A Local Discontinuous Galerkin Method for Two-Dimensional Time Fractional Diffusion Equations

For two-dimensional(2D)time fractional diffusion equations,we construct a numerical method based on a local discontinuous Galerkin(LDG)method in space and a finite differ-ence scheme in time.We investigate the numerical stability and convergence of the method for both rectangular and triangular meshes and show that the method is unconditionally stable.Numerical results indicate the effectiveness and accuracy of the method and con-firm the analysis.

Discontinuous bifurcation and coexistence of attractors in a piecewise linear map with a gap

Coexistence of attractors with striking characteristics is observed in this work, where a stable period-5 attractor coexists successively with chaotic band-ll, period-6, chaotic band-12 and band-6 attractors. They are induced by dif- ferent mechanisms due to the interaction between the discontinuity and the non-invertibility. A characteristic boundary collision bifurcation, is observed. The critical conditions are obtained both analytically and numerically.

Interpreting a period-adding bifurcation scenario in neural bursting patterns using border-collision bifurcation in a discontinuous map of a slow control variable

To further identify the dynamics of the period-adding bifurcation scenarios observed in both biological experiment and simulations with differential Chay model, this paper fits a discontinuous map of a slow control variable of Chay model based on simulation results. The procedure of period adding bifurcation scenario from period k to period k ＋ 1 bursting （k = 1, 2, 3, 4） involved in the period-adding cascades and the stochastic effect of noise near each bifurcation point is also reproduced in the discontinuous map. Moreover, dynamics of the border-collision bifurcation is identified in the discontinuous map, which is employed to understand the experimentally observed period increment sequence. The simple discontinuous map is of practical importance in modeling of collective behaviours of neural populations like synchronization in large neural circuits.

Partial and complete periodic synchronization in coupled discontinuous map lattices

The partial and complete periodic synchronization in coupled discontinuous map lattices consisting of both discon- tinuous and non-invertible maps are discussed. We classify three typical types of periodic synchronization states, which give rise to different spatiotemporal patterns including static partial periodic synchronization, dynamically periodic syn- chronization, and complete periodic synchronization patterns. A special prelude dynamics of partial and complete periodic synchronization motion, which is shown by five separated concave curves in the time series plots of the order parameters, is observed. The detailed analysis shows that the special prelude dynamics is induced by the competition between two synchronized clusters, and the analytical expression for the corresponding order parameter is obtained.

A Prelude Staircase to a Type V Intermittency in Two—Dimensional Discontinuous Maps

A sequence of periodic attractors has been observed in a two-dimensional discontinuous map, which canbe considered as a model of impact oscillator. The so-called 'transfer number', which is defined as the mean numberof transfer from non-impact state to impact state per iteration, is locked onto a lot of rational values to form a curveconsisting of many steps. Our numerical investigation confirms that every step is confined by conditions created by thecollision between the periodic orbit and the discontinuous boundary of the system. After the last collision the systemshows a chaotic motion with intermittent characteristics. Therefore the staircase can be addressed as a 'prelude staircaseto type V intermittency'. The similar phenomenon has also been observed in a model of electric circuit. These resultsof our study suggest that this kind of staircases is common in two (or even higher) dimensional discontinuous maps.

A novel class of two-dimensional chaotic maps with infinitely many coexisting attractors

We study a novel class of two-dimensional maps with infinitely many coexisting attractors.Firstly,the mathematical model of these maps is formulated by introducing a sinusoidal function.The existence and the stability of the fixed points in the model are studied indicating that they are infinitely many and all unstable.In particular,a computer searching program is employed to explore the chaotic attractors in these maps,and a simple map is exemplified to show their complex dynamics.Interestingly,this map contains infinitely many coexisting attractors which has been rarely reported in the literature.Further studies on these coexisting attractors are carried out by investigating their time histories,phase trajectories,basins of attraction,Lyapunov exponents spectrum,and Lyapunov(Kaplan–Yorke)dimension.Bifurcation analysis reveals that the map has periodic and chaotic solutions,and more importantly,exhibits extreme multi-stability.

