Bifurcation of Equilibria in a Class of Planar Piecewise Smooth Systems with 3 Parameters

The goal of this paper is to investigate the bifurcation properties of stationary points of a class of planar piecewise smooth systems with 3 parameters using the theory of differential inclusions. We especially study the existence of the stationary points on the line of discontinuity of this kind of planar piecewise smooth system.

Bifurcation in a Class of Planar Piecewise Smooth Systems with 3-parameters

This paper is concerned with the bifurcation properties on the line of discontinuity of planar piecewise smooth systems. The existence of equilibria and periodic solutions with sliding motion in a class of planar piecewise smooth systems with 3-parameters is investigated in this paper using the theory of differential inclu-sion and tools of Poincar′e maps.

On Stability Discrimination of Limit Cycles for Piecewise Smooth Systems

This paper is concerned with the problem of stability discrimination of limit cycles for piecewise smooth systems.We first establish the Poincaré map near a periodic orbit,and deduce the first order derivative of the map for general piecewise smooth systems on the plane.Then,we obtain a sufficient condition for determining the stability of limit cycles for these systems.

PIECEWISE SMOOTH SOLUTION OF THE FIRST ORDER SEMILINEAR HYPERBOLIC SYSTEMS IN HIGHER SPACE DIMENSION

It is important to study the propagation and interaction of progressing waves of nonlinear equations in the class of piecewise smooth function. However, there has not been many works on that in multidimensional case. In 1985, J, Rauch & M. Reed have provad the existence and uniqueness of piecewise smooth solution for

New family of piecewise smooth support vector machine

Support vector machines （SVMs） have been extensively studied and have shown remarkable success in many applications. A new family of twice continuously differentiable piecewise smooth functions are used to smooth the objective function of uncon- strained SVMs. The three-order piecewise smooth support vector machine （TPWSSVMd） is proposed. The piecewise functions can get higher and higher approximation accuracy as required with the increase of parameter d. The global convergence proof of TPWSSVMd is given with the rough set theory. TPWSSVMd can efficiently handle large scale and high dimensional problems. Nu- merical results demonstrate TPWSSVMa has better classification performance and learning efficiency than other competitive base- lines.

Dynamical analysis in a piecewise smooth predator-prey model with predator harvesting

The aim of this paper is to study the dynamical behaviors of a piecewise smooth predator-prey model with predator harvesting.We consider a harvesting strategy that allows constant catches if the population size is above a certain threshold value(to obtain predictable yield)and no catches if the population size is below the threshold(to protect the population).It is shown that boundary equilibrium bifurcation and sliding grazing bifurcation can happen as the threshold value varies.We provide analytical analysis to prove the existence of sliding limit cycles and sliding homoclinic cycles,the coexistence of them with standard limit cycles.Some numerical simulations are given to demonstrate ourresults.

BIFURCATION ANALYSIS OF A CLASS OF PLANAR PIECEWISE SMOOTH LINEAR-QUADRATIC SYSTEM

In this paper,we consider a planar piecewise smooth differential system consisting of a linear system and a quadratic Hamiltonian system.The quadratic system has some folds on the discontinuity line.The linear system may have a focus,saddle or node.Our results show that this piecewise smooth differential system will have two limit cycles and a sliding cycle.Moreover,this piecewise smooth system will undergo pseudo-homoclinic bifurcation,Hopf bifurcation and critical crossing bifurcation CC.Some examples are given to illustrate our results.

ESTIMATION AND UNCERTAINTY QUANTIFICATION FOR PIECEWISE SMOOTH SIGNAL RECOVERY

This paper presents an application of the sparse Bayesian learning(SBL)algorithm to linear inverse problems with a high order total variation(HOTV)sparsity prior.For the problem of sparse signal recovery,SBL often produces more accurate estimates than maximum a posteriori estimates,including those that useℓ1 regularization.Moreover,rather than a single signal estimate,SBL yields a full posterior density estimate which can be used for uncertainty quantification.However,SBL is only immediately applicable to problems having a direct sparsity prior,or to those that can be formed via synthesis.This paper demonstrates how a problem with an HOTV sparsity prior can be formulated via synthesis,and then utilizes SBL.This expands the class of problems available to Bayesian learning to include,e.g.,inverse problems dealing with the recovery of piecewise smooth functions or signals from data.Numerical examples are provided to demonstrate how this new technique is effectively employed.

Mather theory for piecewise smooth Lagrangian systems

We establish the Mather theory for a type of piecewise smooth and positive definite Lagrangian systems.It models a mechanical system subject to external impulsive forcing.We show the existence of the minimal measure and the Lipschitz property of Aubry set.In addition,the weak KAM solution to this kind of piecewise smooth Lagrangian is also established.

