WELL-POSEDNESS AND PEAKON SOLUTIONS FOR A HIGHER ORDER CAMASSA-HOLM TYPE EQUATION

In this paper,we delve into a generalized higher order Camassa-Holm type equation,(or,an ghmCH equation for short).We establish local well-posedness for this equation under the condition that the initial data uo belongs to the Sobolev space H'(R)for some s>2.In addition,we obtain the weak formulation of this equation and prove the existence of both single peakon solution and a multi-peakon dynamic system.

Control system design for a pressure-tube-type supercritical water-cooled nuclear reactor via a higher order sliding mode method

Nuclear power plants exhibit non-linear and time-variable dynamics.Therefore,designing a control system that sets the reactor power and forces it to follow the desired load is complicated.A supercritical water reactor(SCWR)is a fourth-generation conceptual reactor.In an SCWR,the non-linear dynamics of the reactor require a controller capable of control-ling the nonlinearities.In this study,a pressure-tube-type SCWR was controlled during reactor power maneuvering with a higher order sliding mode,and the reactor outgoing steam temperature and pressure were controlled simultaneously.In an SCWR,the temperature,pressure,and power must be maintained at a setpoint(desired value)during power maneuvering.Reactor point kinetics equations with three groups of delayed neutrons were used in the simulation.Higher-order and classic sliding mode controllers were separately manufactured to control the plant and were compared with the PI controllers speci-fied in previous studies.The controlled parameters were reactor power,steam temperature,and pressure.Notably,for these parameters,the PI controller had certain instabilities in the presence of disturbances.The classic sliding mode controller had a higher accuracy and stability;however its main drawback was the chattering phenomenon.HOSMC was highly accurate and stable and had a small computational cost.In reality,it followed the desired values without oscillations and chattering.

Opti mal higher order learning adaptive control approach for a class of SISO nonlinear systems

In this paper, an optimal higher order learning adaptive control approach is developed for a class of SISO nonlinear systems. This design is model-free and depends directly on pseudo-partial-derivatives derived on-line from the input and output information of the system. A novel weighted one-step-ahead control criterion function is proposed for the control law. The convergence analysis shows that the proposed control law can guarantee the convergence under the assumption that the desired output is a set point. Simulation examples are provided for nonlinear systems to illustrate the better performance of the higher order learning adaptive control.

Mei symmetries and Mei conserved quantities for higher-order nonholonomic constraint systems

This paper presents the Mei symmetries and new types of non-Noether conserved quantities for a higher-order nonholonomic constraint mechanical system. On the basis of the form invariance of differential equations of motion for dynamical functions under general infinitesimal transformation, the determining equations, the constraint restriction equations and the additional restriction equations of Mei symmetries of the system are constructed. The criterions of Mei symmetries, weak Mei symmetries and strong Mei symmetries of the system are given. New types of conserved quantities, i.e. the Mei symmetrical conserved quantities, the weak Mei symmetrical conserved quantities and the strong Mei symmetrical conserved quantities of a higher-order nonholonomic system, are obtained. Then, a deduction of the first-order nonholonomic system is discussed. Finally, two examples are given to illustrate the application of the method and then the results.

BIFURCATION OF A CLASS OF CODIMENSION-TWO NONLINEAR HIGHER ORDER SYSTEM

We discuss a kind of codimension-tvro nonlinear higher order system, by normai form theory the system can be reduced into system equivalent toBogdanov, Takens, Carr have been studied it's local bifurcations for n - 2, after that, Wang Mingshu, Luo Dingjun, Li Jibin, Wang Xian and others studied the global bifurcations for n = 3. In this paper, we study the bifurcations for n - 4, and give all the local bifurcation curves of the system.

Mei symmetry and new conserved quantities of Tzénoff equations for higher-order nonholonomic system

In this paper,the Mei symmetry of Tzénoff equations for the higher-order nonholonomic system and the new conserved quantities derived from that are researched,and the function expression of new conserved quantities and criterion equation which deduces these conserved quantities are presented.This result establishes the theory basis for further researches on conservation laws of Tzénoff equations of the higher-order nonholonomic constraint system.

