alternately because of uneven temporal and spatial distributions of water resources. An optimization model for the in-series reservoir (ISR) with replenishment pumping stations was developed to obtain the minimum annual sum of water shortage and systematically considered the reservoir operation rule of water spill and replenishment. This model features multiple dimensions; dynamic programming (DP) may cause a ‘curse of dimensions’, while the decomposition-coordination method has difficulty in judging logic conditions in the reservoir operation rules. So, an improved decomposition and DP aggregation (DDPA) method was proposed. The proposed model and the method were applied to a real case in the humid region of southern China. Compared to a conventional scheduling method, the water supply was increased by 0.8% and replenishment was reduced by 2.5%. Moreover, a comparison between DDPA and six heuristic algorithms was discussed. All heuristic algorithms' objective function values only obtained local optimal solutions, and the water shortage of the system was 0.12–20.5%. The obtained results demonstrated that DDPA was the better choice for highly complex multi-reservoir systems. The proposed optimization algorithm enriched the optimization theory of multi-dimensional and multi-variable complex systems.

Development of a Control System for a Technological Line for the Production of Products from Foamed Polysty

Purpose of research. When designing new and modernizing existing control systems for the polymer drawing process, an important role is played by solving the problems of creating mathematical or simulation models and synthesizing control algorithms that reduce the share of manual labor. The article provides an overview of approaches to the development and modeling of automated control systems for extrusion production lines for the production of polymer products. The possibility of developing and modeling a multidimensional control system for the process of stretching foamed polystyrene based on the Matlab mathematical analysis package is considered. The relevance of the research topic is related to the trend of growth in the consumption market and the implementation of management in the context of creating more complex forms of packaging from polymers.Methods. For the development of the control system and its analysis, the fundamentals of the theory of multidimensional systems in control problems, the theory of plastics processing, methods of computer simulation of control systems were applied.Results. The article presents an algorithm for controlling the molding process and its implementation in the Simulink environment. Graphs of changes in the pulling speed and winding speed, changes in the thickness and tension of the web over time are presented and analyzed. A scheme of process line automation based on the developed algorithm is presented.Conclusion. The results of testing the model show that the use of mathematical modeling of the control system for stretching the expanded polystyrene web is possible to solve the problem of increasing production efficiency on existing equipment, which will bring their characteristics closer to new production lines, leaving the mechanical part unchanged. The significance of the work is due to the fact that the use of modern control systems can improve the quality of manufactured products, reduce the consumption of raw materials. Materials and research methods can be applied in the development of projects for other installations, for example, in the production of an optical sheet.

Brief review of the development of theories of robustness, roughness and bifurcations of dynamic systems

The development issues of theories of robustness, roughness and bifurcations of dynamic systems are considered. In the modern theory of dynamic systems and automatic control systems, researches of the properties of roughness and robustness of systems are becoming more and more important. The work considers methods of research and ensuring robust stability of interval dynamic systems of both algebraic and frequency directions of robust stability. The main results of the original algebraic method of robust stability for continuous and discrete time are given. In the frequency direction of robust stability, the issues of a frequency-robust method to the analysis and synthesis of robust multidimensional control systems based on the use of the frequency condition number of the transfer matrix of the “input-output” ratio are considered. The main provisions of the theory and method of topological roughness of dynamic systems based on the concept of roughness according to Andronov-Pontryagin are presented with the introduction of a measure of roughness of systems in the form of a condition number of matrices of reduction to a diagonal (quasi-diagonal) basis at special points of phase space. Criteria for dynamic systems bifurcations are formulated. Applications of the topological roughness method to synergetic systems and chaos have been used to investigate many systems, such as Lorenz and Rössler attractors, Belousov-Jabotinsky, Chua systems, “predator-prey” and “predator-prey-food”, Hopf bifurcation, Schumpeter and Caldor economic systems, Henon mapping, and others. For research of weakly formalized and non-formalized systems, the use of the approach of analogies of theoretical-multiple topology and the abstract method to such systems is proposed. Further research suggests the development of roughness and bifurcation theories for complex nonlinear dynamical systems.

Construction, stochastization and computer study of dynamic population models “two competitors - two migration areas”

When studying deterministic and stochastic population models, the actual problems are the formalization of processes, taking into account new effects caused by the interaction of species, and the development of computer research methods. Computer research methods make it possible to analyze the trajectories of multidimensional population systems. We consider the “two competitors - two migration areas” model, which takes into account intraspecific and interspecific competition in two populations, as well as bidirectional migration of both populations. For this model, we take into account the variability of the reproduction rates of species. A formalized description of the four-dimensional model “two competitors - two migration areas” and its modifications is proposed. Using the implementation of the evolutionary algorithm, a set of parameters is obtained that ensure the coexistence of populations under conditions of competition between two species in the main area, taking into account the migration of these species. Taking into account the obtained set of parameters, a positive stationary state is found. Two-dimensional and three-dimensional projections of phase portraits are constructed. Stochastization of the model “two competitors - two migration areas” is carried out based on the method of self-consistent one-step models constructing. The Fokker-Planck equations are used to describe the structure of the model. A transition to a four-dimensional stochastic differential equation in the Langevin form is performed. To carry out numerical experiments, a specialized software package is used to construct and study stochastic models, and a computer program based on differential evolution is developed. Algorithms for generating trajectories of the Wiener process and multipoint distributions and modifications of the Runge-Kutta method are used. In the deterministic and stochastic cases, the dynamics of the trajectories of populationmigration systems is studied. A comparative analysis of deterministic and stochastic models is carried out. The results can be used in modeling of different classes of dynamic systems.

