Piecewise Maps Research Articles

Dynamical Aspects of Piecewise Conformal Maps

We study the dynamics of piecewise conformal maps in the Riemann sphere. The normality and chaotic regions are defined and we state several results and properties of these sets. We show that the stability of these piecewise maps is related to the Kleinian group generated by their transformations under certain hypotheses. The general motivation of the article is to compare the dynamics of piecewise conformal maps and those of the Kleinian groups and iterations of rational maps.

The dynamics of a business game: A 2D-piecewise smooth nonlinear map

In this paper, we give a proper investigation of the problem of remanufacturing duopoly game.11In [Physica A 514 (2019) 658-670] the authors have introduced the problem of remanufacturing duopoly game. The analytical and numerical analysis are wrong. They had to study their system as a piecewise one as shown in the current paper. A smooth piecewise map is introduced to describe such game. A rich analysis either analytical or numerical is presented in this paper in order to investigate the stability of the fixed points of the game’s model. Our obtained results show that those fixed points lose its stability due to flip and Neimark-Sacker border bifurcations. Furthermore, we study the stability of those points when there are periodic cycles with points jump from region to other region.

Piecewise Classifier Mappings: Learning Fine-Grained Learners for Novel Categories With Few Examples.

Humans are capable of learning a new fine-grained concept with very little supervision, e.g., few exemplary images for a species of bird, yet our best deep learning systems need hundreds or thousands of labeled examples. In this paper, we try to reduce this gap by studying the fine-grained image recognition problem in a challenging few-shot learning setting, termed few-shot fine-grained recognition (FSFG). The task of FSFG requires the learning systems to build classifiers for the novel fine-grained categories from few examples (only one or less than five). To solve this problem, we propose an end-to-end trainable deep network, which is inspired by the state-of-the-art fine-grained recognition model and is tailored for the FSFG task. Specifically, our network consists of a bilinear feature learning module and a classifier mapping module: while the former encodes the discriminative information of an exemplar image into a feature vector, the latter maps the intermediate feature into the decision boundary of the novel category. The key novelty of our model is a "piecewise mappings" function in the classifier mapping module, which generates the decision boundary via learning a set of more attainable sub-classifiers in a more parameter-economic way. We learn the exemplar-to-classifier mapping based on an auxiliary dataset in a meta-learning fashion, which is expected to be able to generalize to novel categories. By conducting comprehensive experiments on three fine-grained datasets, we demonstrate that the proposed method achieves superior performance over the competing baselines.

Pulse Wave Imaging in Carotid Artery Stenosis Human Patients in Vivo

Carotid stenosis involves narrowing of the lumen in the carotid artery potentially leading to a stroke, which is the third leading cause of death in the United States. Several recent investigations have found that plaque structure and composition may represent a more direct biomarker of plaque rupture risk compared with the degree of stenosis. In this study, pulse wave imaging was applied in 111 (n = 11, N = 13 plaques) patients diagnosed with moderate (>50%) to severe (>80%) carotid artery stenosis to investigate the feasibility of characterizing plaque properties based on the pulse wave-induced arterial wall dynamics captured by pulse wave imaging. Five (n = 5 patients, N = 20 measurements) healthy volunteers were also imaged as a control group. Both conventional and high-frame-rate plane wave radiofrequency imaging sequences were used to generate piecewise maps of the pulse wave velocity (PWV) at a single depth along stenotic carotid segments, as well as intra-plaque PWV mapping at multiple depths. Intra-plaque cumulative displacement and strain maps were also calculated for each plaque region. The Bramwell-Hill equation was used to estimate the compliance of the plaque regions based on the PWV and diameter. Qualitatively, wave convergence, elevated PWV and decreased cumulative displacement around and/or within regions of atherosclerotic plaque were observed and may serve as biomarkers for plaque characterization. Intra-plaque mapping revealed the potential to capture wave reflections between calcified inclusions and differentiate stable (i.e., calcified) from vulnerable (i.e., lipid) plaque components based on the intra-plaque PWV and cumulative strain. Quantitatively, one-way analysis of variance indicated that the pulse wave-induced cumulative strain was significantly lower (p < 0.01) in the moderately and severely calcified plaques compared with the normal controls. As expected, compliance was also significantly lower in the severely calcified plaques regions compared with the normal controls (p < 0.01). The results from this pilot study indicated the potential of pulse wave imaging coupled with strain imaging to differentiate plaques of varying stiffness, location and composition. Such findings may serve as valuable information to compensate for the limitations of currently used methods for the assessment of stroke risk.