Image encryption technique based on new two-dimensional fractional-order discrete chaotic map and Menezes–Vanstone elliptic curve cryptosystem

We propose a new fractional two-dimensional triangle function combination discrete chaotic map(2D-TFCDM)with the discrete fractional difference.Moreover,the chaos behaviors of the proposed map are observed and the bifurcation diagrams,the largest Lyapunov exponent plot,and the phase portraits are derived,respectively.Finally,with the secret keys generated by Menezes-Vanstone elliptic curve cryptosystem,we apply the discrete fractional map into color image encryption.After that,the image encryption algorithm is analyzed in four aspects and the result indicates that the proposed algorithm is more superior than the other algorithms.

Geophysical Significance of the Senegalo-Mali Discontinuity: Evidence from Secondary Structures, Kédougou-Kéniéba Inlier, Western Mali

The present study focuses on the analysis and description of lineaments interpreted as secondary structures to describe the nature of Senegalo Malian Discontinuity. These lineaments cross-cut the large north-south oriented transcurrent lithospheric structure known as the Senegalo Malian Discontinuity (SMD). Two lineaments were selected oriented NNE (N15˚ to N25˚), one at Dialafara and one at Sadiola. Four profiles on each lineament of these 2 zones, so that there were 2 on each side of the SMD. The ground data collected were processed using proper parameter and software. Some filters were applied to enhance the signal level. These ground data were later compared to the existing airborne magnetic data for consistency and accuracy using the upward continuation filter. The results show that the quality of ground data is good. In addition, the ground magnetic data show the presence of certain local anomalies that are not visible in the regional data. The analytical signal was also used to determine domain boundaries or possible contact zones. The contact zone can be highlighted on certain profiles such as L300 and L600. The study showed that the west and east sides of the SMD are not the same. Secondary structures become wide when approaching the SMD on both sides. They are also duplicated to the east of the SMD when we move progressively away. In the Dialafara area, the ground magnetic data intersect an interpreted fold. The results of this work confirm the presence of the secondary structures and their evolution in relation to the SMD. The relationships between the secondary structures in the Dailafara and Sadiola zones and their relations with the SMD are highlighted. The technique used in this study, is an important approach to better description and interpreting of regional structures using the secondary structures and proposing a structural model.

Phase order in one-dimensional piecewise linear discontinuous map

The phase order in a one-dimensional（1 D） piecewise linear discontinuous map is investigated. The striking feature is that the phase order may be ordered or disordered in multi-band chaotic regimes, in contrast to the ordered phase in continuous systems. We carried out an analysis to illuminate the underlying mechanism for the emergence of the disordered phase in multi-band chaotic regimes, and proved that the phase order is sensitive to the density distribution of the trajectories of the attractors. The scaling behavior of the net direction phase at a transition point is observed. The analytical proof of this scaling relation is obtained. Both the numerical and analytical results show that the exponent is 1, which is controlled by the feature of the map independent on whether the system is continuous or discontinuous. It extends the universality of the scaling behavior to systems with discontinuity. The result in this work is important to understanding the property of chaotic motion in discontinuous systems.

Extremely hidden multi-stability in a class of two-dimensional maps with a cosine memristor

We present a class of two-dimensional memristive maps with a cosine memristor. The memristive maps do not have any fixed points, so they belong to the category of nonlinear maps with hidden attractors. The rich dynamical behaviors of these maps are studied and investigated using different numerical tools, including phase portrait, basins of attraction,bifurcation diagram, and Lyapunov exponents. The two-parameter bifurcation analysis of the memristive map is carried out to reveal the bifurcation mechanism of its dynamical behaviors. Based on our extensive simulation studies, the proposed memristive maps can produce hidden periodic, chaotic, and hyper-chaotic attractors, exhibiting extremely hidden multistability, namely the coexistence of infinite hidden attractors, which was rarely observed in memristive maps. Potentially,this work can be used for some real applications in secure communication, such as data and image encryptions.

