Discontinuous Maps Research Articles

Comparative Study on Continuous and Discontinuous Dorling Maps

ABSTRACTThe Dorling map, a special type of thematic map, is a widely employed tool for visualizing geospatial statistical data. It transforms regions into circles with areas proportionate to statistical values while endeavoring to maintain regional spatial relationships. Notably, two types of Dorling maps emerge, distinguished by their consideration of regional continuity: continuous and discontinuous Dorling maps. However, as of yet, the discrepancies between these two types remain inconclusive. In this paper, we employ an eye‐tracking method to investigate the efficacy of Dorling maps in two common application scenarios, namely unpurposed browsing tasks and purposeful reading tasks. To this end, we administer tasks involving region search, attribute comparison/recognition/memory, conditional selection, relationship judgment, summary, and subjective evaluation. Subsequently, we perform a statistical analysis of the eye movement data of participants when they complete the above tasks in the continuous and discontinuous Dorling maps. The results indicate that the discontinuous Dorling maps are significantly better than the continuous ones in interpretation time for forward and reverse region search, selecting conditions, and judging adjacent relationships. Continuous Dorling maps significantly outperform discontinuous maps in terms of search efficiency during attribute comparison. Moreover, continuous maps significantly outperformed discontinuous maps in terms of cognitive supplementation or reprocessing of previous regions during conditional selection. This study can help users choose the right form of Dorling map visualization according to their needs.

Grazing-sliding bifurcation in a dry-friction oscillator on a moving belt under periodic excitation.

In this paper, we consider the grazing-sliding bifurcations in a dry-friction oscillator on a moving belt under periodic excitation. The system is a nonlinear piecewise smooth system defined in two zones whose analytical expressions of the solutions are not available. Thus, we obtain conditions of the existence of grazing-sliding orbits numerically by the shooting method. Then, we compute the lower and higher order approximations of the stroboscopic Poincaré map, respectively, near the grazing-sliding bifurcation point by the method of local zero-time discontinuity mapping. The results of computing the bifurcation diagrams obtained by the lower and higher order maps, respectively, are compared with those from direct simulations of the original system. We find that there are big differences between the lower order map and the original system, while the higher order map can effectively reduce such disagreements. By using the higher order map and numerical simulations, we find that the system undergoes very complicated dynamical behaviors near the grazing-sliding bifurcation point, such as period-adding cascades and chaos.

Remarkable results obtained when studying a question concerning the invariant set of an IFS

Some years ago, a question was raised about whether an iterated function system consisting of a finite number of Reich (or Kannan) contractions (generally discontinuous contractive mappings) has a unique compact invariant set. In this article, we give remarkable results obtained by studying previous research articles that tried to answer this question positively and previous counterexamples that demonstrate that this question is negatively answered. Our results strongly encourage the development of fractal theory because the existence, uniqueness, compactness and boundedness of invariant sets with respect to functional families are one of the main research topics in the theory of fractals.

Application of deep learning reduced-order modeling for single-phase flow in faulted porous media

AbstractOur research is positioned within the framework of subsurface resource utilization for sustainable economies. We concentrate on modeling the underground single-phase fluid flow affected by geological faults using numerical simulations. The study of such flows is characterized by strong uncertainites in the data defing the problem due to the difficulty of taking precise measurements in the subsoil. We aim to demonstrate the feasibility of a reduced order model that is both reliable and computationally efficient, thereby facilitating the incorporation of uncertainties. We account for the uncertainities of the properties of the rock and the geometry of the fault. The latter is achieved by using a radial basis function mesh deformation method. This approach benefits from a mixed-dimensional framework to model the rock matrix and faults as n and $${n-1}$$ n - 1 dimensional domains, allowing for non-conforming meshes. Our primary focus is on a reduced-order model capable of reproducing flow variables across the entire domain. We utilize the Deep Learning Reduced Order Model (DL-ROM), a nonintrusive neural network-based technique, and we compare it against the traditional Proper Orthogonal Decomposition (POD) method across various scenarios. The most relevant contributions of this work are: the proof of concept of the use of neural network for reduced order models for subsoil flow, dealing with non-affine problems and mixed dimensional domain. Additionally, we generalize an existing mesh deformation method for discontinuous deformation maps. Our analysis highlights the capability of reduced order model, highlighting DL-ROM’s capacity to expedite complex analyses with promising accuracy and efficiency, making multi-query analyses with various quantities of interest affordable.

