Research on Chaos of Nonlinear Singular Integral Equation

In this paper, one class of nonlinear singular integral equation is discussed through Lagrange interpolation method. We research the connections between numerical solutions of the equations and chaos in the process of solving by iterative method.

A Kind of Cover Layer Phenomenon of Solution for the Singularly Perturbed Problem with Third Order Nonlinear Equation

利用微分不等式理论研究了一类三阶非线性奇摄动问题解的套层现象。得到了问题解的一致有效的渐近展开式。

A singularly perturbed reaction diffusion problem for the nonlinear boundary condition with two parameters

A class of singularly perturbed initial boundary value problems of reaction diffusion equations for the nonlinear boundary condition with two parameters is considered. Under suitable conditions, by using the theory of differential inequalities, the existence and the asymptotic behaviour of the solution for the initial boundary value problem are studied. The obtained solution indicates that there are initial and boundary layers and the thickness of the boundary layer is less than the thickness of the initial layer.

THE NONLINEAR SINGULARLY PERTURBED PROBLEMS FOR REACTION DIFFUSION EQUATIONS WITH TIME DELAY

A class of nonlinear for singularly perturbed problems for reaction diffusion equations with time delays are considered. Under suitable conditions, using theory of differential inequalities the asymptotic behavior of solution for the initial boundary value problems are studied.

A CLASS OF NONLINEAR SINGULARLY PERTURBED PROBLEMS FOR REACTION DIFFUSION EQUATIONS

A class of nonlinear singularly perturbed problems for reaction diffusion equations are considered. Under suitable conditions, by using the theory of differential inequalities, the asymptotic behavior of solutions for the initial boundary value problems are studied, reduced problems of which possess two intersecting solutions.

THE NONLINEAR NONLOCAL SINGULARLY PERTURBED PROBLEMS FOR REACTION DIFFUSION EQUATIONS

A class of nonlinear nonlocal for singularly perturbed Robin initial boundary value problems for reaction diffusion equations is considered. Under suitable conditions, firstly, the outer solution of the original problem is obtained, secondly, using the stretched variable, the composing expansion method and the expanding theory of power series the initial layer is constructed, finally, using the theory of differential inequalities the asymptotic behavior of solution for the initial boundary value problems are studied and educing some relational inequalities the existence and uniqueness of solution for the original problem and the uniformly valid asymptotic estimation is discussed.

Singular perturbation for the weakly nonlinear reaction diffusion equation with boundary perturbation

In this paper, a class of nonlinear singularly perturbed initial boundary value problems for reaction diffusion equations with boundary perturbation are considered under suitable conditions. Firstly, by dint of the regular perturbation method, the outer solution of the original problem is obtained. Secondly, by using the stretched variable and the expansion theory of power series the initial layer of the solution is constructed. And then, by using the theory of differential inequalities, the asymptotic behavior of the solution for the initial boundary value problems is studied. Finally, using some relational inequalities the existence and uniqueness of solution for the original problem and the uniformly valid asymptotic estimation are discussed.

NONLINEAR SINGULARLY PERTURBED PREDATOR-PREY REACTION DIFFUSION SYSTEMS

A class of nonlinear predator prey reaction diffusion systems for singularly pe rturbed problems are considered.Under suitable conditions, by using theory of di fferential inequalities the existence and asymptotic behavior of solution for in itial boundary value problems are studied.

Singularly perturbed solution for weakly nonlinear equations with two parameters

A class of singularly perturbed boundary value problems of weakly non linear equation for fourth order on the interval [a, b] with two parameters is considered. Under suitable conditions, firstly, the reduced solution and formal outer solution are constructed using the expansion method of power series. Secondly, using the transformation of stretched variable, the first boundary layer corrective term near x = a is constructed which possesses exponential attenuation behavior. Then, using the stronger transformation of stretched variable, the second boundary layer corrective term near x = a is constructed, wtfich also possesses exponential attenuation behavior. The thickness of second boundary layer is smaller than the first one and forms a cover layer near x = a. Finally, using the theory of differential inequalities, the existence, uniform validity in the whole interval [a, b] and asymptotic behavior of solution for the original boundary value problem are proved. Satisfying results are obtained.

Design of a fault diagnosis scheme for a class of singular nonlinear systems

A new fault detection and diagnosis approach is developed in this paper for a class of singular nonlinear systems via the use of adaptive updating rules. Both detection and diagnostic observers are established, where Lyapunov stability theory is used to obtain the required adaptive tuning rules for the estimation of the process faults. This has led to stable observation error systems for both fault detection and diagnosis. A simulated numerical example is included to demonstrate the use of the proposed approach and encouraging results have been obtained.

A CLASS OF SINGULARLY PERTURBED NONLINEAR BOUNDARY VALUE PROBLEM

The singularly perturbed boundary value problem for the nonlinear boundary conditions is considered.Under suitable conditions,the asymptotic behavior of solution for the original problems is studied by using theory of differential inequalities.

The Nonlinear Nonlocal Singularly Perturbed Problems for Elliptic Equations with Boundary Perturbation

The nonlinear nonlocal singularly perturbed boundary value problems for elliptic equation with boundary perturbation was considered.Under suitable conditions,firstly,the outer solution of the original problem is obtained,secondly,using the stretched variable,the composing expansion method and the expanding theory of power series the boundary layer is constructed,finally,using the theory of differential inequalities the asymptotic behavior of solution for the boundary value problems is studied and educing some relational inequalities the existence and uniqueness of solution for the original problem and the uniformly valid asymptotic estimation is discussed.

