Studies On Bifurcation Solutions And Their Stability Of Infinite-Dimensional Dynamical Systems

Bifurcation problem in time evolution of infinite-dimensional dynamical system exists widely in physics, chemistry and biology, including fluid mechanics, solid mechanics, fracture mechanics, atmospheric dynamics, chemical reaction systems, biological evolution system and so on. For example, the non-equilibrium phase transitions in convection and heat conduction, condensed matter physics and interface motion lead to the formation of stable patterns far from equilibrium. In addition, for the nonlinear vibration problem modeling axially moving viscoelastic beams, rods and other structures, the buckling and instability occur under a certain external load or force. These problems are closely related to space-time bifurcation of infinite dimensional dynamical systems.Global attractor and approximate inertial manifold of the infinite-dimensional dynamical systems proposed by R. Temam have become the effective tools for studying the dynamic behavior of systems. On this basis, by using center manifold reduction method of infinite dimensional systems, the attractor bifurcation theory is obtained. It is a most important development of the bifurcation research in infinite dimensional dynamical systems. This article also briefly introduces some other methods, such as the topological method, analytical approximation and numerical methods, and gives the definition of asymptotic stability of equilibrium solutions in an infinite dimensional space.In the Hilbert space, the bifurcation problems of some infinite-dimensional dynamic models will be considered in this article, including Kuramoto-Sivashinsky equation describing the non-equilibrium interface growth in one-dimensional and two-dimensional spatial domain, the Kirchhoff type nonlinear beam vibration equation and the Burgers-Fisher equation describing the reaction diffusion problems. For the homogeneous problems, the use of center manifold theorem and the attractor bifurcation theorem directly shows the approximate bifurcation solutions and their stability. When an external constant force acts on the system, perturbation method first is applied for bifurcation analysis of steady-state equation. Then by using a priori estimate of infinite-dimensional dynamical system, the stability of steady-state bifurcation solution is found. The results show that the critical bifurcation point shifted. A time periodically forced complex Ginzburg-Landau model also is considered, which shows a bifurcation near a limit cycle and obtains a special approximation quasi-periodic solution and its stability. These results will contribute to track the new steady state, such as spatial pattern formation structure, and facilitate the bifurcation control.Numerical analysis on bifurcation problem of the infinite dimensional system is the leading subject. Utilizing the difference method, the phase transition in a interface evolution model is verified, and the bifurcation diagram and the numerical simulation of bifurcation solution are constructed. In addition, considering a external driven and strongly damped nonlinear wave equation, by using multi-scale method and numerical simulation, the steady-state amplitude frequency response of the reduced system based on a finite dimensional approximate inertial manifold of the original system are studied. The saddle-node bifurcation diagram and bifurcation sets are also obtained.