Impact of discontinuity data acquisition methods on rockfall susceptibility assessment using high-resolution 3D point cloud

This research evaluates the influence of various discontinuity data acquisition methods on three-dimensional rockfall susceptibility assessments, employing a high-resolution 3D point cloud of the Špičunak rock slope in Croatia. Discontinuity mapping was conducted utilising manual and semi-automated methods. Manual mapping was done in open-source software CloudCompare, while semi-automated mapping was done using Discontinuity Set Extractor and two CloudCompare plug-ins, qFacet Fast Marching and qFacet Kd-Tree algorithm. Significant differences were observed between the methods in terms of the number of identified discontinuity planes, sets, mean orientation values, and weighted densities associated with specific sets. Based on the mapping results, three-dimensional rockfall susceptibility maps were generated. The 3D rockfall susceptibility assessment employed a Kinematic Hazard Index (KHI) for planar, wedge, and flexural toppling failure mechanisms, which determines the percentage of discontinuity planes that meet the geometrical conditions specific to each type of failure. The findings underscore that the mapping results greatly affect the 3D rockfall susceptibility assessment, impacting the identification of potential rockfall source areas. The study emphasizes the importance of integrating manual and semi-automated mapping for reliable 3D rockfall susceptibility assessment. Additionally, this study evaluated the effect of varying the number of discontinuity orientation input data on rockfall susceptibility by comparing nine scenarios, each involving a reduction in the number of discontinuity orientation data. The results indicate that accurate discontinuity mapping is more critical than the sheer number of mapped discontinuities for reliable rockfall susceptibility assessment. Therefore, prioritizing the precise determination of discontinuity set numbers and their associated weighted densities is essential for reliable susceptibility evaluations.

The mapping of shallow upper mantle discontinuities using teleseismic P-wave autocorrelation

The mapping of shallow upper mantle discontinuities using teleseismic P-wave autocorrelation

Geotechnical Study for Assessing Slope Stability at the Proposed Weito Dam Site in Ethiopia: Implications for Environmental Sustainability and Resilience

There is a proposed dam at Weito in Ethiopia’s Southern Nations Nationalities and People Regional State. This embankment-type dam primarily serves irrigation purposes. Weito’s proposed dam’s slope stability is the primary focus of this investigation. The objective is to assess the geological and geotechnical conditions influencing slope stability using the slope mass rating (SMR) classification system. This study examined various slope stability parameters. Uniaxial compressive strength, rock quality designation, joint condition, discontinuity spacing, joint orientation, and groundwater conditions were measured. An analysis of field data, including geological structures and lithology, was used to generate a structural discontinuity map. The slope mass rating was calculated to assess rock mass stability. The study area was examined for faults, joints, fractures, and shear zones during fieldwork. Schmidt hammer tests indicated a range of 10.5–50 MPa uniaxial compressive strength. Rock quality designation values were also within 72.5% to 95%. Additionally, the joint spacing of rocks varied from 3.95 cm to 47.5 cm. Rock mass ratings ranged from 39% to 62%. The study contributes to the understanding of the geological conditions at the Weito dam site and ensures the dam’s safe design and construction.

The use of digital technology for rock mass discontinuity mapping: review of benchmarking exercise

The use of digital technology for rock mass discontinuity mapping: review of benchmarking exercise

Bifurcation structures of a two-dimensional piecewise linear discontinuous map: analysis of a cobweb model with regime-switching expectations

We consider the bifurcations occurring in a two-dimensional piecewise-linear discontinuous map that describes the dynamics of a cobweb model in which firms rely on a regime-switching expectation rule. In three different partitions of the phase plane, separated by two discontinuity lines, the map is defined by linear functions with the same Jacobian matrix, having two real eigenvalues, one of which is negative and one equal to 0. This leads to asymptotic dynamics that can belong to two or three critical lines. We show that when the basic fixed point is attracting, it may coexist with at most three attracting cycles. We have determined their existence regions, in the two-dimensional parameter plane, bounded by border collision bifurcation curves. At parameter values for which the basic fixed point is repelling, chaotic attractors may exist - either one that is symmetric with respect to the basic fixed point, or, if not symmetric, the symmetric one also exists. The homoclinic bifurcations of repelling cycles leading to the merging of chaotic attractors are commented by using the first return map on a suitable line. Moreover, four different kinds of homoclinic bifurcations of a saddle 2-cycle, leading to divergence of the generic trajectory, are determined.