A class of two-dimensional rational maps with self-excited and hidden attractors

This paper studies a new class of two-dimensional rational maps exhibiting self-excited and hidden attractors. The mathematical model of these maps is firstly formulated by introducing a rational term. The analysis of existence and stability of the fixed points in these maps suggests that there are four types of fixed points, i.e., no fixed point, one single fixed point, two fixed points and a line of fixed points. To investigate the complex dynamics of these rational maps with different types of fixed points, numerical analysis tools, such as time histories, phase portraits, basins of attraction, Lyapunov exponent spectrum, Lyapunov(Kaplan–Yorke) dimension and bifurcation diagrams, are employed. Our extensive numerical simulations identify both self-excited and hidden attractors, which were rarely reported in the literature. Therefore, the multi-stability of these maps, especially the hidden one, is further explored in the present work.

A modified method of discontinuity trace mapping using three-dimensional point clouds of rock mass surfaces

This paper presents an automated method for discontinuity trace mapping using three-dimensional point clouds of rock mass surfaces.Specifically,the method consists of five steps:(1)detection of trace feature points by normal tensor voting theory,(2)co ntraction of trace feature points,(3)connection of trace feature points,(4)linearization of trace segments,and(5)connection of trace segments.A sensitivity analysis was then conducted to identify the optimal parameters of the proposed method.Three field cases,a natural rock mass outcrop and two excavated rock tunnel surfaces,were analyzed using the proposed method to evaluate its validity and efficiency.The results show that the proposed method is more efficient and accurate than the traditional trace mapping method,and the efficiency enhancement is more robust as the number of feature points increases.

Interfacial fracture analysis for a two-dimensional decagonal quasi-crystal coating layer structure

The interface crack problems in the two-dimensional(2D)decagonal quasicrystal(QC)coating are theoretically and numerically investigated with a displacement discontinuity method.The 2D general solution is obtained based on the potential theory.An analogy method is proposed based on the relationship between the general solutions for 2D decagonal and one-dimensional(1D)hexagonal QCs.According to the analogy method,the fundamental solutions of concentrated point phonon displacement discontinuities are obtained on the interface.By using the superposition principle,the hypersingular boundary integral-differential equations in terms of displacement discontinuities are determined for a line interface crack.Further,Green’s functions are found for uniform displacement discontinuities on a line element.The oscillatory singularity near a crack tip is eliminated by adopting the Gaussian distribution to approximate the delta function.The stress intensity factors(SIFs)with ordinary singularity and the energy release rate(ERR)are derived.Finally,a boundary element method is put forward to investigate the effects of different factors on the fracture.

A Local Discontinuous Galerkin Method with Generalized Alternating Fluxes for 2D Nonlinear Schrödinger Equations

In this paper,we consider the local discontinuous Galerkin method with generalized alter-nating numerical fluxes for two-dimensional nonlinear Schrödinger equations on Carte-sian meshes.The generalized fluxes not only lead to a smaller magnitude of the errors,but can guarantee an energy conservative property that is useful for long time simulations in resolving waves.By virtue of generalized skew-symmetry property of the discontinuous Galerkin spatial operators,two energy equations are established and stability results con-taining energy conservation of the prime variable as well as auxiliary variables are shown.To derive optimal error estimates for nonlinear Schrödinger equations,an additional energy equation is constructed and two a priori error assumptions are used.This,together with properties of some generalized Gauss-Radau projections and a suitable numerical initial condition,implies optimal order of k+1.Numerical experiments are given to demonstrate the theoretical results.