Computing Complex Horseshoes by Means of Piecewise Maps

A systematic procedure to numerically compute a horseshoe map is presented. This new method uses piecewise functions and expresses the required operations by means of elementary transformations, such as translations, scalings, projections and rotations. By repeatedly combining such transformations, arbitrarily complex folding structures can be created. We show the potential of these horseshoe piecewise maps to illustrate several central concepts of nonlinear dynamical systems, as for example, the Wada property.

Changes in motor performance and mental workload duringpractice of reaching movements: a team dynamics perspective.

Few investigations have examined mental workload during motor practice or learning in a context of team dynamics. This study examines the underlying cognitive-motor processes of motor practice by assessing the changes in motor performance and mental workload during practice of reaching movements. Individuals moved a robotic arm to reach targets as fast and as straight as possible while satisfying the task requirement of avoiding a collision between the end-effector and the workspace limits. Individuals practiced the task either alone (HA group) or with a synthetic teammate (HRT group), which regulated the effector velocity to help satisfy the task requirements. The findings revealed that the performance of both groups improved similarly throughout practice. However, when compared to the individuals of the HA group, those in the HRT group (1) had a lower risk of collisions, (2) exhibited higher performance consistency, and (3) revealed a higher level of mental workload while generally perceiving the robotic teammate as interfering with their performance. As the synthetic teammate changed the effector velocity in specific regions near the workspace boundaries, individuals may have been constrained to learn a piecewise visuomotor map. This piecewise map made the task more challenging, which increased mental workload and perception of the synthetic teammate as a burden. The examination of both motor performance and mental workload revealed a combination of both adaptive and maladaptive team dynamics. This work is a first step to examine the human cognitive-motor processes underlying motor practice in a context of team dynamics and contributes to inform human-robot applications.

On the finiteness of attractors for piecewise maps of the interval

We consider piecewise $C^{2}$ non-flat maps of the interval and show that, for Lebesgue almost every point, its omega-limit set is either a periodic orbit, a cycle of intervals or the closure of the orbits of a subset of the critical points. In particular, every piecewise $C^{2}$ non-flat map of the interval displays only a finite number of non-periodic attractors.

Semi-conjugacies between m-horseshoe maps and n-horseshoe maps

The conjugacy problem is one of the central questions in dynamical systems. Many works have devoted to conjugacy problems of horseshoe maps with the same number of laps, which is a kind of continuous, piecewise monotone interval maps. This paper considers semi-conjugacies from m-horseshoe maps to n-horseshoe maps for . We first give new orientation-preserving/reversing partition on their domains. Then by Matkowski’s fixed point principle we prove the existence of semi-conjugacies from m-horseshoe maps to n-horseshoe maps. We construct some sequences of functions to approximate these semi-conjugacies. Finally, an example is given for a piecewise linear 4-horseshoe maps and a piecewise linear 3-horseshoe map.

Ultraslow diffusion and weak ergodicity breaking in right triangular billiards.

We investigate the diffusion behavior of a right triangular billiard system by transforming its dynamics to a two-dimensional piecewise map. We find that the diffusion in the momentum space is ultraslow, i.e., the mean squared displacement grows asymptotically as the square of the logarithm of time. The mechanism of the ultraslow diffusion behavior is explained and numerical evidence corroborating our conclusion is provided. The weak ergodicity breaking of the system is also discussed.

Image Enhancement Using Adaptive Piecewise Mapping

Image Enhancement Using Adaptive Piecewise Mapping

Bifurcation analysis of Lu system with the periodic parameter-switching scheme

In this paper, switches related to the parameters altered periodically are introduced, upon which a typical switched dynamical model between two Lu systems is established. To explore the dynamical behaviors and the mechanism of the switching oscillations, Poincare´ map of the whole system is defined by the switching conditions and piecewise maps. The location of the fixed points corresponding to various periodic switching oscillations and associated Floquet multipliers are calculated, based on which bifurcation scenarios, such as saddle-note bifurcation, pitchfork bifurcation and period-doubling bifurcation, are determined in the switched Lu system.

Piecewise Mapping in HEVC Lossless Intra-Prediction Coding.

