Approximate Inertial Manifolds (AIMs) is approached by multilevel finite element method, which can be referred to as a Post-processed nonlinear Galerkin finite element method, and is applied to the model reduction for fluid dynamics, a typical kind of nonlinear continuous dynamic system from viewpoint of nonlinear dynamics. By this method, each unknown variable, namely, velocity and pressure, is divided into two components, that is the large eddy and small eddy components. The interaction between large eddy and small eddy components, which is negligible if standard Galerkin algorithm is used to approach the original governing equations, is considered essentially by AIMs, and consequently a coarse grid finite element space and a fine grid incremental finite element space are introduced to approach the two components. As an example, the flow field of incompressible flows around airfoil is simulated numerically and discussed, and velocity and pressure distributions of the flow field are obtained accurately. The results show that there exists less essential degrees-offreedom which can dominate the dynamic behaviors of the discretized system in comparison with the traditional methods, and large computing time can be saved by this efficient method. In a sense, the small eddy component can be captured by AIMs with fewer grids, and an accurate result can also be obtained.

From the viewpoint of nonlinear dynamics, the stability and bifurcation of the rotor dynamical system supported in gas bearings are investigated. First, the dynamical model of gas bearing-Jeffcott rotor system is given, and the finite element method is used to approach the unsteady Reynolds equation in order to obtain gas film forces. Then, the method for stability analysis of the unbalance response of the rotor system is developed in combination with the Newmark-based direct integral method and Floquet theory. Finally, a numerical example is presented, and the complex behaviors of the nonlinear dynamical system are simulated numerically, including the trajectory of the journal and phase portrait. In particular, the stabilities of the system’s equilibrium position and unbalance responses are studied via the orbit diagram, phase space, Poincaré mapping, bifurcation diagram, and power spectrum. The results show that the numerical method for solving the unsteady Reynolds equation is efficient, and there exist a rich variety of nonlinear phenomena in the system. The half-speed whirl encountered in practice is the result from Hopf bifurcation of equilibrium, and the numerical method presented is available for the stability and bifurcation analysis of the complicated gas film-rotor dynamic system.

The focus of this paper is on the investigation of the mathematical nature of buckling from the point of view of bifurcation theory. For the doubly curved orthotropic panels subjected to quasi-static uniform load and with hinged boundary conditions, the solution to the non-linear partial di6erential equation is partitioned into two parts and projected onto the complete space spanned by the eigenfunctions of the linear operator of the governing equation. Furthermore, the fundamental branch, from which a new solution will emanate, is approximated by the 8rst single mode pair which is close to the real membrane state. Whereas the ensuing bifurcated branch is approximated by the other single mode pair, under the assumption that the coupling between modes can be neglected. The present analysis could give a deep insight into the mechanism of the instability of panel structures, and show that there exists a mode transition at the critical point and the snap-through, then results from saddle-node bifurcation on the bifurcated branch. As a conclusion, the buckling of the system studied can be stated as: a bifurcated branch emanates from the fundamental branch at a critical point, and a saddle-node bifurcation, behaving as jumping, then occurs on the ensuing bifurcated branch.