NECESSARY AND SUFFICIENT CONDITIONS OF EXISTENCE AND UNIQUENESS OF LIMIT CYCLES FOR A CLASS OF POLYNOMIAL SYSTEM

In this paper, we discuss the limit cycles of the systemdx/dt=y·[1+(A(x)]oy/dt=(-x+δy+α_1x^2+α_2xy+α_5x^2y)[1+B(x)] (1)where A(x)=sum form i=1 to n(a_ix~), B(x)=sum form j=1 to m(β_jx^j) and 1+B(x)>0. We prove that (1) possesses at most one limit cycle and give out the necessary and sufficient conditions of existence and uniqueness of limit cycles.

CENTER CONDITIONS AND BIFURCATION OF LIMIT CYCLES FOR A CLASS OF FIFTH DEGREE SYSTEMS

The center conditions and bifurcation of limit cycles for a class of fifth degree systems are investigated.Two recursive formulas to compute singular quantities at infinity and at the origin are given.The first nine singular point quantities at infinity and first seven singular point quantities at the origin for the system are given in order to get center conditions and study bifurcation of limit cycles.Two fifth degree systems are constructed.One allows the appearance of eight limit cycles in the neighborhood of infinity,which is the first example that a polynomial differential system bifurcates eight limit cycles at infinity.The other perturbs six limit cycles at the origin.

Describing failure in geomaterials using second-order work approach

Geomaterials are known to be non-associated materials. Granular soils therefore exhibit a variety of failure modes, with diffuse or localized kinematical patterns. In fact, the notion of failure itself can be confusing with regard to granular soils, because it is not associated with an obvious phenomenology. In this study, we built a proper framework, using the second-order work theory, to describe some failure modes in geomaterials based on energy conservation. The occurrence of failure is defined by an abrupt increase in kinetic energy. The increase in kinetic energy from an equilibrium state, under incremental loading, is shown to be equal to the difference between the external second-order work,involving the external loading parameters, and the internal second-order work, involving the constitutive properties of the material. When a stress limit state is reached, a certain stress component passes through a maximum value and then may decrease. Under such a condition, if a certain additional external loading is applied, the system fails, sharply increasing the strain rate. The internal stress is no longer able to balance the external stress, leading to a dynamic response of the specimen. As an illustration, the theoretical framework was applied to the well-known undrained triaxial test for loose soils. The influence of the loading control mode was clearly highlighted. It is shown that the plastic limit theory appears to be a particular case of this more general second-order work theory. When the plastic limit condition is met, the internal second-order work is nil. A class of incremental external loadings causes the kinetic energy to increase dramatically, leading to the sudden collapse of the specimen, as observed in laboratory.

UBM STUDY ON DOUBLE-DIE IRONING PROCESS

The double-die ironing process is studied by means of UBM. The formulas of deformation load.contact stress on die surface, and tensile stress which acts on workpiece is obtained. Taking account of dirnensional accuracy, a new critical condition of limit reduction in cross section area is put forward for the flrst time. The test experiment indicats that results of theoretical analysis well accord with the actual conditions.[0]

The masking condition for the quantum state in two-dimensional Hilbert space

This paper focuses on quantum information masking for the quantum state in two-dimensional Hilbert space.We present a system of equations as the condition of quantum information masking.It is shown that quantum information contained in a single qubit state can be masked,if and only if the coefficients of the quantum state satisfy the given system of equations.By observing the characteristics of non-orthogonal maskable quantum states,we obtain a related conclusion,namely,if two non-orthogonal two-qubit quantum states can mask a single qubit state,they have the same number of terms and the same basis.Finally,for maskable orthogonal quantum states,we analyze two special examples and give their images for an intuitive description.