Natural Exponential and Three-Dimensional Chaotic System.

Existing chaotic system exhibits unpredictability and nonrepeatability in a deterministic nonlinear architecture, presented as a combination of definiteness and stochasticity. However, traditional two-dimensional chaotic systems cannot provide sufficient information in the dynamic motion and usually feature low sensitivity to initial system input, which makes them computationally prohibitive in accurate time series prediction and weak periodic component detection. Here, a natural exponential and three-dimensional chaotic system with higher sensitivity to initial system input conditions showing astonishing extensibility in time series prediction and image processing is proposed. The chaotic performance evaluated theoretically and experimentally by Poincare mapping, bifurcation diagram, phase space reconstruction, Lyapunov exponent, and correlation dimension provides a new perspective of nonlinear physical modeling and validation. The complexity, robustness, and consistency are studied by recursive and entropy analysis and comparison. The method improves the efficiency of time series prediction, nonlinear dynamics-related problem solving and expands the potential scope of multi-dimensional chaotic systems.

Admissible transformation approach to Roesser state‐space model realization of singular multidimensional systems

AbstractThe problem of the singular Roesser (state‐space) model realization of non‐causal multivariate transfer (function) matrices is investigated. Specifically, the notion of so‐called admissible transformation is introduced, which allows to introduce and investigate a novel invariant matrix relationship on polynomial matrices. The singular multidimensional Roesser model (SNDRM) realization is transformed to the problem of how to reduce an initial polynomial matrix associated with the given transfer matrix into the objective polynomial matrix with a specific structure. Construction procedures are developed to obtain SNDRMs by applying admissible transformations on the initial polynomial matrix. Moreover, a separated standard form is introduced for an SNDRM so that the proposed approach can effectively treat both the left and right matrix fractional descriptions of a non‐causal multivariate transfer matrix in a unified manner. Non‐trivial examples are provided to illustrate the details and effectiveness of the proposed approach.

Engineering isospectrality in multidimensional photonic systems

Abstract Selective manipulation of energy levels plays an essential role in realizing multichannel wave devices. One of the representative examples is to utilize the concept of quasi-isospectrality: a family of wave systems with an almost identical spectrum except for a part of energy levels. Most approaches toward quasi-isospectrality have employed analytical methods based on symmetry or tridiagonalization, such as supersymmetry, Householder, or Lanczos transformations. Although such analytical approaches provide deterministic and stable designs based on operator factorizations, the mathematical strictness in the factorizations, at the same time, hinders isospectral engineering in a given multidimension. Here we develop the semi-analytical method for engineering isospectrality in multidimensional photonic systems. The method provides the systematic perturbation for the target energy level shifts by decomposing the allowed form of system changes into the perturbation basis. We demonstrate the isospectrality of lower-, higher-, and random-order states while imposing the designed shifts on the other states. The stability analysis shows that the accuracy of the method is determined by the ranges of isospectral state numbers and perturbation strength. The systematic, free-form, and multidimensional natures of the proposed method show great potential for the platform-transparent design of multichannel devices.

Novel methods for reliability study of multi-dimensional non-linear dynamic systems

This research presents two unique techniques for engineering system reliability analysis of multi-dimensional non-linear dynamic structures. First, the structural reliability technique works best for multi-dimensional structural responses that have been either numerically simulated or measured over a long enough length to produce an ergodic time series. Second, a novel extreme value prediction method that can be used in various engineering applications is proposed. In contrast to those currently used in engineering reliability methodologies, the novel method is easy to use, and even a limited amount of data can still be used to obtain robust system failure estimates. As demonstrated in this work, proposed methods also provide accurate confidence bands for system failure levels in the case of real-life measured structural response. Additionally, traditional reliability approaches that deal with time series do not have the benefit of being able to handle a system's high dimensionality and cross-correlation across several dimensions readily. Container ship that experiences significant deck panel pressures and high roll angles when travelling in bad weather was selected as the example for this study. The main concern for ship transportation is the potential loss of cargo owing to violent movements. Simulating such a situation is difficult since waves and ship motions are non-stationary and complicatedly non-linear. Extreme movements greatly enhance the role of nonlinearities, activating effects of second and higher order. Furthermore, laboratory testing may also be called into doubt due to the scale and the choice of the sea state. Therefore, data collected from actual ships during difficult weather journeys offer a unique perspective on the statistics of ship movements. This work aims to benchmark state-of-the-art methods, making it possible to extract necessary information about the extreme response from available on-board measured time histories. Both suggested methods can be used in combination, making them attractive and ready to use for engineers. Methods proposed in this paper open up possibilities to predict simply yet efficiently system failure probability for non-linear multi-dimensional dynamic structure.