Abstract:
This paper is concerned with a delayed predator-prey diffusion model with Neumann boundary conditions. We study the asymptotic stability of the positive constant steady state and the conditions for the existence of Hopf bifurcation. In particular, we show that large diffusivity has no effect on the Hopf bifurcation, while small diffusivity can lead to the fact that spatially nonhomogeneous periodic solutions bifurcate from the positive constant steady-state solution when the system parameters are all spatially homogeneous. Meanwhile, we study the properties of the spatially nonhomogeneous periodic solutions applying normal form theory of partial functional differential equations (PFDEs).

Abstract:
This paper is intended to apply the potential integration method to the differential equations of the Birkhoffian system. The method is that, for a given Birkhoffian system, its differential equations are first rewritten as 2n first-order differential equations. Secondly, the corresponding partial differential equations are obtained by potential integration method and the solution is expressed as a complete integral. Finally, the integral of the system is obtained.

Abstract:
This paper studies the Mei symmetry and Mei conserved quantity for nonholonomic systems of unilateral Chetaev type in Nielsen style. The differential equations of motion of the system above are established. The definition and the criteria of Mei symmetry, loosely Mei symmetry, strictly Mei symmetry for the system are given in this paper. The existence condition and the expression of Mei conserved quantity are deduced directly by using Mei symmetry. An example is given to illustrate the application of the results.

Abstract:
This paper studies the form invariance of Maggi equation. Its definition and criterion are presented. A Hojman conserved quantity can be deduced using the form invariance. An example is given to illustrate the application of the result.

Abstract:
A new conserved quantity is investigated by utilizing the definition anddiscriminant equation of Mei symmetry of Tzenoff equations fornonholonomic systems. In addition, the expression of this conservedquantity, and the determining condition induced new conserved quantityare also presented.

Abstract:
In this paper, the dissipative and the forced terms of the Duffing equation are considered as the perturbations of nonlinear Hamiltonian equations and the perturbational effect is indicated by parameter $\varepsilon $. Firstly, based on the gradient-Hamiltonian decomposition theory of vector fields, by using splitting methods, this paper constructs structure-preserving algorithms (SPAs) for the Duffing equation. Then, according to the Liouville formula, it proves that the Jacobian matrix determinants of the SPAs are equal to that of the exact flow of the Duffing equation. However, considering the explicit Runge--Kutta methods, this paper finds that there is an error term of order $pIn this paper, the dissipative and the forced terms of the Duffing equation are considered as the perturbations of nonlinear Hamiltonian equations and the perturbational effect is indicated by parameter $\varepsilon $. Firstly, based on the gradient-Hamiltonian decomposition theory of vector fields, by using splitting methods, this paper constructs structure-preserving algorithms (SPAs) for the Duffing equation. Then, according to the Liouville formula, it proves that the Jacobian matrix determinants of the SPAs are equal to that of the exact flow of the Duffing equation. However, considering the explicit Runge--Kutta methods, this paper finds that there is an error term of order $pIn this paper, the dissipative and the forced terms of the Duffing equation are considered as the perturbations of nonlinear Hamiltonian equations and the perturbational effect is indicated by parameter $\varepsilon $. Firstly, based on the gradient-Hamiltonian decomposition theory of vector fields, by using splitting methods, this paper constructs structure-preserving algorithms (SPAs) for the Duffing equation. Then, according to the Liouville formula, it proves that the Jacobian matrix determinants of the SPAs are equal to that of the exact flow of the Duffing equation. However, considering the explicit Runge--Kutta methods, this paper finds that there is an error term of order $pIn this paper, the dissipative and the forced terms of the Duffing equation are considered as the perturbations of nonlinear Hamiltonian equations and the perturbational effect is indicated by parameter $\varepsilon $. Firstly, based on the gradient-Hamiltonian decomposition theory of vector fields, by using splitting methods, this paper constructs structure-preserving algorithms (SPAs) for the Duffing equation. Then, according to the Liouville formula, it proves that the Jacobian matrix determinants of the SPAs are equal to that of the exact flow of the Duffing equation. However, considering the explicit Runge--Kutta methods, this paper finds that there is an error term of order $p$+1 for the Jacobian matrix determinants. The volume evolution law of a given region in phase space is discussed for different algorithms, respectively. As a result, the sum of Lyapunov exponents is exactly invariable for the SPAs propo

Abstract:
Based on the weak Noether symmetry proposed by Mei F X, this paper discusses the weak Noether symmetry for nonholonomic system of non-Chetaev type, and presents expressions of three kinds of conserved quantities by weak Noether symmetry. Finally, the application of this new results is showed by a practical example.