The lossless intra-prediction coding modality of the High Efficiency Video Coding standard provides high coding performance while allowing frame-by-frame basis access to the coded data. This is of interest in many professional applications, such as medical imaging, automotive vision, and digital preservation in libraries and archives. Various improvements to lossless intra-prediction coding have been proposed recently, most of them based on sample-wise prediction using differential pulse code modulation (DPCM). Other recent proposals aim at further reducing the energy of intra-predicted residual blocks. However, the energy reduction achieved is frequently minimal due to the difficulty of correctly predicting the sign and magnitude of residual values. In this paper, we pursue a novel approach to this energy-reduction problem using piecewise mapping (pwm) functions. In particular, we analyze the range of values in residual blocks and apply accordingly a pwm function to map specific residual values to unique lower values. We encode the appropriate parameters associated with the pwm functions at the encoder, so that the corresponding inverse pwm functions at the decoder can map values back to the same residual values. These residual values are then used to reconstruct the original signal. This mapping is, therefore, reversible and introduces no losses. We evaluate the pwm functions on 4 × 4 residual blocks computed after DPCM-based prediction for lossless coding of a variety of camera-captured and screen content sequences. Evaluation results show that the pwm functions can attain the maximum bitrate reductions of 5.54% and 28.33% for screen content material compared with DPCM-based and block-wise intra-prediction, respectively. Compared with intra-block copy, piecewise mapping can attain the maximum bit-rate reductions of 11.48% for a camera-captured material.

Synchronization transition with coexistence of attractors in coupled discontinuous system

The studies of extended dynamics systems are relevant to the understanding of spatiotemporal patterns observed in diverse fields. One of the well-established models for such complex systems is the coupled map lattices, and several features of pattern formation including synchronization, unsynchronization, traveling waves and clustering synchronization are found. Among the above-mentioned patterns, chaotic synchronization has been intensively investigated in recent years. It has been demonstrated that two or more chaotic systems can be synchronized by linking them with mutual coupling or a common signal or some signals. Over the last decade, a number of theoretical methods have been presented to deal with this problem, such as the methods of master stability functions and eigenvalue analysis. While much effort has been devoted to the networks with different topological structures in continuous systems. The coupled discontinuous maps have been investigated with increasing interest in recent years, they showed that the complete synchronization in coupled discontinuous systems is more complicated than in coupled continuous systems. However, a similar problem of synchronization transition in coupled discontinuous systems is much less known.The synchronization transition in coupled discontinuous map lattices is studied. The average order parameter and maximal Lyapunov exponent are calculated to diagnose the synchronization of coupled piecewise maps. The results indicate that there exist the periodic clusters and the synchronization state, and a new transition style from periodic cluster states to complete synchronization states is found. The periodic cluster states consist of two kinds of periodic orbits: symmetric periodic orbits and asymmetric periodic orbits.Based on the pattern analysis, the common features of the patterns are constructed by the two periodic attractors, and the periodic orbit is an unstable periodic orbit of the isolate map. The discontinuities in a system can divide the phase space into individual zones of different dynamical features. The interactions between the local nonlinearity and the spatial coupling confine orbit into certain spaces and form a dynamic balance between two periodic clusters. The system can reach complete synchronization states when the balance is off. It is shown that synchronization transition of the coupled discontinuous map can exhibit the different processes, which depends on coupling strength. Four transition modes are found in coupled discontinuous map: 1) the transition, from the coexistence of chaotic synchronization and chaotic un-synchronization states to the coexistence of chaotic synchronization, chaotic un-synchronization, symmetric periodic orbits and asymmetric periodic orbits; 2) the transition from the coexistence of chaotic synchronization, chaotic un-synchronization, symmetric periodic orbits and asymmetric periodic orbits to the coexistence of chaotic synchronization, symmetric periodic orbits and asymmetric periodic orbits; 3) the transition from the coexistence of chaotic synchronization, symmetric periodic orbits and asymmetric periodic orbits to the coexistence of chaotic synchronization and symmetric periodic orbits; 4) the transition from the coexistence of chaotic synchronization and symmetric periodic orbits to the chaotic synchronization. Because the local dynamics has discontinuous points, the coupled system shows a riddle basin characteristic in the phase space, and the synchronization transition of coupled piecewise maps looks more complex than continuous system.