Stratified burn severity assessment by integrating spaceborne spectral and waveform attributes in Great Xing'an Mountain

Burn severity assessment is critical for understanding the pattern of post-fire vegetation recovery and ecosystem resilience. Previous studies proposed various field criteria (e.g., Composite Burn Index (CBI)) to quantify burn severity from strata level to total site level, yet suffering from surveyors' subjective interpretation across site conditions. High-resolution passive remote sensing allows for more objective assessment based on the strong relation between fire damages and spectral features. Importantly, burn severity generally characterizes differences between forest overstory and understory layers due to their discrepancies in vegetation structures and environmental conditions. Spatially explicit mapping of strata-level burn severity is vital for post-fire forest management and ecological function evaluation. Until now, almost no available spaceborne remote sensing method can concurrently assess overstory and understory burn severity over heterogeneous forests. In this study, we proposed a Hybrid Composite Burn Index (HCBI) that comprehensively indicates the fire-induced spectral and structural changes along the vertical profile by integrating spectral and waveform attributes from both active and passive remote sensing data. Firstly, we introduced two remote sensing-based rating factors namely relative spectral change (RSC) and relative waveform change (RWC), and established HCBI through weighting and scoring the rating factors. Subsequently, we evaluated the effectiveness and generality of overstory, understory, and total site HCBI based on various pairs of simulated remote sensing data of pre-fire and post-fire forest scenes. Thirdly, we derived the spatially discontinuous maps of overstory, understory, and total site HCBI of the Xiushan fire site in Great Xing'an Mountain using WorldView-2 (WV-2) multispectral imagery and Global Ecosystem Dynamics Investigation (GEDI) full-waveform LiDAR data. Finally, we produced wall-to-wall HCBI maps using multiple predictive variables from pre-fire and post-fire Sentinel-2 MultiSpectral Instrument (MSI) images with a Random Forest (RF) algorithm. The predicted overstory, understory, and total site HCBI were validated by field-surveyed CBI. The assessment results based on simulation data showed HCBI was sensitive to the fire damage regardless of burn severity levels and vegetation cover levels (R2 of larger than 0.97 and RMSE of <0.15). The prediction results based on RF models achieved reliable HCBI of the total site, overstory, and understory levels (R2 of around 0.85 and RMSE of around 0.30). We found the wall-to-wall maps of HCBI captured the subtle horizontal and vertical variation even in the case of understory burn alone. We concluded that the newly proposed HCBI can advance the remote sensing-based assessment of stratified burn severity, offering opportunities for making fine-scale forest management decisions.

On Forced Oscillations in a Relay System with Hysteresis

This paper discusses the phenomenon associated with the forced synchronization (“entrainment of a self-sustained oscillator by an external force”) in a relay system with hysteresis, which manifests itself in the occurrence of periodic motions close to the rhythmic activity of neurons, when packets of fast oscillations are interspersed with intervals of the slow dynamics. To study this phenomenon, we introduce a circle mapping, which, depending on the parameters, can be a circle diffeomorphism or discontinuous map (“gap map”). In both cases, this mapping demonstrates the so-called period-adding bifurcation structure. It is demonstrated that packets number of fast oscillations in the period of periodic motion is determined by the rotation number, and the length of the intervals between the packets may be found of the boundaries of the absorbing interval. The change in the number of pulses in the packet occurs through the border-collision bifurcation.

A human-machine interaction method for rock discontinuities mapping by three-dimensional point clouds with noises

Rock discontinuities control rock mechanical behaviors and significantly influence the stability of rock masses. However, existing discontinuity mapping algorithms are susceptible to noise, and the calculation results cannot be fed back to users timely. To address this issue, we proposed a human-machine interaction (HMI) method for discontinuity mapping. Users can help the algorithm identify the noise and make real-time result judgments and parameter adjustments. For this, a regular cube was selected to illustrate the workflows: (1) point cloud was acquired using remote sensing; (2) the HMI method was employed to select reference points and angle thresholds to detect group discontinuity; (3) individual discontinuities were extracted from the group discontinuity using a density-based cluster algorithm; and (4) the orientation of each discontinuity was measured based on a plane fitting algorithm. The method was applied to a well-studied highway road cut and a complex natural slope. The consistency of the computational results with field measurements demonstrates its good accuracy, and the average error in the dip direction and dip angle for both cases was less than 3°. Finally, the computational time of the proposed method was compared with two other popular algorithms, and the reduction in computational time by tens of times proves its high computational efficiency. This method provides geologists and geological engineers with a new idea to map rapidly and accurately rock structures under large amounts of noises or unclear features.

