Abstract:The harmony finite element method for solving the evolution partial differential equation requires different dimension on spacetime domain, which does not satisfy the restrictions on the same dimension of symplectic matrix group. In this paper, according to variational principle and using the virtual work principle, a principle for dealing with the different dimension problem of symplectic transfer matrix was proposed. The numerical example shows that the proposed method is effective.

Abstract:The generalized Birkhoff′s equations are more general equations of constrained mechanical systems. The stability of equilibrium for generalized Birkhoff equations was addressed. The equations of equilibrium and energy change were established. The criterion of stability of equilibrium was obtained by using the property of definite sign of the Birkhoffian. An example was given to illustrate the application of the result.

Abstract:This paper studied the conformal invariance and conserved quantity of the holonomic system, which corresponds to a higherorder nonholonomic system. Firstly, the definition and determining equation of conformal invariance of the system were presented. The conformal factor, which is the necessary and sufficient condition that conformal invariance of the system would be Lie symmetry was deduced from conformal invariance and Lie symmetry. The conformal invariance of weak and strong Lie symmetry for the highorder nonholonomic system was given using restriction equations and additional restriction equations. Secondly, the Noether conserved quantity of conformal invariance of the system was derived. Lastly, an example was given to illustrate the application of the results.

Abstract:Based on the analysis of the structure of Lagrangions, a new general method and 6 special methods for the construction of Lagrangians for one dimensional system from the motion equation directly were presented. By use of the metthods, some Lagrangians of the differetial equations were derived. By means of the direct approach, it is verified that a system may have many different equivalent Lagrangians, even an infinite family of Lagrangians. The approach presented is a way to obtain Lagrange symmetries and the conserved quantities.

Abstract:This paper studies the Perturbation to Mei symmetry and adiabatic invariants of general discrete holonomic systems. The exact invariants of Mei symmetry for general discrete holonomic systems without perturbation are given. The perturbation to Mei symmetry is discussed under the effect of small quantities and the adiabatic invariants induced from the perturbation to Mei symmetry of general discrete holonomic systems are obtained. In the end, an example is discussed to show the applications of the results.

Abstract:This paper studied the nonlinear vibration of an axially moving beam with coupled transverse and longitudinal motions, and the focus was on the internal resonance in the neighborhood of a 1:3 between the first two transverse modes. First, the motion equations of the axially moving beam are derived through the Hamilton’s principle, and were discretized by the Galerkin’s method to obtain the quadratic and cubic nonlinearities. Then, the IHB method was applied to analyze the complicated frequencyamplitude response curves, thus obtaining the nonlinear vibration of the axially moving beam near the first two transverse natural frequencies with coupled transverse and longitudinal motions, which reveals the rich and interesting nonlinear phenomena.

Abstract:In the environment of earth gravitation and magnetic field, the influence of aerodynamic drag was considered. The chaotic phenomenon was discussed when the magnetic rigid spacecraft without internal damping was on a circular orbit round the earth. The dynamical model was established by the law of moment of momentum. The Melnikov analysis was carried out to verify that the chaotic phenomenon is possible, and the numerical simulations were used to analysize the systematic dynamical motion. The theory analysis results agree with numerical simulations.

Abstract:Through a Generalized Hamiltonian system and observer approach, the chaotic synchronization of nonsmooth systems was transformed into the stability of zero solution of smooth systems, thus the condition of the chaotic synchronization was obtained. Moreover the Duffing Oscillator with dry friction and impact and their chaotic synchronization were studied. The numerical results agreed well with the theoretical analysis.

Abstract:Based on the theory of Hopf bifurcation of timedelayed system, a single delayed state feedback controller was designed to control the chaotic motion of the Chua’s circuit system. First, the stability of the equilibriums and the chaotic motion of Chua’s circuit system were investigated. Then, the necessary controllable ranges of the feedback gains were determined via detailed analysis for each of the x, y, z single delayed state feedbacks, respectively. Finally, the results of numerical simulation verified the validity of the theoretical analysis.

Abstract:Based on the idea of active control and the Lyapunov stability theory, an active controller is designed to synchronize two different chaotic systems (the unified chaotic system and Qi chaotic system). Moreover, the fast adaptive synchronization is achieved between the drive system and the response system with different structures and unknown parameters. The results are then applied to the secure communication based on chaotic masking and chaotic spread spectrum. The MATLAB is used for the numerical simulations. The simulation results show that the method can realize synchronization of two different chaotic systems and the useful signal can be recovered effectively in the receiver during the secure communication.