On the Number of Limit Cycles in Small Perturbations of a Piecewise Linear Hamiltonian System with a Heteroclinic Loop

In this paper, the authors consider limit cycle bifurcations for a kind of nonsmooth polynomial differential systems by perturbing a piecewise linear Hamiltonian system with a center at the origin and a heteroclinic loop around the origin. When the degree of perturbing polynomial terms is n(n ≥ 1), it is obtained that n limit cycles can appear near the origin and the heteroclinic loop respectively by using the first Melnikov function of piecewise near-Hamiltonian systems, and that there are at most n + [(n+1)/2] limit cycles bifurcating from the periodic annulus between the center and the heteroclinic loop up to the first order in ε. Especially, for n = 1, 2, 3 and 4, a precise result on the maximal number of zeros of the first Melnikov function is derived.

Retrieval of Single-Doppler Radar Wind Field by Nonlinear Approximation

The methods employed in recent years to retrieve vector wind information from single-Doppler radar observation are reviewed briefly. These methods are based on a linearity hypothesis for the wind field, so the retrieved wind field is sometimes negatively affected by the non-linearity of wind. This paper proposes a new method based on a non-linear approximation technique. This method, which relies on the piecewise smooth property of the wind field and makes full use of the radar velocity data, is applied to two cases of the Huaihe River Basin Energy and Water Cycle Experiment (HUBEX) in 1998. Checked against the wind field observed by dual-Doppler radar, the retrieved wind field by the method presented in this paper yields a relatively accurate horizontal vector wind field with high resolution, as well as a reasonable estimate of the magnitude of vertical velocity.

Smoothing Approximations for Some Piecewise Smooth Functions

In this paper,we study smoothing approximations for some piecewise smooth functions.We first present two approaches for one-dimensional case:a global approach is to construct smoothing approximations over the whole domain and a local approach is to construct smoothing approximations within appropriate neighborhoods of the nonsmooth points.We obtain some error estimate results for both approaches and discuss whether the smoothing approximations can inherit the convexity of the original functions.Furthermore,we extend the global approach to some multiple dimensional cases.

Study on the static properties of spiral springs under static loading

Spiral springs have a wide range of applications in various fields.As a result of the complexity of friction,few theoretical analyses of spring belts under static loading have been carried out.Considering the piecewise smooth property of the whole contact area,a simplified static model of spiral springs under loading is established in this paper.Besides,three main stress and friction distribution areas of the spring belt are proposed,namely,internal,transitional,and external regions.Since the outermost side of the spring is not subject to any pressure,a recursive method is adopted from the outside to the inside.The model provides the parameter conditions,i.e.,the internal and external forces are independent or dependent.Therefore,the case that the whole contact region of the spring belt has one subregion,two subregions,and three subregions is obtained.The model gives a theoretical basis for the parameter optimization of spiral springs.

REGULARITY OF ENTROPY SOLUTIONS TO NONCONVEX SCALAR CONSERVATION LAWS WITH MONOTONE INITIAL DATA

We prove that for a given strictly increasing initial datum in Ck, the solution of the initial value problem is piecewise Ck smooth except for flux functions of nonconvex conservation laws in a certain subset of C^k＋1 of first category, defined in the range of the initial datum.

APPROXIMATION PROPERTIES OF LAGRANGE INTERPOLATION POLYNOMIAL BASED ON THE ZEROS OF (1-x^2)cosnarccosx

We study some approximation properties of Lagrange interpolation polynomial based on the zeros of （1-x^2）cosnarccosx. By using a decomposition for f（x） ∈ C^τC^τ＋1 we obtain an estimate of ‖f（x） -Ln＋2（f, x）‖ which reflects the influence of the position of the x＇s and ω（f^（r＋1）,δ）j,j = 0, 1,... , s,on the error of approximation.

Principal Value and Its Applications for One Type of Cauchy–FantappièIntegral and the System of Equations

It is known that principal value plays vital roles in the study of singular integral.In this paper,one type of Cauchy–Fantappièintegral with multiple indexes is introduced.Since the structure of the kernel is complexity,the Jacobian determinant included in the kernel is expanded for making clearly the expression of the kernel.Moreover,one differential operator is utilized for setting up relations between integrals with higher and usual orders.The work also concerns the convergent properties of the integral.In order to study Hadamard principal value and composite formula of this integral,finite and divergent parts will be estimated and separated.As an application,solvability of the system of integral equations with higher order singularity kernel is discussed.

Further study on Horozov-Iliev's method of estimating the number of limit cycles

In the study of the number of limit cycles of near-Hamiltonian systems,the first order Melnikov function plays an important role.This paper aims to generalize Horozov-Iliev’s method to estimate the upper bound of the number of zeros of the function.

A High Order Method for Determining the Edges in the Gradient of a Function

Detection of edges in piecewise smooth functions is important in many applications.Higher order reconstruction algorithms in image processing and post processing of numerical solutions to partial differential equations require the identification of smooth domains,creating the need for algorithms that will accurately identify discontinuities in a given function as well as those in its gradient.This work expands the use of the polynomial annihilation edge detector,(Archibald,Gelb and Yoon,2005),to locate discontinuities in the gradient given irregularly sampled point values of a continuous function.The idea is to preprocess the given data by calculating the derivative,and then to use the polynomial annihilation edge detector to locate the jumps in the derivative.We compare our results to other recently developed methods.