Discontinuous Maps Research Articles

On the relaxed area of the graph of discontinuous maps from the plane to the plane taking three values with no symmetry assumptions

In this paper, we estimate from above the area of the graph of a singular map u taking a disk to three vectors, the vertices of a triangle, and jumping along three $${\mathcal {C}}^2$$-embedded curves that meet transversely at only one point of the disk. We show that the singular part of the relaxed area can be estimated from above by the solution of a Plateau-type problem involving three entangled nonparametric area-minimizing surfaces. The idea is to “fill the hole” in the graph of the singular map with a sequence of approximating smooth two-codimensional surfaces of graph-type, by imagining three minimal surfaces, placed vertically over the jump of u, coupled together via a triple point in the target triangle. Such a construction depends on the choice of a target triple point, and on a connection passing through it, which dictate the boundary condition for the three minimal surfaces. We show that the singular part of the relaxed area of u cannot be larger than what we obtain by minimizing over all possible target triple points and all corresponding connections.

Method of Limiting Differential Inclusions for Nonautonomous Discontinuous Systems with Delay

Functional-differential equations ẋ = f (t, ϕ(·)) with piecewise continuous right-hand sides are studied. It is assumed that the set M of discontinuity points of the right-hand side has empty interior in contrast to being a measure zero sets, as in the case of differential equations without delay. This assumption is made largely because the domain of the function f is infinite-dimensional. Solutions to the equations under consideration are understood in A. F. Filippov’s sense. The main results are theorems on the asymptotic behavior of solutions formulated with the use of invariantly differentiable Lyapunov functionals with constant-sign derivatives. Nonautonomous systems are difficult to deal with because ω-limit sets of their solutions do not possess invariance-type properties, whereas sets of zeros of derivatives of Lyapunov functionals may depend on the variable t and extend beyond the space of variables ϕ(·). For discontinuous nonautonomous systems, there arises the issue of constructing the limiting differential equations with the use of shifts fτ(t + τ, ϕ(·)) of the function f. We introduce the notion of limiting differential inclusion without employing limit passages on sequences of shifts of discontinuous or multivalued mappings. The properties of such inclusions are studied. Invariance-type properties of ω-limit sets of solutions and analogs of LaSalle’s invariance principle are established.

Augmented Reality Mapping of Rock Mass Discontinuities and Rockfall Susceptibility Based on Unmanned Aerial Vehicle Photogrammetry

In rockfall hazard management, the investigation and detection of potential rockfall source areas on rock cliffs by remote-sensing-based susceptibility analysis are of primary importance. However, when the rockfall analysis results are used as feedback to the fieldwork, the irregular slope surface morphology makes it difficult to objectively locate the risk zones of hazard maps on the real slopes, and the problem of straightforward on-site visualization of rockfall susceptibility remains a research gap. This paper presents some of the pioneering studies on the augmented reality (AR) mapping of geospatial information from cyberspace within 2D screens to the physical world for on-site visualization, which directly recognizes the rock mass and superimposes corresponding rock discontinuities and rockfall susceptibility onto the real slopes. A novel method of edge-based tracking of the rock mass target for mobile AR is proposed, where the model edges extracted from unmanned aerial vehicle (UAV) structure-from-motion (SfM) 3D reconstructions are aligned with the corresponding actual rock mass to estimate the camera pose accurately. Specifically, the visually prominent edges of dominant structural planes were first explored and discovered to be a robust visual feature of rock mass for AR tracking. The novel approaches of visual-geometric synthetic image (VGSI) and prominent structural plane (Pro-SP) were developed to extract structural planes with identified prominent edges as 3D template models which could provide a pose estimation reference. An experiment verified that the proposed Pro-SP template model could effectively improve the edge tracking performance and quality, and this approach was relatively robust to the changes of sunlight conditions. A case study was carried out on a typical roadcut cliff in the Mentougou District of Beijing, China. The results validate the scalability of the proposed mobile AR strategy, which is applicable and suitable for cliff-scale fieldwork. The results also demonstrate the feasibility, efficiency, and significance of the geoinformation AR mapping methodology for on-site zoning and locating of potential rockfalls, and providing relevant guidance for subsequent detailed site investigation.