Regular and chaotic dynamics in a 2D discontinuous financial market model with heterogeneous traders

We develop a financial market model where three types of traders operate simultaneously: fundamentalists and chartists of two types, namely, trend followers and contrarians. The dynamics of this model is described by a two-dimensional discontinuous map defined by two linear functions, where one acts in the partition between two (parallel) discontinuity lines and the other one acts outside this partition. Our analysis shows that despite the linearity of the map components, its dynamics can be quite complex, with various, possibly coexisting attracting cycles and chaotic attractors. As a first step towards the understanding how the overall bifurcation structure observed in the parameter space of the map is organized, we obtain analytically the boundaries of periodicity regions related to the simplest attracting period-n cycles, n≥3, with one point in the middle partition and n−1 points outside it. These boundaries can be related to border-collision bifurcations (when a point of the cycle collides with a discontinuity line) as well as to degenerate bifurcations (associated with eigenvalues on the unit circle). We show also some elements of period-adding and period-incrementing bifurcation structures for which the cycles mentioned above are basic. From an economic point of view, our study confirms that a fairly simple financial market model with heterogeneous agents is able to produce complicated boom-bust dynamics typical of real financial markets.

Bifurcation, chaos and multi-stability regions in an asset pricing model with three subsystems

An asset pricing model with two types of chartists and fundamentalists and trend followers is considered, it is driven by a two-dimensional piecewise linear discontinuous map with three subsystems. There are great differences in the dynamic behaviour between expected offset (the expectations of trend followers offset the difference between the expectations of Type 1 traders in bull and bear markets) and expected non-offset. It is proven that there is no chaos in the dynamic of system with expected offset. However, chaos may exist in the dynamic of system with expected non-offset. We present a systematic approach to the problem of analysing the bifurcation phenomena associated with the appearance/disappearance of cycles, which may be related to several bifurcations. The multi-stability regions in parameter plane and related basins of multi-attractors in phase space are investigated. This paper aims to uncover the endogenous law of the unpredictability and excessive volatility in financial markets.

Discontinuity characterization of rock strata using borehole wall imagery and scanline

Rock discontinuity mapping techniques, such as scanlines on slopes or exposed vertical surfaces, can collect data on discontinuity characteristics required to assess the rock’s stability. When slope conditions are highly weathered, or it is challenging to identify discontinuity characteristics using a scanline mapping survey, other methods must be used to obtain this information. One of them employs borehole wall acoustic imaging. This method is sophisticated, fast, and accurate for getting discontinuity characteristic data, but it has a disadvantage in that it requires preliminary drilling work. This issue can be solved if it is done in conjunction with the work of exploration or geotechnical drilling. This study aims to compare the discontinuity characterization data recorded using borehole wall acoustic imaging and scanline methods by considering the advantages and disadvantages of both approaches. The characterization data is essential for assessing rock mass quality. When these two methods are combined in mining activities, they will provide added value by utilizing the right technology and ensuring high accuracy while following good mining practices.

A 2D piecewise-linear discontinuous map arising in stock market modeling: Two overlapping period-adding bifurcation structures

We consider a 2D piecewise-linear discontinuous map defined on three partitions that drives the dynamics of a stock market model. This model is a modification of our previous model associated with a map defined on two partitions. In the present paper, we add more realistic assumptions with respect to the behavior of sentiment traders. Sentiment traders optimistically buy (pessimistically sell) a certain amount of stocks when the stock market is sufficiently rising (falling); otherwise they are inactive. As a result, the action of the price adjustment is represented by a map defined by three different functions, on three different partitions. This leads, in particular, to families of attracting cycles which are new with respect to those associated with a map defined on two partitions. We illustrate how to detect analytically the periodicity regions of these cycles considering the simplest cases of rotation number 1/n,n≥3, and obtaining in explicit form the bifurcation boundaries of the corresponding regions. We show that in the parameter space, these regions form two different overlapping period-adding structures that issue from the center bifurcation line. In particular, each point of this line, associated with a rational rotation number, is an issue point for two different periodicity regions related to attracting cycles with the same rotation number but with different symbolic sequences. Since these regions overlap with each other and with the domain of a locally stable fixed point, a characteristic feature of the map is multistability, which we describe by considering the corresponding basins of attraction. Our results contribute to the development of the bifurcation theory for discontinuous maps, as well as to the understanding of the excessively volatile boom-bust nature of stock markets.