Border Collision Bifurcations in a Generalized Model of Population Dynamics

We analyze the dynamics of a generalized discrete time population model of a two-stage species with recruitment and capture. This generalization, which is inspired by other approaches and real data that one can find in literature, consists in considering no restriction for the value of the two key parameters appearing in the model, that is, the natural death rate and the mortality rate due to fishing activity. In the more general case the feasibility of the system has been preserved by posing opportune formulas for the piecewise map defining the model. The resulting two-dimensional nonlinear map is not smooth, though continuous, as its definition changes as any border is crossed in the phase plane. Hence, techniques from the mathematical theory of piecewise smooth dynamical systems must be applied to show that, due to the existence of borders, abrupt changes in the dynamic behavior of population sizes and multistability emerge. The main novelty of the present contribution with respect to the previous ones is that, while using real data, richer dynamics are produced, such as fluctuations and multistability. Such new evidences are of great interest in biology since new strategies to preserve the survival of the species can be suggested.

Coexistence of attractors and effects of noise on coupled piecewise maps

The synchronization transition in coupled piecewise map lattices is studied. The average order parameter is calculated to diagnose synchronization of coupled piecewise maps. As a result, it shows that coexistence of a period cluster and synchronization states in the synchronization transition process. A detailed research shows that the period clusters consist of two coexisting period attractors, and their period orbit is an unstable periodic orbit of the isolate map. The unstable periodic orbit is stabilized again by the mutual interactions of the local nonlinearity and the spatial coupling. The influence of noise on the complete synchronization (CS) in coupled piecewise map systems is also studied in this work. Synchronization probability densities and average order parameters as a function of the coupling strength with various noise intensities are calculated. When the noise intensity is small, the noise can postpone synchronization transition processes. However, if the noise intensity is larger, the noise can destroy the period clusters states, shorten the synchronization transition and enhance the CS in coupled piecewise maps.

Dynamics in two-elevator traffic system with real-time information

Abstract We study the dynamics of traffic system with two elevators using a elevator choice scenario. The two-elevator traffic system with real-time information is similar to the two-route vehicular traffic system. The dynamics of two-elevator traffic system is described by the two-dimensional nonlinear map. An elevator runs a neck-and-neck race with another elevator. The motion of two elevators displays such a complex behavior as quasi-periodic one. The return map of two-dimensional map shows a piecewise map.

Nonlinear-map model for the control of an airplane schedule

We study the dynamics and control of an airplane in air transportation between two airports. The dynamic models are presented for the airplane schedule. The dynamics of an airplane is described by the piecewise map and delayed map models. The characteristics of the nonlinear maps are studied analytically and numerically. The airplane displays the complex motion in the unstable region. The motion of the airplane depends on the quantity of goods, the control method, and the delay. It is shown that the control delay has an important effect on the motion.

Complex motion of elevators in piecewise map model combined with circle map

Abstract We study the dynamic behavior in the elevator traffic controlled by capacity when the inflow rate of passengers into elevators varies periodically with time. The dynamics of elevators is described by the piecewise map model combined with the circle map. The motion of the elevators depends on the inflow rate, its period, and the number of elevators. The motion in the piecewise map model combined with the circle map shows a complex behavior different from the motion in the piecewise map model.

NONSMOOTH AND SMOOTH BIFURCATIONS IN A DISCONTINUOUS PIECEWISE MAP

In this paper, a piecewise map with singularity of the power (-1/2) is introduced. For this piecewise map, there is an infinite discontinuous gap on the origin. The conditions of nonsmooth border-collision bifurcation and smooth fold or flip bifurcation are analytically derived. For period-1 fixed point, two-parameter-plane can be divided into seven ranges according to different bifurcation structures. For period-n orbits, codimension-2 bifurcation point may lead to different period-increment sequence, and a peculiar feature is found that there are two different period-increment sequences in the same bifurcation diagram.

Escape distribution for an inclined billiard

Hénon [8] used an inclined billiard to investigate aspects of chaotic scattering which occur in satellite encounters and in other situations. His model consisted of a piecewise mapping which described the motion of a point particle bouncing elastically on two disks. A one parameter family of orbits, named h-orbits, was obtained by starting the particle at rest from a given height. We obtain an analytical expression for the escape distribution of the h-orbits, which is also compared with results from numerical simulations. Finally, some discussion is made about possible applications of the h-orbits in connection with Hill’s problem.