Abstract:Based on the stability theory of fractional order linear systems,a novel method combining feedback control with active control was proposed for the lag synchronization of fractional order chaotic systems. By designing a proper controller the lag synchronization of fractional order chaotic systems was converted to the asymptotic stability of the fractional order linear error systems at origin. The numerical simulation results on fractional order Chen system verify the effectiveness of the proposed method.

Abstract:A modified conjugate method for unconstrained optimization problem was presented on the basis of DY conjugate gradient method.It is proved that the new formula is of full descent under the condition of the strong wolfe line search. At the same time ,the new formula can support the global convergence.The numerical results show that the method is of great value. And the algorithm was applied to nonlinear parameter estimation of buring anteisodynamics model of sulfur dioxide acted on by caesiumrubidiumvaradium low temperature sulfur acid catalyst, and oblaine satisfactory results were obtained.

Abstract:This paper, addressed the stability problem of discretetime switched positive systems with time delay. By using the switched linear copositive Lyapunov functional and common linear copositive Lyapunov functional, a number of stability criteria(LP and LMI) for the discretetime switched positive systems with time delay were presented.

Abstract:This paper, under the excitations of correlated multiplicative and additive noises, the mean firstpassage times (MFPTs) in a bistable system with two different kinds of time delays are investigated. First, based on an approximated method, the analytical expressions of the MFPTs are obtained. Then, the effects of the multiplicative noise intensity, the delays and the noise coupling strength on the MFPTs are studied. For the positive coupling strength, the MFPT T1(x-→x+) is a nonlinear function of the multiplicative noise intensity and the two delays, but an increasing function of the noise coupling strength. Moreover, the delays contained in the determined and random forces respectively influence the maximum and the corresponding noise level on T1(x-→x+). The MFPT T2(x+→x-) is a nonlinear function of the delay that contained in the determined force, but a decreasing function of the multiplicative noise intensity, the other kind of delay and the noise coupling strength.

Abstract:Differential feedback control in random parameter model of forced Brusselator was considered. First, this model was transformed into its equivalent deterministic nonlinear systems by the Chebyshev polynomial approximation. Then, a nonlinear differential feedback controller was designed in order to make the chaotic model of forced Brusselator stable in a certain unstable periodic orbit. Numerical results show the effectiveness of the method.

Abstract:The repulsive forces exist between the two rails when the armature is thrusted forward along the rail. At the same time the armature expands due to Joule heat, which causes rail’s vibration.The electromagnetic rail was modeled as a simply supported beam of finite length sitting on an elastic foundation. According to the theory of vibration, the dynamic response of the rail was governed by a fourthorder differential equation with an extra term of elastic support subjected to armature’s thrust. Then the parameters, such as the current intensity, the elastic constant and damping of the elastic foundation, were analyzed. The theoretical basis of the design of rail was obtained from the analysis result.

Abstract:We investigated the dynamics of the simplest kind of switched Hamiltonian systemswitched simple pendulum, on which the only force acting is gravity. Taking advantage of the Hamiltonian function of each subsystem, we find that the dynamics of the switched pendulum is much more complex than that of the simple pendulum. The switched pendulum is able to rotate more and more rapidly or settle down except oscillating periodically.

Abstract:The effects of pacemaker on synchronization in bidirectional coupling Hindmarsh-Rose neurons with gap junction were studied. Results show that, with proper parameters, pacemaker can enhance or inhibit the complete synchronization of two identical neurons, while it can always enhance the complete synchronization of three identical neurons. For three un-identical neurons with different firing patterns, pacemaker can induce phase synchronization and nearly complete synchronization if the controlling strength is big enough. When time delay was considered, the pacemaker was more likely to trigger such synchronization.

Abstract:Control of desynchronization in discrete neural networks by nonlinear delayed feedback was studied. The network consisting of Mapbased neurons could get chaotic bursting synchronization through adjusting the coupling strength between the cells. Once nonlinear delayed feedback signals were applied into the fast variables of neurons, a complete desynchronization was achieved and neuron’s inherent bursting characteristic was restored. Compared with linear delayed feedback control, the method was effective even in the case of strong coupling and robust against parameter variations.

Abstract:This paper, studied the stability and Hopf bifurcation of an economic model. According to its characteric roots, the critical condition on which the system loses its stability was derived. Then pseudooscillator analysis and iteration method were conducted on the system. In this way, the direction of the Hopf bifurcation and the amplitude of bifurcated periodic solution can be obtained in a simple way. Besides, numerical examples were given to show the effectiveness of the methods. Specially, compared with the pseudooscillator analysis, the solution from iteration method is more accurate as the amplitude of bifurcated periodic solution increases. Though the system loses stability when the Hopf bifurcation occurs, the bifurcated periodic solution still keeps stable by varying the parameters within a certain range, which means that the system can reach a beneficial economical cycle.