相位测量的不连续位置自适应分割

为了消除不连续位置对基于相位测量轮廓术三维重建精度的影响,提出了一种相位测量的不连续位置自适应分割方法。对该方法所采用的基于自适应方向相干性的相位不连续分割算法进行研究。首先,采用自适应方向相干图和平滑相位图确定不连续位置,即获得不连续区域与连续区域的灰度映射图。接着,将灰度映射图后处理为使用0-1表示的不连续区域与连续区域的掩码图。最后,将掩码图作为加权最小二乘相位展开方法的权重图将相对相位展开为绝对相位,所求解权重主要用以衡量包裹相位图中每个像素位置相位的质量。仿真结果表明:该自适应方向相干法对噪声方差为0.8的包裹相位图中直线和矩形不连续位置的分割平均误差分别为1.6783和3.0002个像素。实际相位测量轮廓实验也证明所提出方法可以有效分割包裹相位图不连续位置。所提出方法可实现对不连续位置的自适应分割以用于高精度的三维重建。

Symbolic dynamics of Belykh-type maps

The symbolic dynamics of a Belykh-type map （a two-dimensional discon- tinuous piecewise linear map） is investigated. The admissibility condition for symbol sequences named the pruning front conjecture is proved under a hyperbolicity condition. Using this result, a symbolic dynamics model of the map is constructed according to its pruning front and primary pruned region. Moreover, the boundary of the parameter region in which the map is chaotic of a horseshoe type is given.

Local Discontinuous Galerkin Methods for the Two-Dimensional Camassa–Holm Equation Dedicated to Celebrate the Sixtieth Anniversary of USTC

In this paper,the local discontinuous Galerkin method is developed to solve the two-dimensional Camassa–Holm equation in rectangular meshes.The idea of LDG methods is to suitably rewrite a higher-order partial differential equations into a firstorder system,then apply the discontinuous Galerkin method to the system.A key ingredient for the success of such methods is the correct design of interface numerical fluxes.The energy stability for general solutions of the method is proved.Comparing with the Camassa–Holm equation in one-dimensional case,there are more auxiliary variables which are introduced to handle high-order derivative terms.The proof of the stability is more complicated.The resulting scheme is high-order accuracy and flexible for arbitrary h and p adaptivity.Different types of numerical simulations are provided to illustrate the accuracy and stability of the method.

Depth Distribution of Moho Discontinuity Beneath Beijing and Its Adjacent Area

Based on the method of "two-dimensional depth structure of the crust" proposed by Horiuchi et al., about 5000 arrival times of 303 local shallow earthquakes recorded by the Beijing Seismographic Network from 1990 ~ 1993 are used to investigate the depth distribution of Moho discontinuity beneath Beijing and its adjacent area. We simultaneously determined the hypocenter parameters and P- and S-wave station corrections. The data of the North China Network were also investigated. The results are as follows: (1) The depth distribution of Moho discontinuity becomes shallower from the northwest to the southeast, i.e., in Zhangjiakou area, the Moho discontinuity is located at a depth range from 40~42 km. In the Beijing area, it is 36~39 km. However, at the eastern and southeastern part of this area, it is only 28-30 km and 30~32 km, respectively. (2) Beneath the Tangshan area, there is another elliptic interface shallower than the Moho discontinuity. Separately, its major and minor axis is approximately

二维不连续映射中心分叉的动力学分析

研究了二维分段不连续线性映射中心分叉的一些性质.类比光滑映射Neimark-Sacker分叉分析了中心分叉.首先,当定义分段线性映射的两个线性矩阵中有一个的特征值为共轭复根并且通过单位圆时,会发生中心分叉;其次,当定义分段线性映射的两个线性矩阵的特征值均为共轭复根并且通过单位圆时,会发生中心分叉.证明了发生中心分叉时在相空间中存在两个与不动点有关的不变区域,并且给出了寻找不变区域并确定其位置和大小的计算方法.讨论了相空间中在不变区域外的动力学行为,最后说明了系统在中心分叉后在不动点附近可能存在不变闭曲线.

The best approximation theorems and variational inequalities for discontinuous mappings in Banach spaces

We discuss Ky Fan's theorem and the variational inequality problem for discontinuous mappings f in a Banach space X. The main tools of analysis are the variational characterizations of the metric projection operator and the order-theoretic fixed point theory. Moreover, we derive some properties of the metric projection operator in Banach spaces. As applications of our best approximation theorems, three fixed point theorems for non-self maps are established and proved under some conditions. Our results are generalizations and improvements of various recent results obtained by many authors.