On the High-Order Quasi Exactly Solvable Differential Equations

In this paper, we present a new method for solving a class of high-order quasi exactly solvable ordinary differential equations. With this method, the computed solution is expressed as a linear combination of the canonical polynomials associated with the given differential operator. An iterative algorithm summarizing the procedure is presented and its efficiency is demonstrated through considering two applied problems.

Sliding-mode observers for systems with unknown inputs and measurement disturbances

The method to design sliding-mode observers for systems with unknown inputs and measurement disturbances is presented in the paper. An augmented system is constructed by viewing the measurement disturbances as unknow inputs. For such an augmented system, the so-called observer matching condition is not satisfied. Based on the construction of auxiliary outputs, the observer matching condition may be satisfied. High-order sliding-mode differentiators are developed to obtain the estimates of those unmeasurable variables contained in the auxiliary output vector. Employing the estimate of auxiliary output vector, a sliding-mode observer is designed. The simulation results to a real model show that the proposed method is effective.

Numerical Analysis of a High-Order Scheme for Nonlinear Fractional Differential Equations with Uniform Accuracy

We introduce a high-order numerical scheme for fractional ordinary differential equations with the Caputo derivative.The method is developed by dividing the domain into a number of subintervals,and applying the quadratic interpolation on each subinterval.The method is shown to be unconditionally stable,and for general nonlinear equations,the uniform sharp numerical order 3−νcan be rigorously proven for sufficiently smooth solutions at all time steps.The proof provides a gen-eral guide for proving the sharp order for higher-order schemes in the nonlinear case.Some numerical examples are given to validate our theoretical results.

J-selfadjointness of a class of high-order differential operators with transmission conditions

This paper is to investigate the J-selfadjointness of a class of high-order complex coefficients differential operators with transmission conditions.Using the Lagrange bilinear form of J-symmetric differential equations,the definition of J-selfadjoint differential operators and the method of matrix representation,we prove that the operator is J-selfadjoint operator,and the eigenvectors and eigen-subspaces corresponding to different eigenvalues are C-orthogonal.

New Stability Criteria for High-Order Neural Networks with Proportional Delays

This paper is concerned with high-order neural networks with proportional delays. The proportional delay is a time-varying unbounded delay which is different from the constant delay, bounded time-varying delay and distributed delay. By the nonlinear transformation yi(t) = ui( et)(i = 1, 2,..., n), we transform a class of high-order neural networks with proportional delays into a class of high-order neural networks with constant delays and timevarying coefficients. With the aid of Brouwer fixed point theorem and constructing the delay differential inequality, we obtain some delay-independent and delay-dependent sufficient conditions to ensure the existence, uniqueness and global exponential stability of equilibrium of the network. Two examples with their simulations are given to illustrate the theoretical findings. Our results are new and complement previously known results.

Hyper-chaos encryption using convolutional masking and model free unmasking

In this paper, during the masking process the encrypted message is convolved and embedded into a Qi hyper-chaotic system characterizing a high disorder degree. The masking scheme was tested using both Qi hyper-chaos and Lorenz chaos and indicated that Qi hyper-chaos based masking can resist attacks of the filtering and power spectrum analysis, while the Lorenz based scheme fails for high amplitude data. To unmask the message at the receiving end, two methods are proposed. In the first method, a model-free synchronizer, i.e. a multivariable higher-order differential feedback controller between the transmitter and receiver is employed to de-convolve the message embedded in the receiving signal. In the second method, no synchronization is required since the message is de-convolved using the information of the estimated derivative.

Energy-Work Connection Integration Scheme for Nonholonomic Hamiltonian Systems

This paper focuses on studying a new energy-work relationship numericM integration scheme of nonholonomic Hamiltonian systems. The signal-stage numerical, multi-stage and parallel composition numerical integration schemes are presented. The high-order energy-work relation scheme of the system is constructed by a parallel connection of n multi-stage schemes of order 2, its order of accuracy is 2n. The connection, which is discrete analogue of usual case, between the change of energy and work of nonholonomic constraint forces is obtained for nonholonomie Hamiltonian systems. This paper also gives that there is smaller error of the scheme when taking a large number of stages than a less one. Finally, an applied example is discussed to illustrate these results.