CLASSIFICATION OF SOLUTIONS TO HIGHER FRACTIONAL ORDER SYSTEMS

Let 0<α,β<n and f,g∈ C([0,∞)×[0,∞))be two nonnegative functions.We study nonnegative classical solutions of the system{(-△)^(α/2)u=f(u,v)in R^(n),(-△)^(β/2)v=g(u,v)in R^(n),and the corresponding equivalent integral system.We classify all such solutions when f(s,t)is nondecreasing in s and increasing in t,g(s,t)is increasing in s and nondecreasing in i,and f(μ^(n-α)s,μ^(n-β)t)/μ^(n-α),g(μ^(n-α)s,μ^(n-β)t)/μ^(n-β)are nonincreasing in μ>0 for all s,t≥0.The main technique we use is the method of moving spheres in integral forms.Since our assumptions are more general than those in the previous literature,some new ideas are introduced to overcome this difficulty.

EQUATION OF MATHEMATICAL PHYSICS AND OTHER AREAS OF APPLICATION: A CLASSICAL INTEGRABLE SYSTEM AND THE INVOLUTIVE REPRESENTATION OF SOLUTIONS OF THE HIGHER ORDER KAUP-NEWELL EQUATION

Under the constrained condition induced by the eigenfunction expresson of the potential (u, v)T = (-[A2q, q], [A2p, p])T = f (q, p), the spatial part of the Lax pair of the Kaup-Newell equation is non linearized to be a completely integrable system (R2N, Adp AND dq, H = H-1) with the Hamiltonian H-1 = -[A3q, p]-1/2[A2p, p][A2q, q]. while the nonlinearization of the time part leads to its N-involutive system {H(m)}. The involutive solution of the compatible fsystem (H-1), (H(m)) is mapped by into the solution of the higher order Kaup-Newell equation.

The permanence of a higher order discrete time system

The permanence of a nonlinear higher order discrete time system from macroeconomics is studied, and a sufficient condition is proposed for the permanence of the system described by 11(,...,)nnnnkxrxfxx---=+ where :kfRR, the initial values 01,,kxx-are real numbers and [0,1)r is constant after exploring the relationship between this equation and 1(,...,)nnnkxfxx--= for certain classes of function f. As an application a short proof is given to a known result in a simpler way than ever reported.

ONE TYPE OF INTEGRALS FOR THE EQUATIONS OF MOTION OF HIGHER-ORDER NONHOLONOMIC SYSTEMS

This paper presents one type of integrals and its condition of existence for the equations of motion of higher-order nonholonomic systems, including l-order integral (generalized energy integral), 2-order integral and p-order integral (p>2)All of these integrals can be constructed by the Lagrangian function of the system. Two examples are given to illustrate the application of the suggested method.

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

HIGHER ORDER SPECTRA AND ITS APPLICATION IN MACHINERY FAULT DIAGNOSIS

The application of higher order spectra to machinery faults diagnosis is studied in this paper.A brief review of bispectra is presented,and more emphasis is placed on the ability of higher order spectra to extract diagnostic information from fault signals.Furthermore,by use of the algorithm of higher order spectra,two kinds of typical mechanical faults are analyzed.Results show that the high order spectra analysis is a more efficient method in machinery diagnosis compared with the FFT based spectral analysis.

PARTIAL REGULARITY FOR MINIMIZERS OF HIGHER ORDER QUASICONVEX FUNCTIONALS

We consider the following quasiconvex functional I(u)=∫ Gf(x,δu,D mu) d x where u is a vector valued function in W m,p (G),m>1 and p>2. The partial C m,a —regularity is proved for minimizers of I(u) under weaker conditions.

Robust Adaptive Gain Higher Order Sliding Mode Observer Based Control-constrained Nonlinear Model Predictive Control for Spacecraft Formation Flying