Degenerate grazing bifurcations in a three-degree-of-freedom impact oscillator

This paper presents the analysis of the degenerate grazing bifurcation in a three-degree-of-freedom impact oscillator by studying the bifurcations of near-grazing period-one impact motion near the degenerate grazing point. Actually, this paper extends the higher-order zero time discontinuity mapping to perform the perturbation analysis of characteristic equation of period-one impact motion and obtains feasible eigenvalue approximation to study the potential bifurcations. The shooting method is applied to verify the validity of the derived approximation and corresponding computation results. In addition to the known bifurcation scenarios of saddle-node and period-doubling, novel Neimark–Sacker bifurcation and related co-dimension two bifurcation points of near-grazing period-one impact motion are also found to arise near the degenerate grazing point in a three-degree-of-freedom impact oscillator. For the in-depth understanding of near-grazing dynamics, the obtained results are compared with the reported results in the single- and two-degree-of-freedom impact oscillators.

Automatic characterization of rock mass discontinuities using 3D point clouds

Direct contact measurements for rock mass discontinuities are generally difficult to perform, time consuming, biased, and often dangerous. This paper presents an automatic characterization method for rock mass discontinuities that uses 3D point clouds, which can be obtained through non-contact measuring techniques such as photogrammetry and Light Detection and Ranging (LiDAR). In this method, five discontinuity parameters, namely, the orientation, trace, spacing, roughness, and aperture, are extracted automatically. The overall methodology is as follows: (1) orientation is determined by using an improved K-means clustering method; (2) trace segments are detected by using the Normal Tensor Voting Theory, and four post-processing techniques are employed to compute the trace length; (3) spacing is calculated by plotting virtual normal scan lines on the projected traces; (4) roughness is evaluated by the correlation between the Joint Roughness Coefficient (JRC) and the root mean square of the discontinuity surface profile; and (5) aperture is obtained by computing the average minimum width based on sub-pixel edge detection. The proposed method was applied to a drill-and-blast rock tunnel, where the extracted discontinuity parameters were used to calculate the Rock Mass Rating (RMR) value and Geological Strength Index (GSI) of the rock mass. The application results showed that photogrammetry was more objective and efficient for acquiring rock mass discontinuity information, and that it could be used as a potential alternative to the traditional discontinuity mapping method.

Invariant property for discontinuous mean-type mappings

It is known that if $M,\,N$ are continuous two-variable means such that $|M(x,y)-N(x,y)| < |x-y|$ for every $x,\ y$ with $x\ne y$, then there exists a unique invariant mean (which is continuous too). We are looking for invariant means for pairs satisfying the inequality above, but continuity of means is not assumed. In this setting the invariant mean is no longer uniquely defined, but we prove that there exist the smallest and the biggest one. Furthermore it is shown that there exists at most one continuous invariant mean related to each pair.

Complex near-grazing dynamics in impact oscillators

Impact oscillators frequently appear in various physical and engineering systems with non-smooth characteristics and can exhibit different dynamic behavior from the smooth nonlinear systems, including grazing bifurcation in which an impact with zero velocity occurs. This paper investigates the near-grazing dynamics of the multi-degree-of-freedom impact oscillators in the small neighborhood of degenerate grazing points, with a focus on the stability and potential bifurcations of near-grazing period-one impact motions. The high order zero time discontinuity mapping method is applied to perform the prospective analyses of stability and bifurcations. Particularly, this paper shows that the peculiar Neimark-Sacker bifurcations regaining the stability of near-grazing period-one impact motion can be induced by two different ways, either through the interaction between the singular and regular real eigenvalues or via a grazing bifurcation directly. A two degree-of-freedom impact oscillator is taken as an example to present detailed numerical results for the verification of proposed analysis.