ON THE CAUCHY PROBLEM FOR A REACTION-DIFFUSION SYSTEM WITH SINGULAR NONLINEARITY

We consider the growth rate and quenching rate of the following problem with singular nonlinearityfor some positive constants b：, b2 （see Theorem 3.3 for the parametersfor some constantsHence, the solution （u, v） quenches at the originx = 0 at the same time ＇1＇ （see Theorem 4.3）. We also tind various other conditions tor the solution to quench in a finite time and obtain the corresponding decay rate of the solution near the quenching time.

Characteristics and Nonlinear Growth of the Singular Vector Related to a Heavy Rainfall Case over the Korean Peninsula

In this study, singular vectors related to a heavy rainfall case over the Korean Peninsula were calculated using the fifth-generation Pennsylvania State University-National Center for Atmospheric Research Mesoscale Model （MM5） adjoint modeling system. Tangent linear and adjoint models include moist physical processes, and a moist basic state and a moist total energy norm were used for the singular-vector calculations. The characteristics and nonlinear growth of the first singular vector were analyzed, focusing on the relationship between the basic state and the singular vector. The horizontal distribution of the initial singular vector was closely related to the baroclinicity index and the moisture availability of the basic state. The temperature-component energy at a lower level was dominant at the initial time, and the kinetic energy at upper levels became dominant at the final time in the energy profile of the singular vector. The nonlinear growth of the singular vector appropriately reflects the temporal variations in the basic state. The moisture-component energy at lower levels was dominant at earlier times, indicating continuous moisture transport in the basic state. There were a large amount of precipitation and corresponding latent heat release after that period because the continuous moisture transport created favorable conditions for both convective and nonconvective precipitation. The vertical propagation of the singular-vector energy was caused by precipitation and the corresponding latent heating in the basic state.

On Generating Expected Kinetostatic Nonlinear Stiffness Characteristics by the Kinematic Limb-Singularity of a Crank-Slider Linkage with Springs

Being different from avoidance of singularity of closed-loop linkages, this paper employs the kinematic singularity to construct compliant mechanisms with expected nonlinear stiffness characteristics to enrich the methods of compliant mechanisms synthesis. The theory for generating kinetostatic nonlinear stiffness characteristic by the kinematic limb-singularity of a crank-slider linkage is developed. Based on the principle of virtual work, the kinetostatic model of the crank-linkage with springs is established. The influences of spring stiffness on the toque-position angle relation are analyzed. It indicates that corresponding spring stiffness may generate one of four types of nonlinear stiffness characteristics including the bi-stable, local negative-stiffness, zero-stiffness or positive-stiffness when the mechanism works around the kinematic limb-singularity position. Thus the compliant mechanism with an expected stiffness characteristic can be constructed by employing the pseudo rigid-body model of the mechanism whose joints or links are replaced by corresponding flexures. Finally, a tri-symmetrical constant-torque compliant mechanism is fabricated, where the curve of torque-position angle is obtained by an experimental testing. The measurement indicates that the compliant mechanism can generate a nearly constant-torque zone.

A CLASS OF NONLINEAR NONLOCAL SINGULARLY PERTURBED PROBLEMS FOR REACTION DIFFUSION EQUATIONS WITH BOUNDARY PERTURBATION

A class of nonlinear nonlocal for singularly perturbed Robin initial boundary value problems for reaction diffusion equations with boundary perturbation is considered. Under suitable conditions, first, the outer solution of the original problem was obtained. Secondly, using the stretched variable, the composing expansion method and the expanding theory of power series the initial layer was constructed. Finally, using the theory of differential inequalities the asymptotic behavior of solution for the initial boundary value problems was studied, and educing some relational inequalities the existence and uniqueness of solution for the original problem and the uniformly valid asymptotic estimation were discussed.

ASYMPTOTIC SOLUTION OF A SINGULARLY PERTURBED NONLINEAR STATE REGULATOR PROBLEM

This paper, will seek the optimal control and corresponding trajectories of the singularly perturbed nonlinear state regulator problem. Under appropriate hypotheses. it will be possible to complete an asymptotic solution which is uniformly valid when e→0.

SINGULAR ANALYSIS OF BIFURCATION OF NONLINEAR NORMAL MODES FOR A CLASS OF SYSTEMS WITH DUAL INTERNAL RESONANCES

The nonlinear normal modes (NNMs) associated with integrnal resonance can be classified into two kinds: uncoupled and coupled. The bifurcation problem of the coupled NNM of system with 1 : 2 : 5 dual internal resonance is in two variables. The singular analysis of it is presented after separating the two variables by taking advantage of Maple algebra, and some new bifurcation patterns are found. Different from the NNMs of systems with single internal resonance, the number of the NNMs of systems with dual internal resonance may be more or less than the number of the degrees of freedom. At last, it is pointed out that bifurcation problems in two variables can be conveniently solved by separating variables as well as using coupling equations.

A CLASS OF NONLINEAR SINGULARLY PERTURBED INITIAL BOUNDARY VALUE PROBLEMS FOR REACTION DIFFUSION EQUATIONS WITH BOUNDARY PERTURBATION

A class of nonlinear initial boundary value problems for reaction diffusion equations with boundary perturbation is considered. Under suitable conditions and using the theory of differential inequalities the asymptotic solution of the initial boundary value problems is studied.