A Laboratory for the Integration of Geomatic and Geomechanical Data: The Rock Pinnacle “Campanile di Val Montanaia”

This work describes a procedure for building a high-quality 3D model of a rocky pinnacle in the Dolomites, Italy, using Structure from Motion (SfM) techniques. The pinnacle, known as “Campanile di Val Montanaia”, is challenging to survey due to its high elevation and sub-vertical cliffs. The construction of the 3D model is the first step in a multi-disciplinary approach to characterize the rock mass and understand its behavior and evolution. This paper discusses the surveying operations, which involved climbing the pinnacle to collect Ground Control Points (GCPs) and using a UAV to capture aerial imagery. The photographs were processed using SfM software to generate point clouds, mesh, and texture, which were then used for rock mass discontinuity mapping. The study compares models of different qualities and point densities to determine the optimal trade-off between processing time and accuracy in terms of discontinuity mapping. The results show that higher quality models allow for more detailed mapping of discontinuities, with some drawbacks due to noise in the case of the densest solution (e.g., increase in frequency of outliers across the point cloud). These pros and cons are also discussed in relation to the computational cost necessary to build the models. The study also examines the limitations and challenges of performing discontinuity mapping in the different models, including subjectivity in interpretation. A further element of interest is the publication of a high-quality 3D georeferenced model of the “Campanile di Val Montanaia” to be used for several potential further applications, such as stability analyses and numerical modeling.

Can a border collision bifurcation of a chaotic attractor lead to its expansion?

Recently, we reported that a chaotic attractor in a discontinuous one-dimensional map may undergo a so-called exterior border collision bifurcation, which causes additional bands of the attractor to appear. In the present paper, we suppose that the chaotic attractor’s basin boundary contains a chaotic repeller, and discuss a bifurcation pattern consisting of an exterior border collision bifurcation and an expansion bifurcation (interior crisis). In the generic case, where neither the border point the chaotic attractor collides with, nor any of its images belong to the chaotic repeller, the exterior border collision bifurcation is followed by the expansion bifurcation, and the distance between both bifurcations may be arbitrarily small but positive. In the non-generic (codimension-2) case, these bifurcations occur simultaneously, so that a border collision bifurcation of a chaotic attractor leads directly to its expansion.

Long transients in discontinuous time-discrete models of population dynamics

Traditionally, mathematical modelling of population dynamics was focused on long-term, asymptotic behaviour (systems attractors), whereas the effects of transient regimes were largely disregarded. However, recently there has been a growing appreciation of the role of transients both in empirical ecology and theoretical studies. Among the main challenges are identification of the mechanisms triggering transients in various dynamical systems and understanding of the corresponding scaling law of the transient’s lifetime; the latter is of a vital practical importance for long-term ecological forecasting and regime shifts anticipation. In this study, we reveal and investigate various patterns of long transients occurring in two generic time-discrete population models which are mathematically described by discontinuous (piece-wise) maps. In particular, we consider a single-species population model and a predator–prey system, in each model we assume that the dispersal of species at the end of each season is density dependent. For both models, we demonstrate transients due to crawl-by dynamics, chaotic repellers, chaotic saddles, ghost attractors, and a rich variety of intermittent regimes. For each type of transient, we investigate the corresponding scaling law of the transient’s lifetime. We explore the space of key model parameters, to find where particular types of long transients can be expected, and we show that long transients are omnipresent since they can be observed within a wide range of model parameters. We also reveal the possibility of complex patterns occurring as a cascade of transients of different types. We also considered a stochastic version of the model where some parameters exhibit random fluctuations. We show that stochasticity can reduce, extend or produce new patterns of long transients. We conclude that the discontinuity in population models significantly facilitates the emergence of long transients by creating new types and increasing parameter domains of the corresponding transient dynamics. Another important conclusion is that the asymptotic regime of population dynamics is hardly possible to predict based on a finite time course of species densities, which is crucial for ecosystem management and decision making.

Modeling of the Three-Dimensional Point Cloud Based on the Binocular Stereovision and Structure Identification

Abstract The existence of weak planes affects the stability of rock engineering, especially the failure modes of the surrounding rocks. Therefore, investigation of the structural planes of rock mass and the extraction of discontinuities is important for rock mechanics. In this work, the LenaCV binocular stereovision system was studied and applied to establish three-dimensional (3D) surface models. Calibration of the binocular camera parameters, optimal distance discussion, and 3D modeling optimization were conducted. The optimal measuring distance and the intrinsic parameters suitable for the LenaCV stereovision were discussed. Finally, the 3D reconstruction of two case studies was carried out and the 3D point cloud modeling was established. The structural discontinuity mapping in rock faces was also carried out. The feasibility of the binocular stereovision system, as well as the optimum baseline distance, was discussed and verified. The study results can provide a new platform for the economic investigation of 3D rock mass structures and provide a new approach for the efficient extraction of rock mass discontinuities.