Detection and geometric characterization of rock mass discontinuities using a 3D high-resolution digital outcrop model generated from RPAS imagery – Ormea rock slope, Italy

The use of a remotely piloted aircraft system (RPAS) and digital photogrammetry is valuable for the detection of discontinuities in areas where field mapping and terrestrial photogrammetry or laser scanner surveys cannot be employed because the slope is unsafe, inaccessible, or characterized by a complex geometry with areas not visible from the ground. Using the Structure-from-Motion method, the acquired images can be used to create a 3D texturized digital outcrop model (TDOM) and a detailed point cloud representing the rock outcrop. Discontinuity orientations in a complex rock outcrop in Italy were mapped in the field using a geological compass and by manual and automated techniques using a TDOM and point cloud generated from RPAS imagery. There was a good agreement between the field measurements and manual mapping in the TDOM. Semi-automated discontinuity mapping using the point cloud was performed using the DSE, qFacet FM, and qFacet KD-tree methods applied to the same 3D model. Significant discrepancies were found between the semi-automatic and manual methods. In particular, the automatic methods did not adequately detect discontinuities that are perpendicular to the slope face (bedding planes in the case study). These differences in detection of discontinuities can adversely influence the kinematic analysis of potential rock slope failure mechanisms. We use the case study to demonstrate a workflow that can accurately map discontinuities with results comparable to field measurements. The combined use of TDOM and RPAS dramatically increases the discontinuity data because RPAS can supply a good coverage of inaccessible or hidden portions of the slope and TDOM is a powerful representation of the reality that can be used to map discontinuity orientations including those that are oriented perpendicular to the slope.

Application of structural interpretation and simultaneous inversion to reservoir characterization of the Duvernay Formation, Fox Creek, Alberta, Canada

Simultaneous prestack inversion of multicomponent 3D seismic data integrated with structural interpretation can provide an effective workflow to maximize value for unconventional plays. We outline an integrated workflow for characterizing the Duvernay play in western Canada, an emerging world-class low-permeability unconventional resource fairway. This workflow includes the determination of a time-depth relationship using synthetic seismograms, generation of seismic-derived time- and depth-converted structural maps, and calculation of inversion-based parameters of density and P- and S-wave velocity. The model-based procedure includes poststack (acoustic) inversion, amplitude variation with offset prestack inversion, and joint PP-PS inversion. With these rock properties determined, calculations are made to determine Young's modulus, Poisson's ratio, and brittleness. Faults are mapped based on time slices, isochrons, and correlatable vertical displacements of stratigraphic marker reflections. Significant strike-slip movements are identified by lateral displacement on interpreted geologic features, such as channels and reef edges. Seismic-derived attributes, combined with structural mapping, highlight zones that are conducive to hydraulic fracturing as well as areas unfavorable for development. Mapping of structural discontinuities provides a framework for understanding zones of preexisting weakness and induced-seismicity hazards.

Use of seismic attributes and open-hole log data to characterize production variability in a fractured carbonate play: A case study from Madison County, Texas

Understanding natural fracture systems is key for tight carbonate plays, in which production is dependent on secondary interconnected porosity networks. Locating geographic areas and stratigraphic sections with high natural fracture density and optimizing well locations and perforations to connect these fractures can enhance well performance and asset value. There is substantial production variation in the Cretaceous stacked carbonate play in East Texas, despite similarities in well completion and perforated intervals. Petrophysical property models did not explain the significant variation in well production; therefore, we have developed a multidisciplinary workflow combining seismic and log data with the goal of identifying faulting and natural fractures and understanding their effect on production. We used seismic discontinuity to map faults as the main indicator of presence of fractures. We calibrated triple combo logs with an image log to generate an indicator curve to identify natural fractures. The fracture indicator curve provided a good prediction of where natural fractures may occur, and discontinuity maps revealed a good correlation to well production. Furthermore, we concluded that drilling too closely to large faults negatively impacted production and correlated with increased water production. The workflow developed here can be used to optimize well placement in the stacked carbonate play of Madison County, Texas, and it can be applied to other fractured carbonate reservoirs.

BCB Curves and Contact Bifurcations in Piecewise Linear Discontinuous Map Arising in a Financial Market

In this paper, we further study a financial market model established in our earlier paper. The model dynamics is driven by a two-dimensional piecewise linear discontinuous map, which is investigated analytically and numerically for one-sided fixed points being flip saddle and two-sided fixed points being attractors. The existence of chaotic orbit is explained by using the theory of homoclinic intersection between stable and unstable manifolds of the flip saddle invariant set. The structure of chaotic attractor is disclosed. It consists of finite segments rooted on both sides of the [Formula: see text]-axis which are unstable manifolds of flip saddle invariant set. The basins and their structural changes of bounded attractors and coexisting attractors are presented by contact bifurcation theory and numerical simulations. The border collision bifurcation (BCB for short) curves are calculated and coexisting multiattractors are disclosed by overlapping periodicity regions. The results can deepen our understanding of financial markets and dynamical systems.