This work deals with the development of a decentralized optimal control algorithm, along with a robust observer,for the relative motion control of spacecraft in leader-follower based formation. An adaptive gain higher order sliding mode observer has been proposed to estimate the velocity as well as unmeasured disturbances from the noisy position measurements.A differentiator structure containing the Lipschitz constant and Lebesgue measurable control input, is utilized for obtaining the estimates. Adaptive tuning algorithms are derived based on Lyapunov stability theory, for updating the observer gains,which will give enough flexibility in the choice of initial estimates.Moreover, it may help to cope with unexpected state jerks. The trajectory tracking problem is formulated as a finite horizon optimal control problem, which is solved online. The control constraints are incorporated by using a nonquadratic performance functional. An adaptive update law has been derived for tuning the step size in the optimization algorithm, which may help to improve the convergence speed. Moreover, it is an attractive alternative to the heuristic choice of step size for diverse operating conditions. The disturbance as well as state estimates from the higher order sliding mode observer are utilized by the plant output prediction model, which will improve the overall performance of the controller. The nonlinear dynamics defined in leader fixed Euler-Hill frame has been considered for the present work and the reference trajectories are generated using Hill-Clohessy-Wiltshire equations of unperturbed motion. The simulation results based on rigorous perturbation analysis are presented to confirm the robustness of the proposed approach.

Generalizations of Euler Numbers and Euler Numbers of Higher Order

The purpose of this paper is to define the generalized Euler numbers and the generalized Euler numbers of higher order, their recursion formula and some properties were established, accordingly Euler numbers and Euler numbers of higher order were extended.

Visual function and higher order aberration after implantation of aspheric and spherical multifocal intraocular lenses:a meta-analysis

AIMTo assess the visual outcomes of aspheric multifocal intraocular lenses (IOLs) compared with spherical multifocal IOL after cataract surgery.

SINGULAR INTEGRAL EQUATIONS ALONG AN OPEN ARC WITH SOLUTIONS HAVING SINGULARITIES OF HIGHER ORDER

In this paper, the difficulties on calculation in solving singular integral equations are overcome when the restriction of curve of integration to be a closed contour is cancelled. When the curve is an open arc and the solutions for singular integral equations possess singularities of higher order, the solution and the solvable condition for characteristic equations as well as the generalized Noether theorem for complete equations are given.

HIGHER ORDER SPECTRAL ANALYSIS IN FAULT DIAGNOSIS OF ROTORS

The nonlinear properties of rotating machinery vibration signals are presented. The relationship between faults and quadratic phase coupling is discussed. The mechanism that gives rise to quadratic phase coupling is analyzed, and the coupling models are summarized. As a result, higher order spectra analysis is introduced into fault diagnosis of rotors. A brief review of the properties of higher order spectra is presented. Furthermore, the bicoherence spectrum is employed to extract the features that signify the machinery condition. Experiments show that bicoherence spectrum patterns of different faults are quite different, so it is proposed to identify the faults in rotors.

HIGHER ORDER SINGULAR INTEGRAL EQUATIONS ON COMPLEX HYPERSPHERE

A theory of a class of higher order singular integral under the operator （L f） （u） =[u1 σf/σu1（u） - u1σf/σu1（u） ＋ f（u）] is given. We transform the higher order singular integral to a usual Cauchy integral, extend the permutation formula of the higher order singular integral deduced by Qian and Zhong in [4] to a general case, and discuss the regularization problem of the higher order singular integral equations with Cauchy kernel and variable coefficients on complex hypersphere.

Higher order aberration comparison between two aspherical intraocular lenses: MC6125AS and Akreos advanced optics

AIM: To compare higher order aberrations in two aspherical intraocular lenses(IOLs): Akreos advanced optics(AO) and Dr. Schmidt Microcrystalline 6125 aspheric anterior surface(MC6125AS) with each other. METHODS: Forty eyes of 39 patients underwent phacoemulsification and Akreos AO and MC6125 AS were implanted in their eyes in a random manner. Three months post-operatively, higher order aberrations including spherical aberration, coma aberration, and total aberrations were measured and compared.RESULTS: The total aberration was 0.24±0.17 in eyes with Dr. Schmidt and 0.20 ±0.01 in eyes with Akreos AO(P =0.361). The mean of coma aberration was 0.17 ±0.21 and 0.09 ±0.86 in Dr. Schmidt and Akreos lenses,respectively(P =0.825). Total spherical aberration was almost the same in both groups(mean: 0.05, P =0.933).Best corrected visual acuity in Akreos AO(0.10±0.68) and Dr. Schmidt(0.09±0.67) did not differ significantly(P =0.700). CONCLUSION: There is no statistically significant difference in the higher order aberrations between these two aspherical lenses.