Meir-Keeler type and Caristi type fixed point theorems

Agarwal et al [1] have proved some interesting local and global fixed point theorems for Meir-Keeler [7] type and Caristi [2] type maps. We obtain analogues of the main results of Agarwal et al [1] under weaker conditions so as to include continuous as well as discontinuous maps. Our results provide new answers to Rhoades' problem ([15], p. 242) on existence of contractive definitions which admit discontinuity at the fixed point. Several examples are given to illustrate our results.

Magnetic Connectivity in the Corona as a Source of Structure in the Solar Wind

AbstractFive decades of satellite data confirm that the solar wind contains many boundaries separating flow with distinct magnetic and plasma properties. Some speculate the boundaries in the solar wind found at Earth originate at the solar surface and are carried along with the expanding solar wind as fossil structures to 1 AU. This begs the question, is it the physics and magnetic structure above the photosphere that creates well‐defined boundaries between different magnetic flux regions at 1 AU in the solar wind? Magnetic boundaries in the corona exist all the time as topological features of null points in the field. These topological magnetic boundaries seem to be likely locations for plasma boundaries. It can be expected that these boundaries are typical locations where field line‐integrated quantities, such as field‐aligned current, experience large and abrupt changes. We perform three‐dimensional resistive magnetohydrodynamic simulations of the solar corona driven by photospheric foot point motions. We find that large and abrupt changes occur for field line‐integrated quantities across a magnetic topological boundary and the cause for these changes is the discontinuous mapping for magnetic field lines and thus for Alfvén waves across these boundaries. It is also demonstrated via in situ properties that thin layers of field‐aligned and perpendicular currents are frequently located at or close to topological boundaries.

On the Existence of Chaos in a Discontinuous Area-Preserving Map Arising in Financial Markets

In this paper, we further study a discontinuous piecewise-linear financial market model established in our previous paper. The model dynamics is driven by a two-dimensional discontinuous area-preserving map. We exploit its complex dynamics from the viewpoint of open conservative systems. We first give the classification of fixed points, and then theoretically present the existence of periodic saddle orbit and a homoclinic chaos for some parameter setting. We finally apply the Conley–Moser conditions to verify the existence of Smale horseshoe-like dynamics and chaos for another parameter setting. The result is helpful for understanding the internal change rule of the finance market.

Classic crystal plasticity theory vs crystal plasticity theory based on strong discontinuities—Theoretical and algorithmic aspects

SummaryThis paper deals with two different approaches suitable for the description of plasticity in single crystals. The first one is the standard approach that is based on a continuous deformation mapping. Plasticity is driven by a classic Schmid‐type relation connecting the shear stresses to the shear strains at a certain slip system. By way of contrast, the second approach is nonstandard. In this novel model, localized plastic deformation at certain slip planes is approximated by a strong discontinuity (discontinuous deformation mapping). Accordingly, a modified Schmid‐type model relating the shear stresses to the shear displacements (displacement jump) is considered in this model. Although both models are indeed different, it is shown that they can be characterized by almost the same set of equations, eg, by a multiplicative decomposition of the deformation gradient into an elastic part and a plastic part. This striking analogy eventually leads to a unifying algorithmic formulation covering both models. Since the set of active slip systems is not known in advance, its determination is of utmost importance. This problem is solved here by using the nonlinear complementarity problem (NCP) as advocated by Fischer and Burmeister. While this idea is not new, it is shown that the NCP problem is well posed, independent of the number of active slip systems. To be more explicit, the tangent matrix in the return‐mapping scheme is regular even for more than five simultaneously active slip systems. Based on this algorithm, texture evolution in a polycrystal is analyzed by means of both models and the results are compared in detail.

Analytical-numerical studies on the stability and bifurcations of periodic motion in the vibro-impact systems with clearances

This paper investigates the stability and bifurcations of periodic solutions in three-degree-of-freedom vibro-impact systems based on the explicit critical criteria and the discontinuous mapping method. Firstly, a six-dimensional Poincaré map is established by taking the impact surface as the Poincaré section. The explicit criteria including eigenvalue assignment and transversality condition are applied to determine the bifurcation point of co-dimension one pitchfork bifurcation. The stability and direction of the bifurcation solution are then studied by using center manifold reduction theory and normal form approach. Secondly, the bifurcation points of co-dimension-two Hopf–Hopf interaction bifurcation and pitchfork–Neimark–Sacker bifurcation are determined by applying the explicit critical criteria, and the local dynamic behaviors are examined in the neighborhood of these co-dimension-two bifurcation points. Finally, a six-dimensional Poincaré map formed by choosing the constant phase angle as the Poincaré section is used to investigate the existence and stability of grazing bifurcation based on the piecewise compound normal form map. The causes of the discontinuous jump and the coexistence of attractors near the grazing periodic motion are explained for the three-degree-of-freedom vibro-impact system with a moving constraint.

A cobweb model with elements from prospect theory

We present a cobweb model to explain price adjustment in a competitive market with homogeneous firms based on assumptions from Prospect Theory. Price changes are evaluated with respect to a psychological reference price which enters directly into the demand function. Accordingly, firms face a downward-sloping demand curve that is kinked at the consumers’ reference price. Differently from the traditional cobweb model, the economy is described by a discontinuous map. Without assuming specific non-linearities and keeping the essential underlying mechanics of the model intact, we find that the implementation of several features from Prospect Theory into our simple cobweb model may significantly influence the market dynamics. Behavioral parameters play an important role for the market stability by reducing fluctuations and by directly affecting consumers’ demand as well as production decisions.

Variational Fusion of Time-of-Flight and Stereo Data for Depth Estimation Using Edge-Selective Joint Filtering

In this paper, we propose variational fusion of time-of-flight (TOF) and stereo data for depth estimation using edge-selective joint filtering (ESJF). ESJF is able to adaptively select edges for depth upsampling from the TOF depth map, stereo matching-based disparity map, and stereo images. We adopt ESJF to produce high-resolution (HR) depth maps with accurate edge information from low-resolution ones captured by the TOF camera. First, we measure confidences of TOF and stereo data based on a Gaussian function to be used as fusion weights. Then, we upsample the TOF depth map using ESJF and extract vertical and horizontal discontinuity maps from it. Finally, we perform variational fusion of TOF and stereo depth data guided by the discontinuity maps. Experimental results show that the proposed method successfully produces HR depth maps and outperforms the state of the art in preserving edges and removing noise.

Suppression of grazing-induced instability in single degree-of-freedom impact oscillators

As a typical non-smooth bifurcation, grazing bifurcation can induce instability of elementary near-grazing impact periodic motion in impact oscillators. In this paper, the stability for near-grazing period-one impact motion to suppress grazing-induced instabilities is analyzed, based on which, a control strategy is proposed. The commonly-used leading order zero time discontinuity mapping is extended to a higher order one to aid the perturbation analysis of the characteristic equation. It is shown that the degenerate grazing bifurcation can eliminate the singular term in the characteristic equation, leading to bounded eigenvalues. Based on such a precondition, the bounded eigenvalues are further restricted inside the unit circle, and a continuous transition between non-impact and controlled impact motion is observed. One discrete feedback controller that changes the velocity of the oscillator based on the selected Poincare sections is adopted to demonstrate the control procedure.

Gluing and grazing bifurcations in periodically forced 2-dimensional integrate-and-fire models

In this work we consider a general class of 2-dimensional hybrid systems. Assuming that the system possesses an attracting equilibrium point, we show that, when periodically driven with a square-wave pulse, the system possesses a periodic orbit which may undergo smooth and nonsmooth grazing bifurcations. We perform a semi-rigorous study of the existence of periodic orbits for a particular model consisting of a leaky integrate-and-fire model with a dynamic threshold. We use the stroboscopic map, which in this context is a 2-dimensional piecewise-smooth discontinuous map. For some parameter values we are able to show that the map is a quasi-contraction possessing a (locally) unique maximin periodic orbit. We complement our analysis using advanced numerical techniques to provide a complete portrait of the dynamics as parameters are varied. We find that for some regions of the parameter space the model undergoes a cascade of gluing bifurcations, while for others the model shows multistability between orbits of different periods.